# Set

Set
You are encouraged to solve this task according to the task description, using any language you may know.

Data Structure
This illustrates a data structure, a means of storing data within a program.

You may see other such structures in the Data Structures category.

A   set  is a collection of elements, without duplicates and without order.

Show each of these set operations:

• Set creation
• Test m ∈ S -- "m is an element in set S"
• A ∪ B -- union; a set of all elements either in set A or in set B.
• A ∩ B -- intersection; a set of all elements in both set A and set B.
• A ∖ B -- difference; a set of all elements in set A, except those in set B.
• A ⊆ B -- subset; true if every element in set A is also in set B.
• A = B -- equality; true if every element of set A is in set B and vice versa.

As an option, show some other set operations.
(If A ⊆ B, but A ≠ B, then A is called a true or proper subset of B, written A ⊂ B or A ⊊ B.)

As another option, show how to modify a mutable set.

One might implement a set using an associative array (with set elements as array keys and some dummy value as the values).

One might also implement a set with a binary search tree, or with a hash table, or with an ordered array of binary bits (operated on with bit-wise binary operators).

The basic test, m ∈ S, is O(n) with a sequential list of elements, O(log n) with a balanced binary search tree, or (O(1) average-case, O(n) worst case) with a hash table.

## 11l

Translation of: Python
```V s1 = Set([1, 2, 3, 4])
V s2 = Set([3, 4, 5, 6])
print(s1.union(s2))
print(s1.intersection(s2))
print(s1.difference(s2))
print(s1 < s1)
print(Set([3, 1]) < s1)
print(s1 <= s1)
print(Set([3, 1]) <= s1)
print(Set([3, 2, 4, 1]) == s1)
print(s1 == s2)
print(2 C s1)
print(10 !C s1)
print(Set([1, 2, 3, 4, 5]) > s1)
print(Set([1, 2, 3, 4]) > s1)
print(Set([1, 2, 3, 4]) >= s1)
print(s1.symmetric_difference(s2))
print(s1.len)
print(s1)
print(s1)```
Output:
```Set([1, 2, 3, 4, 5, 6])
Set([3, 4])
Set([1, 2])
0B
1B
1B
1B
1B
0B
1B
1B
1B
0B
1B
Set([1, 2, 5, 6])
4
Set([1, 2, 3, 4, 99])
Set([1, 2, 3, 4])
```

## Action!

The user must type in the monitor the following command after compilation and before running the program!

`SET EndProg=*`
```CARD EndProg ;required for ALLOCATE.ACT

INCLUDE "D2:ALLOCATE.ACT" ;from the Action! Tool Kit. You must type 'SET EndProg=*' from the monitor after compiling, but before running this program!

DEFINE PTR="CARD"
DEFINE NODE_SIZE="6"
TYPE SetNode=[PTR data,prv,nxt]
TYPE SetInfo=[PTR name,begin,end]

PROC PrintSet(SetInfo POINTER s)
SetNode POINTER n
CHAR ARRAY a

n=s.begin
PrintF("%S=(",s.name)
WHILE n
DO
Print(n.data)
a=n.data
IF n.nxt THEN
Print(", ")
FI
n=n.nxt
OD
PrintE(")")
RETURN

PROC CreateSet(SetInfo POINTER s CHAR ARRAY n)
s.name=n
s.begin=0
s.end=0
RETURN

PTR FUNC Find(SetInfo POINTER s CHAR ARRAY v)
SetNode POINTER n

n=s.begin
WHILE n
DO
IF SCompare(v,n.data)=0 THEN
RETURN (n)
FI
n=n.nxt
OD
RETURN (0)

BYTE FUNC Contains(SetInfo POINTER s CHAR ARRAY v)
SetNode POINTER n

n=Find(s,v)
IF n=0 THEN
RETURN (0)
FI
RETURN (1)

PROC Append(SetInfo POINTER s CHAR ARRAY v)
SetNode POINTER n,tmp

IF Contains(s,v) THEN RETURN FI

n=Alloc(NODE_SIZE)
n.data=v
n.prv=s.end
n.nxt=0
IF s.end THEN
tmp=s.end tmp.nxt=n
ELSE
s.begin=n
FI
s.end=n
RETURN

PROC Remove(SetInfo POINTER s CHAR ARRAY v)
SetNode POINTER n,prev,next

n=Find(s,v)
IF n=0 THEN RETURN FI

prev=n.prv
next=n.nxt

Free(n,NODE_SIZE)

IF prev THEN
prev.nxt=next
ELSE
s.begin=next
FI
IF next THEN
next.prv=prev
ELSE
s.end=prev
FI
RETURN

PROC AppendSet(SetInfo POINTER s,other)
SetNode POINTER n

n=other.begin
WHILE n
DO
Append(s,n.data)
n=n.nxt
OD
RETURN

PROC RemoveSet(SetInfo POINTER s,other)
SetNode POINTER n

n=other.begin
WHILE n
DO
Remove(s,n.data)
n=n.nxt
OD
RETURN

PROC Clear(SetInfo POINTER s)
SetNode POINTER n

DO
n=s.begin
IF n=0 THEN RETURN FI
Remove(s,n.data)
OD
RETURN

PROC Union(SetInfo POINTER a,b,res)
Clear(res)
AppendSet(res,a)
AppendSet(res,b)
RETURN

PROC Intersection(SetInfo POINTER a,b,res)
SetNode POINTER n

Clear(res)
n=a.begin
WHILE n
DO
IF Contains(b,n.data) THEN
Append(res,n.data)
FI
n=n.nxt
OD
RETURN

PROC Difference(SetInfo POINTER a,b,res)
Clear(res)
AppendSet(res,a)
RemoveSet(res,b)
RETURN

BYTE FUNC IsSubset(SetInfo POINTER s,sub)
SetNode POINTER n

n=sub.begin
WHILE n
DO
IF Contains(s,n.data)=0 THEN
RETURN (0)
FI
n=n.nxt
OD
RETURN (1)

BYTE FUNC AreEqual(SetInfo POINTER a,b)
IF IsSubset(a,b)=0 OR IsSubset(b,a)=0 THEN
RETURN (0)
FI
RETURN (1)

BYTE FUNC IsProperSubset(SetInfo POINTER s,sub)
IF IsSubset(s,sub)=1 AND IsSubset(sub,s)=0 THEN
RETURN (1)
FI
RETURN (0)

PROC TestContains(SetInfo POINTER s CHAR ARRAY v)
IF Contains(s,v) THEN
PrintF("%S contains %S%E",s.name,v)
ELSE
PrintF("%S does not contain %S%E",s.name,v)
FI
RETURN

PROC TestUnion(SetInfo POINTER a,b,res)
Union(a,b,res)
PrintF("Union %S and %S: ",a.name,b.name)
PrintSet(res)
RETURN

PROC TestIntersection(SetInfo POINTER a,b,res)
Intersection(a,b,res)
PrintF("Intersection %S and %S: ",a.name,b.name)
PrintSet(res)
RETURN

PROC TestDifference(SetInfo POINTER a,b,res)
Difference(a,b,res)
PrintF("Difference %S-%S: ",a.name,b.name)
PrintSet(res)
RETURN

PROC TestSubset(SetInfo POINTER s,sub)
IF IsSubset(s,sub) THEN
PrintF("%S is a subset of %S%E",sub.name,s.name)
ELSE
PrintF("%S is not a subset of %S%E",sub.name,s.name)
FI
RETURN

PROC TestEqual(SetInfo POINTER a,b)
IF AreEqual(a,b) THEN
PrintF("%S and %S are equal%E",a.name,b.name)
ELSE
PrintF("%S and %S are not equal%E",a.name,b.name)
FI
RETURN

PROC TestProperSubset(SetInfo POINTER s,sub)
IF IsSubset(s,sub) THEN
PrintF("%S is a proper subset of %S%E",sub.name,s.name)
ELSE
PrintF("%S is not a proper subset of %S%E",sub.name,s.name)
FI
RETURN

PROC TestAppend(SetInfo POINTER s CHAR ARRAY v)
Append(s,v)
PrintF("%S+%S: ",s.name,v)
PrintSet(s)
RETURN

PROC TestRemove(SetInfo POINTER s CHAR ARRAY v)
Remove(s,v)
PrintF("%S-%S: ",s.name,v)
PrintSet(s)
RETURN

PROC Main()
SetInfo s1,s2,s3,s4

Put(125) PutE() ;clear screen

AllocInit(0)
CreateSet(s1,"A")
CreateSet(s2,"B")
CreateSet(s3,"C")
CreateSet(s4,"D")

Append(s1,"Action!") Append(s1,"Basic")
Append(s2,"Pascal") Append(s2,"Action!")
Append(s2,"C++") Append(s2,"C#")
Append(s3,"Basic") Append(s3,"Fortran")

PrintSet(s1) PrintSet(s2) PrintSet(s3)
PutE()

TestContains(s1,"Action!")
TestContains(s2,"Fortran")
TestUnion(s1,s2,s4)
TestIntersection(s1,s2,s4)
TestDifference(s2,s1,s4)
TestSubset(s1,s4)
TestSubset(s2,s4)
TestEqual(s1,s3)
TestEqual(s2,s3)
TestProperSubset(s1,s4)
TestProperSubset(s1,s3)
TestRemove(s3,"Fortran")
TestRemove(s3,"C#")
TestAppend(s3,"Java")
TestAppend(s3,"Java")

Clear(s1)
Clear(s2)
Clear(s3)
Clear(s4)
RETURN```
Output:
```A=(Action!, Basic, Ada, Fortran)
B=(Pascal, Action!, C++, C#)

A contains Action!
B does not contain Fortran
Union A and B: D=(Action!, Basic, Ada, Fortran, Pascal, C++, C#)
Intersection A and B: D=(Action!)
Difference B-A: D=(Pascal, C++, C#)
D is not a subset of A
D is a subset of B
A and C are equal
B and C are not equal
D is not a proper subset of A
C is a proper subset of A
```

This solution uses the generic Ordered_Sets package from the Ada.Containers standard library, which internally is based on red-black trees. An alternative hash-based solution could use the Hashed_Maps package from Ada.Containers.

```with ada.containers.ordered_sets, ada.text_io;

procedure set_demo is
package cs is new ada.containers.ordered_sets (character); use cs;

function "+" (s : string) return set is
(if s = "" then empty_set else Union(+ s(s'first..s'last - 1), To_Set (s(s'last))));

function "-" (s : Set) return string is
(if s = empty_set then "" else - (s - To_Set (s.last_element)) & s.last_element);
s1, s2 : set;
begin
loop
put ("s1= ");
s1 := + get_line;
exit when s1 = +"Quit!";
put ("s2= ");
s2 := + get_line;
Put_Line("Sets [" & (-s1) & "], [" & (-s2) & "] of size"
& S1.Length'img & " and" & s2.Length'img & ".");
Put_Line("Intersection:   [" & (-(Intersection(S1, S2))) & "],");
Put_Line("Union:          [" & (-(Union(s1, s2)))        & "],");
Put_Line("Difference:     [" & (-(Difference(s1, s2)))   & "],");
Put_Line("Symmetric Diff: [" & (-(s1 xor s2)) & "],");
Put_Line("Subset: "  & Boolean'Image(s1.Is_Subset(s2))
& ", Equal: " & Boolean'Image(s1 = s2) & ".");
end loop;
end set_demo;
```
Output:
```set
demo
Sets [est], [demo] of size 3 and 4.
Intersection:   [e],
Union:          [demost],
Difference:     [st],
Symmetric Diff: [dmost],
Subset: FALSE, Equal: FALSE.
quit!
```

## Aime

```record
union(record a, record b)
{
record c;
r_copy(c, a);
return c;
}

record
intersection(record a, record b)
{
record c;
text s;
for (s in a) {
if (r_key(b, s)) {
c[s] = 0;
}
}
return c;
}

record
difference(record a, record b)
{
record c;
r_copy(c, a);
r_vcall(b, r_resign, 1, c);
return c;
}

integer
subset(record a, record b)
{
integer e;
text s;
e = 1;
for (s in a) {
if (!r_key(b, s)) {
e = 0;
break;
}
}
return e;
}

integer
equal(record a, record b)
{
return subset(a, b) && subset(b, a);
}

integer
main(void)
{
record a, b;
text s;

r_fit(a, "apple", 0, "cherry", 0, "grape", 0);
r_fit(b, "banana", 0, "cherry", 0, "date", 0);

s = "banana";

o_(" ", s, " is ", r_key(a, s) ? "" : "not ", "an element of A\n");
o_(" ", s, " is ", r_key(b, s) ? "" : "not ", "an element of B\n");

r_vcall(union(a, b), o_, 1, " ");
o_newline();

r_vcall(intersection(a, b), o_, 1, " ");
o_newline();

r_vcall(difference(a, b), o_, 1, " ");
o_newline();

o_(" ", subset(a, b) ? "yes" : "no", "\n");

o_(" ", equal(a, b) ? "yes" : "no", "\n");

return 0;
}```
Output:
``` banana is not an element of A
banana is an element of B
apple banana cherry date grape
cherry
apple grape
no
no```

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Reuses a lot of code from the Symetric difference task.

```# sets using associative arrays                                              #
# include the associative array code for string keys and values              #

# adds the elements of s to the set a,                                       #
#      the elements will have empty strings for values                       #
OP // = ( REF AARRAY a, []STRING s )REF AARRAY:
BEGIN
FOR s pos FROM LWB s TO UPB s DO
a // s[ s pos ] := ""
OD;
a
END # // # ;
# returns a set containing the elements of a that aren't in b                #
OP - = ( REF AARRAY a, REF AARRAY b )REF AARRAY:
BEGIN
REF AARRAY result := INIT HEAP AARRAY;
REF AAELEMENT e := FIRST a;
WHILE e ISNT nil element DO
IF NOT ( b CONTAINSKEY key OF e ) THEN
result // key OF e := value OF e
FI;
e := NEXT a
OD;
result
END # - # ;
# returns a set containing the elements of a and those of b, i.e. a UNION b  #
PRIO U = 6;
OP   U = ( REF AARRAY a, REF AARRAY b )REF AARRAY:
BEGIN
REF AARRAY result := INIT HEAP AARRAY;
REF AAELEMENT e := FIRST a;
WHILE e ISNT nil element DO
result // key OF e := value OF e;
e := NEXT a
OD;
e := FIRST b;
WHILE e ISNT nil element DO
result // key OF e := value OF e;
e := NEXT b
OD;
result
END # U # ;
# returns a set containing the elements of a INTERSECTION b                  #
PRIO N = 6;
OP   N = ( REF AARRAY a, REF AARRAY b )REF AARRAY:
BEGIN
REF AARRAY result := INIT HEAP AARRAY;
REF AAELEMENT e := FIRST a;
WHILE e ISNT nil element DO
IF b CONTAINSKEY key OF e THEN
result // key OF e := value OF e
FI;
e := NEXT a
OD;
result
END # N # ;
# returns TRUE if all the elements of a are in b, FALSE otherwise            #
OP <= = ( REF AARRAY a, REF AARRAY b )BOOL:
BEGIN
BOOL result := TRUE;
REF AAELEMENT e := FIRST a;
WHILE result AND ( e ISNT nil element ) DO
result := b CONTAINSKEY key OF e;
e := NEXT a
OD;
result
END # <= # ;
# returns TRUE if all the elements of a are in b                             #
#             and all the elements of b are in a, FALSE otherwise            #
OP = = ( REF AARRAY a, REF AARRAY b )BOOL: a <= b AND b <= a;
# returns NOT ( a = b )                                                      #
OP /= = ( REF AARRAY a, REF AARRAY b )BOOL: NOT ( a = b );
# returns TRUE if all the elements of a are in b                             #
#         but not all the elements of b are in a, FALSE otherwise            #
OP < = ( REF AARRAY a, REF AARRAY b )BOOL: a <= b AND b /= a;

# prints the elements of a in no-particlar order                             #
PROC print set = ( REF AARRAY a )VOID:
BEGIN
print( ( "[" ) );
REF AAELEMENT e := FIRST a;
WHILE e ISNT nil element DO
print( ( " ", key OF e ) );
e := NEXT a
OD;
print( ( " ]", newline ) )
END # print set # ;

# construct associative arrays for the task                                  #
REF AARRAY gas giants       := INIT LOC AARRAY;
REF AARRAY ice giants       := INIT LOC AARRAY;
REF AARRAY rocky planets    := INIT LOC AARRAY;
REF AARRAY inner planets    := INIT LOC AARRAY;
REF AARRAY moonless planets := INIT LOC AARRAY;
gas giants       // []STRING( "Jupiter", "Saturn"  );
ice giants       // []STRING( "Uranus",  "Neptune" );
rocky planets    // []STRING( "Mercury", "Venus", "Earth", "Mars" );
inner planets    // []STRING( "Mercury", "Venus", "Earth", "Mars" );
moonless planets // []STRING( "Mercury", "Venus"   );

print( ( "rocky planets   : " ) );print set( rocky planets    );
print( ( "inner planets   : " ) );print set( inner planets    );
print( ( "gas giants      : " ) );print set( gas giants       );
print( ( "ice giants      : " ) );print set( ice giants       );
print( ( "moonless planets: " ) );print set( moonless planets );
print( ( newline ) );

print( ( """Saturn"" is "
, IF gas giants CONTAINSKEY "Saturn" THEN "" ELSE " not" FI
, "in gas giants", newline
)
);
print( ( """Venus"" is "
, IF gas giants CONTAINSKEY "Venus" THEN "" ELSE "not " FI
, "in gas giants", newline
)
);
print( ( "gas giants UNION ice giants                : " ) );
print set( gas giants U ice giants );
print( ( "moonless planets INTERSECTION rocky planets: " ) );
print set( moonless planets N rocky planets );
print( ( "rocky planets \ moonless planets           : " ) );
print set( rocky planets - moonless planets );
print( ( "moonless planets <= rocky planets          : "
, IF moonless planets <= rocky planets THEN "yes" ELSE "no" FI
, newline
)
);
print( ( "moonless planets = rocky planets           : "
, IF moonless planets = rocky planets THEN "yes" ELSE "no" FI
, newline
)
);
print( ( "inner planets = rocky planets              : "
, IF inner planets = rocky planets THEN "yes" ELSE "no" FI
, newline
)
);
print( ( "moonless planets < rocky planets           : "
, IF moonless planets < rocky planets THEN "yes" ELSE "no" FI
, newline
)
);

# REF AARRAYs are mutable                                                    #

REF AARRAY all planets := inner planets U gas giants U ice giants;
print( ( "all planets                   : " ) );
print set( all planets );
print( ( "... after restoration of Pluto: " ) );
all planets // "Pluto";
print set( all planets )```
Output:
```rocky planets   : [ Mercury Mars Earth Venus ]
inner planets   : [ Mercury Mars Earth Venus ]
gas giants      : [ Jupiter Saturn ]
ice giants      : [ Neptune Uranus ]
moonless planets: [ Mercury Venus ]

"Saturn" is in gas giants
"Venus" is not in gas giants
gas giants UNION ice giants                : [ Neptune Jupiter Saturn Uranus ]
moonless planets INTERSECTION rocky planets: [ Mercury Venus ]
rocky planets \ moonless planets           : [ Mars Earth ]
moonless planets <= rocky planets          : yes
moonless planets = rocky planets           : no
inner planets = rocky planets              : yes
moonless planets < rocky planets           : yes
all planets                   : [ Neptune Jupiter Mercury Mars Earth Venus Saturn Uranus ]
... after restoration of Pluto: [ Neptune Jupiter Mercury Mars Earth Venus Pluto Saturn Uranus ]
```

## Apex

In Apex, Sets are unordered collections of elements. Although elements can be anything including primitives, Ids, Apex classes, or sObjects, typically they are used with primitives and Ids.

```public class MySetController{
public Set<String> strSet {get; private set; }
public Set<Id> idSet {get; private set; }

public MySetController(){
//Initialize to an already known collection.  Results in a set of abc,def.
this.strSet = new Set<String>{'abc','abc','def'};

//Initialize to empty set and add in entries.
this.strSet = new Set<String>();
//Results in {'abc','def'}

//You can also get a set from a map in Apex. In this case, the account ids are fetched from a SOQL query.
Map<Id,Account> accountMap = new Map<Id,Account>([Select Id,Name From Account Limit 10]);
Set<Id> accountIds = accountMap.keySet();

//If you have a set, you can also use it with the bind variable syntax in SOQL:
List<Account> accounts = [Select Name From Account Where Id in :accountIds];

//Like other collections in Apex, you can use a for loop to iterate over sets:
for(Id accountId : accountIds){
Account a = accountMap.get(accountId);
//Do account stuffs here.
}
}
}```

## AppleScript

macOS's Foundation framework offers a few classes of set which can be used in AppleScript by means of AppleScriptObjectiveC code. The basic types are NSSet and its mutable equivalent NSMutableSet.

```use AppleScript version "2.4" -- OS X 10.10 (Yosemite) or later
use framework "Foundation"

-- 'set' at the beginnings of lines is an AppleScript command; nothing to do with sets.

set output to {}
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to ", "

set S to current application's class "NSSet"'s setWithArray:({1, 2, 3, 6, 7, 8, 9, 0})
set end of output to "Set S:  " & (S's allObjects() as list)
set end of output to "\"aardvark\" is a member of S:  " & ((S's containsObject:("aardvark")) as boolean)
set end of output to "3 is a member of S:  " & ((S's containsObject:(3)) as boolean)

set A to S's |copy|() -- or: set A to current application's class "NSSet"'s setWithArray:({1, 2, 3, 6, 7, 8, 9, 0})
set end of output to linefeed & "Set A:  " & (A's allObjects() as list)
set B to current application's class "NSSet"'s setWithArray:({2, 2, 2, 3, 4, 5, 6, 7, 7, 7, 8})
set end of output to "Set B:  " & (B's allObjects() as list)

-- Or:
-- set union to A's mutableCopy()
-- tell union to unionSet:(B)
set end of output to "Union of A and B:  " & (union's allObjects() as list)

set intersection to A's mutableCopy()
tell intersection to intersectSet:(B)
set end of output to "Intersection of A and B:  " & (intersection's allObjects() as list)

set difference to A's mutableCopy()
tell difference to minusSet:(B)
set end of output to "Difference of A and B:  " & (difference's allObjects() as list)

set end of output to "A is a subset of B:  " & ((A's isSubsetOfSet:(B)) as boolean)
set end of output to "A is a subset of S:  " & ((A's isSubsetOfSet:(S)) as boolean)

set end of output to "A is equal to B:  " & ((A's isEqualToSet:(B)) as boolean)
set end of output to "A is equal to S:  " & ((A's isEqualToSet:(S)) as boolean)

set AppleScript's text item delimiters to linefeed
set output to output as text
set AppleScript's text item delimiters to astid

return output

```
Output:
```"Set S:  0, 9, 1, 6, 2, 7, 3, 8
\"aardvark\" is a member of S:  false
3 is a member of S:  true

Set A:  0, 9, 1, 6, 2, 7, 3, 8
Set B:  5, 6, 2, 7, 3, 8, 4
Union of A and B:  0, 9, 7, 5, 3, 1, 8, 6, 4, 2
Intersection of A and B:  6, 2, 7, 3, 8
Difference of A and B:  0, 9, 1
A is a subset of B:  false
A is a subset of S:  true
A is equal to B:  false
A is equal to S:  true"
```

## Arturo

```a: [1 2 3 4]
b: [3 4 5 6]

print in? 3 a
print contains? b 3

print union a b
print intersection a b
print difference a b
print difference.symmetric a b

print a = b

print subset? [1 3] a
print subset?.proper [1 3] a
print subset? [1 3] [1 3]
print subset?.proper [1 3] [1 3]

print superset? a [1 3]
print superset?.proper a [1 3]
print superset? [1 3] [1 3]
print superset?.proper [1 3] [1 3]
```
Output:
```true
true
1 2 3 4 5 6
3 4
1 2
1 2 5 6
false
true
true
true
false
true
true
true
false```

## ATS

Library: xxHash

The following demonstrates sets of strings stored as a hash-keyed AVL tree. It requires the xxHash C library.

(Aside: If one were going to write an ATS implementation of sets as linked lists, it might be well to base the implementation on the reference code for Scheme SRFI-1: [1]. This is particularly so if an eq?-like function is possible for set members.)

```(*------------------------------------------------------------------*)

(*------------------------------------------------------------------*)
(* String hashing using XXH3_64bits from the xxHash suite.          *)

#define ATS_EXTERN_PREFIX "hashsets_postiats_"

%{^ /* Embedded C code. */

#include <xxhash.h>

ATSinline() atstype_uint64
hashsets_postiats_mem_hash (atstype_ptr data, atstype_size len)
{
return (atstype_uint64) XXH3_64bits (data, len);
}

%}

extern fn mem_hash : (ptr, size_t) -<> uint64 = "mac#%"

fn
string_hash (s : string) :<> uint64 =
let
val len = string_length s
in
mem_hash (\$UNSAFE.cast{ptr} s, len)
end

(*------------------------------------------------------------------*)
(* A trimmed down version of the AVL trees from the AVL Tree task.  *)

datatype bal_t =
| bal_minus1
| bal_zero
| bal_plus1

datatype avl_t (key_t  : t@ype+,
data_t : t@ype+,
size   : int) =
| avl_t_nil (key_t, data_t, 0)
| {size_L, size_R : nat}
avl_t_cons (key_t, data_t, size_L + size_R + 1) of
(key_t, data_t, bal_t,
avl_t (key_t, data_t, size_L),
avl_t (key_t, data_t, size_R))
typedef avl_t (key_t  : t@ype+,
data_t : t@ype+) =
[size : int] avl_t (key_t, data_t, size)

extern fun {key_t : t@ype}
avl_t\$compare (u : key_t, v : key_t) :<> int

#define NIL avl_t_nil ()
#define CONS avl_t_cons
#define LNIL list_nil ()
#define :: list_cons
#define F false
#define T true

typedef fixbal_t = bool

prfn
lemma_avl_t_param {key_t : t@ype} {data_t : t@ype} {size : int}
(avl : avl_t (key_t, data_t, size)) :<prf>
[0 <= size] void =
case+ avl of NIL => () | CONS _ => ()

fn {}
minus_neg_bal (bal : bal_t) :<> bal_t =
case+ bal of
| bal_minus1 () => bal_plus1
| _ => bal_zero ()

fn {}
minus_pos_bal (bal : bal_t) :<> bal_t =
case+ bal of
| bal_plus1 () => bal_minus1
| _ => bal_zero ()

fn
avl_t_is_empty {key_t : t@ype} {data_t : t@ype} {size   : int}
(avl : avl_t (key_t, data_t, size)) :<>
[b : bool | b == (size == 0)] bool b =
case+ avl of
| NIL => T
| CONS _ => F

fn
avl_t_isnot_empty {key_t : t@ype} {data_t : t@ype} {size   : int}
(avl : avl_t (key_t, data_t, size)) :<>
[b : bool | b == (size <> 0)] bool b =
~avl_t_is_empty avl

fn {key_t : t@ype} {data_t : t@ype}
avl_t_search_ref {size  : int}
(avl   : avl_t (key_t, data_t, size),
key   : key_t,
data  : &data_t? >> opt (data_t, found),
found : &bool? >> bool found) :<!wrt>
#[found : bool] void =
let
fun
search (p     : avl_t (key_t, data_t),
data  : &data_t? >> opt (data_t, found),
found : &bool? >> bool found) :<!wrt,!ntm>
#[found : bool] void =
case+ p of
| NIL =>
{
prval _ = opt_none {data_t} data
val _ = found := F
}
| CONS (k, d, _, left, right) =>
begin
case+ avl_t\$compare<key_t> (key, k) of
| cmp when cmp < 0 => search (left, data, found)
| cmp when cmp > 0 => search (right, data, found)
| _ =>
{
val _ = data := d
prval _ = opt_some {data_t} data
val _ = found := T
}
end
in
end

fn {key_t : t@ype} {data_t : t@ype}
avl_t_search_opt {size : int}
(avl  : avl_t (key_t, data_t, size),
key  : key_t) :<>
Option (data_t) =
let
var data : data_t?
var found : bool?
val _ = \$effmask_wrt avl_t_search_ref (avl, key, data, found)
in
if found then
let
prval _ = opt_unsome data
in
Some {data_t} data
end
else
let
prval _ = opt_unnone data
in
None {data_t} ()
end
end

fn {key_t : t@ype} {data_t : t@ype}
avl_t_insert_or_replace {size : int}
(avl  : avl_t (key_t, data_t, size),
key  : key_t,
data : data_t) :<>
[sz : pos] (avl_t (key_t, data_t, sz), bool) =
let
fun
search {size   : nat}
(p      : avl_t (key_t, data_t, size),
fixbal : fixbal_t,
found  : bool) :<!ntm>
[sz : pos]
(avl_t (key_t, data_t, sz), fixbal_t, bool) =
case+ p of
| NIL => (CONS (key, data, bal_zero, NIL, NIL), T, F)
| CONS (k, d, bal, left, right) =>
case+ avl_t\$compare<key_t> (key, k) of
| cmp when cmp < 0 =>
let
val (p1, fixbal, found) = search (left, fixbal, found)
in
case+ (fixbal, bal) of
| (F, _) => (CONS (k, d, bal, p1, right), F, found)
| (T, bal_plus1 ()) =>
(CONS (k, d, bal_zero (), p1, right), F, found)
| (T, bal_zero ()) =>
(CONS (k, d, bal_minus1 (), p1, right), fixbal, found)
| (T, bal_minus1 ()) =>
let
val+ CONS (k1, d1, bal1, left1, right1) = p1
in
case+ bal1 of
| bal_minus1 () =>
let
val q = CONS (k, d, bal_zero (), right1, right)
val q1 = CONS (k1, d1, bal_zero (), left1, q)
in
(q1, F, found)
end
| _ =>
let
val p2 = right1
val- CONS (k2, d2, bal2, left2, right2) = p2
val q = CONS (k, d, minus_neg_bal bal2,
right2, right)
val q1 = CONS (k1, d1, minus_pos_bal bal2,
left1, left2)
val q2 = CONS (k2, d2, bal_zero (), q1, q)
in
(q2, F, found)
end
end
end
| cmp when cmp > 0 =>
let
val (p1, fixbal, found) = search (right, fixbal, found)
in
case+ (fixbal, bal) of
| (F, _) => (CONS (k, d, bal, left, p1), F, found)
| (T, bal_minus1 ()) =>
(CONS (k, d, bal_zero (), left, p1), F, found)
| (T, bal_zero ()) =>
(CONS (k, d, bal_plus1 (), left, p1), fixbal, found)
| (T, bal_plus1 ()) =>
let
val+ CONS (k1, d1, bal1, left1, right1) = p1
in
case+ bal1 of
| bal_plus1 () =>
let
val q = CONS (k, d, bal_zero (), left, left1)
val q1 = CONS (k1, d1, bal_zero (), q, right1)
in
(q1, F, found)
end
| _ =>
let
val p2 = left1
val- CONS (k2, d2, bal2, left2, right2) = p2
val q = CONS (k, d, minus_pos_bal bal2,
left, left2)
val q1 = CONS (k1, d1, minus_neg_bal bal2,
right2, right1)
val q2 = CONS (k2, d2, bal_zero (), q, q1)
in
(q2, F, found)
end
end
end
| _ => (CONS (key, data, bal, left, right), F, T)
in
if avl_t_is_empty avl then
(CONS (key, data, bal_zero, NIL, NIL), F)
else
let
prval _ = lemma_avl_t_param avl
val (avl, _, found) = \$effmask_ntm search (avl, F, F)
in
(avl, found)
end
end

fn {key_t : t@ype} {data_t : t@ype}
avl_t_insert {size : int}
(avl  : avl_t (key_t, data_t, size),
key  : key_t,
data : data_t) :<>
[sz : pos] avl_t (key_t, data_t, sz) =
(avl_t_insert_or_replace<key_t><data_t> (avl, key, data)).0

fun {key_t : t@ype} {data_t : t@ype}
push_all_the_way_left (stack : List (avl_t (key_t, data_t)),
p     : avl_t (key_t, data_t)) :
List0 (avl_t (key_t, data_t)) =
let
prval _ = lemma_list_param stack
in
case+ p of
| NIL => stack
| CONS (_, _, _, left, _) =>
push_all_the_way_left (p :: stack, left)
end

fun {key_t : t@ype} {data_t : t@ype}
update_generator_stack (stack     : List (avl_t (key_t, data_t)),
right     : avl_t (key_t, data_t)) :
List0 (avl_t (key_t, data_t)) =
let
prval _ = lemma_list_param stack
in
if avl_t_is_empty right then
stack
else
push_all_the_way_left<key_t><data_t> (stack, right)
end

fn {key_t : t@ype} {data_t : t@ype}
avl_t_make_data_generator {size : int}
(avl  : avl_t (key_t, data_t, size)) :
() -<cloref1> Option data_t =
let
typedef avl_t = avl_t (key_t, data_t)

val stack = push_all_the_way_left<key_t><data_t> (LNIL, avl)
val stack_ref = ref stack

(* Cast stack_ref to its (otherwise untyped) pointer, so it can be
enclosed within ‘generate’. *)
val p_stack_ref = \$UNSAFE.castvwtp0{ptr} stack_ref

fun
generate () :<cloref1> Option data_t =
let
(* Restore the type information for stack_ref. *)
val stack_ref =
\$UNSAFE.castvwtp0{ref (List avl_t)} p_stack_ref

var stack : List0 avl_t = !stack_ref
var retval : Option data_t
in
begin
case+ stack of
| LNIL => retval := None ()
| p :: tail =>
let
val- CONS (_, d, _, left, right) = p
in
retval := Some d;
stack :=
update_generator_stack<key_t><data_t> (tail, right)
end
end;
!stack_ref := stack;
retval
end
in
generate
end

(*------------------------------------------------------------------*)
(* Sets implemented with a hash function, AVL trees and association *)
(* lists.                                                           *)

(* The interface  - - - - - - - - - - - - - - - - - - - - - - - - - *)

(* For simplicity, let us support only 64-bit hashes. *)

typedef hashset_t (key_t : t@ype+) =
avl_t (uint64, List1 key_t)

extern fun {key_t : t@ype}  (* Implement a hash function with this. *)
hashset_t\$hashfunc : key_t -<> uint64

extern fun {key_t : t@ype}     (* Implement key equality with this. *)
hashset_t\$key_eq : (key_t, key_t) -<> bool

extern fun
hashset_t_nil :
{key_t : t@ype}
() -<> hashset_t key_t

extern fun {key_t : t@ype}
(hashset_t key_t, key_t) -<> hashset_t key_t

(*
"remove_member" is not implemented here, because the trimmed down AVL
tree implementation above does not include deletion. We shall
implement everything else without using a member deletion routine.

extern fun {key_t : t@ype}
hashset_t_remove_member :
(hashset_t key_t, key_t) -<> hashset_t key_t

Of course you can remove a member by using hashset_t_difference.
*)

extern fun {key_t : t@ype}
hashset_t_has_member :
(hashset_t key_t, key_t) -<> bool

typedef hashset_t_binary_operation (key_t : t@ype) =
(hashset_t key_t, hashset_t key_t) -> hashset_t key_t

extern fun {key_t : t@ype}
hashset_t_union : hashset_t_binary_operation key_t

extern fun {key_t : t@ype}
hashset_t_intersection : hashset_t_binary_operation key_t

extern fun {key_t : t@ype}
hashset_t_difference : hashset_t_binary_operation key_t

extern fun {key_t : t@ype}
hashset_t_subset :
(hashset_t key_t, hashset_t key_t) -> bool

extern fun {key_t : t@ype}
hashset_t_equal :
(hashset_t key_t, hashset_t key_t) -> bool

(* Note: generators for hashset_t produce their output in unspecified
order. *)
extern fun {key_t : t@ype}
hashset_t_make_generator :
hashset_t key_t -> () -<cloref1> Option key_t

(* The implementation - - - - - - - - - - - - - - - - - - - - - - - *)

(* I make no promises that these are the most efficient
implementations I could devise. They certainly are not! But they
were easy to write and will work. *)

implement
hashset_t_nil () =
avl_t_nil ()

fun {key_t  : t@ype}
find_key {n : nat} .<n>.
(lst : list (key_t, n),
key : key_t) :<>
List0 key_t =
(* This implementation is tail recursive. It will not build up the
stack. *)
case+ lst of
| list_nil () => lst
lst
else
find_key (tail, key)

implement {key_t}
(* The following implementation assumes equal keys are
interchangeable. *)
let
implement
avl_t\$compare<uint64> (u, v) =
if u < v then ~1 else if v < u then 1 else 0
typedef lst_t = List1 key_t
val hash = hashset_t\$hashfunc<key_t> key
val lst_opt = avl_t_search_opt<uint64><lst_t> (set, hash)
in
case+ lst_opt of
| Some lst =>
begin
case+ find_key<key_t> (lst, key) of
| list_cons _ => set
| list_nil () =>
avl_t_insert<uint64><lst_t>
(set, hash, list_cons (key, lst))
end
| None () =>
avl_t_insert<uint64><lst_t>
(set, hash, list_cons (key, list_nil ()))
end

implement {key_t}
hashset_t_has_member (set, key) =
let
implement
avl_t\$compare<uint64> (u, v) =
if u < v then ~1 else if v < u then 1 else 0
typedef lst_t = List1 key_t
val hash = hashset_t\$hashfunc<key_t> key
val lst_opt = avl_t_search_opt<uint64><lst_t> (set, hash)
in
case+ lst_opt of
| None () => false
| Some lst =>
begin
case+ find_key<key_t> (lst, key) of
| list_nil () => false
| list_cons _ => true
end
end

implement {key_t}
hashset_t_union (u, v) =
let
val gen_u = hashset_t_make_generator<key_t> u
val gen_v = hashset_t_make_generator<key_t> v
var w : hashset_t key_t = hashset_t_nil ()
var k_opt : Option key_t
in
for (k_opt := gen_u (); option_is_some k_opt; k_opt := gen_u ())
w := hashset_t_add_member (w, option_unsome k_opt);
for (k_opt := gen_v (); option_is_some k_opt; k_opt := gen_v ())
w := hashset_t_add_member (w, option_unsome k_opt);
w
end

implement {key_t}
hashset_t_intersection (u, v) =
let
val gen_u = hashset_t_make_generator<key_t> u
var w : hashset_t key_t = hashset_t_nil ()
var k_opt : Option key_t
in
for (k_opt := gen_u (); option_is_some k_opt; k_opt := gen_u ())
let
val+ Some k = k_opt
in
if hashset_t_has_member<key_t> (v, k) then
end;
w
end

implement {key_t}
hashset_t_difference (u, v) =
let
val gen_u = hashset_t_make_generator<key_t> u
var w : hashset_t key_t = hashset_t_nil ()
var k_opt : Option key_t
in
for (k_opt := gen_u (); option_is_some k_opt; k_opt := gen_u ())
let
val+ Some k = k_opt
in
if ~hashset_t_has_member<key_t> (v, k) then
end;
w
end

implement {key_t}
hashset_t_subset (u, v) =
let
val gen_u = hashset_t_make_generator<key_t> u
var subset : bool = true
var done : bool = false
in
while (~done)
case+ gen_u () of
| None () => done := true
| Some k =>
if ~hashset_t_has_member<key_t> (v, k) then
begin
subset := false;
done := true
end;
subset
end

implement {key_t}
hashset_t_equal (u, v) =
hashset_t_subset<key_t> (u, v)
&& hashset_t_subset<key_t> (v, u)

implement {key_t}
hashset_t_make_generator (set) =
let
typedef lst_t = List1 key_t
typedef lst_t_0 = List0 key_t

val avl_gen = avl_t_make_data_generator<uint64><lst_t> (set)

val current_list_ref : ref lst_t_0 = ref (list_nil ())
val current_list_ptr =
\$UNSAFE.castvwtp0{ptr} current_list_ref
in
lam () =>
let
val current_list_ref =
\$UNSAFE.castvwtp0{ref lst_t_0} current_list_ptr
in
case+ !current_list_ref of
| list_nil () =>
begin
case+ avl_gen () of
| None () => None ()
| Some lst =>
begin
case+ lst of
begin
!current_list_ref := tail;
end
end
end
begin
!current_list_ref := tail;
end
end
end

(*------------------------------------------------------------------*)

implement
hashset_t\$hashfunc<string> (s) =
string_hash s

implement
hashset_t\$key_eq<string> (s, t) =
s = t

typedef strset_t = hashset_t string

fn {}
strset_t_nil () :<> strset_t =
hashset_t_nil ()

fn
member : string) :<> strset_t =

fn {}
set    : strset_t) :<> strset_t =

#define SNIL strset_t_nil ()
infixr ( :: ) ++        (* Right associative, same precedence as :: *)

fn
strset_t_has_member (set    : strset_t,
member : string) :<> bool =
hashset_t_has_member<string> (set, member)

fn
strset_t_union (u : strset_t, v : strset_t) : strset_t =
hashset_t_union<string> (u, v)

fn
strset_t_intersection (u : strset_t, v : strset_t) : strset_t =
hashset_t_intersection<string> (u, v)
infixl ( + ) ^

fn
strset_t_difference (u : strset_t, v : strset_t) : strset_t =
hashset_t_difference<string> (u, v)

fn
strset_t_subset (u : strset_t, v : strset_t) : bool =
hashset_t_subset<string> (u, v)

fn
strset_t_equal (u : strset_t, v : strset_t) : bool =
hashset_t_equal<string> (u, v)

fn
strset_t_make_generator (set : strset_t) :
() -<cloref1> Option string =
hashset_t_make_generator<string> set

fn
strset_t_print (set : strset_t) : void =
let
val gen = strset_t_make_generator set
var s_opt : Option string
var separator : string = ""
in
print! ("#<strset_t ");
for (s_opt := gen (); option_is_some s_opt; s_opt := gen ())
case+ s_opt of
| Some s =>
begin
(* The following quick and dirty implemenetation does not
insert escape sequences. *)
print! (separator, "\"", s, "\"");
separator := " "
end;
print! (">")
end

implement
main0 () =
let
val set1 =
"one" ++ "two" ++ "three" ++ "guide" ++ "design" ++ SNIL
val set2 =
"ett" ++ "två" ++ "tre" ++ "guide" ++ "design" ++ SNIL
in
print! ("set1 = ");
strset_t_print set1;

println! ();
println! ();
println! ("set1[\"one\"] = ", set1["one"]);
println! ("set1[\"two\"] = ", set1["two"]);
println! ("set1[\"three\"] = ", set1["three"]);
println! ("set1[\"four\"] = ", set1["four"]);

println! ();
print! ("set2 = ");
strset_t_print set2;

println! ();
println! ();
println! ("set2[\"ett\"] = ", set2["ett"]);
println! ("set2[\"två\"] = ", set2["två"]);
println! ("set2[\"tre\"] = ", set2["tre"]);
println! ("set2[\"fyra\"] = ", set2["fyra"]);

println! ();
print! ("Union\nset1 + set2 = ");
strset_t_print (set1 + set2);
println! ();

println! ();
print! ("Intersection\nset1 ^ set2 = ");
strset_t_print (set1 ^ set2);
println! ();

println! ();
print! ("Difference\nset1 - set2 = ");
strset_t_print (set1 - set2);
println! ();

println! ();
println! ("Subset");
println! ("set1 <= set1: ", set1 <= set1);
println! ("set2 <= set2: ", set2 <= set2);
println! ("set1 <= set2: ", set1 <= set2);
println! ("set2 <= set1: ", set2 <= set1);
println! ("(set1 ^ set2) <= set1: ", (set1 ^ set2) <= set1);
println! ("(set1 ^ set2) <= set2: ", (set1 ^ set2) <= set2);

println! ();
println! ("Equal");
println! ("set1 = set1: ", set1 = set1);
println! ("set2 = set2: ", set2 = set2);
println! ("set1 = set2: ", set1 = set2);
println! ("set2 = set1: ", set2 = set1);
println! ("(set1 ^ set2) = (set2 ^ set1): ",
(set1 ^ set2) = (set2 ^ set1));
println! ("(set1 ^ set2) = set1: ", (set1 ^ set2) = set1);
println! ("(set1 ^ set2) = set2: ", (set1 ^ set2) = set2)
end

(*------------------------------------------------------------------*)```
Output:
```\$ patscc -O2 -DATS_MEMALLOC_GCBDW hashsets-postiats.dats -lxxhash -lgc && ./a.out
set1 = #<strset_t "guide" "design" "two" "one" "three">

set1["one"] = true
set1["two"] = true
set1["three"] = true
set1["four"] = false

set2 = #<strset_t "två" "guide" "design" "ett" "tre">

set2["ett"] = true
set2["två"] = true
set2["tre"] = true
set2["fyra"] = false

Union
set1 + set2 = #<strset_t "två" "guide" "design" "two" "ett" "one" "three" "tre">

Intersection
set1 ^ set2 = #<strset_t "guide" "design">

Difference
set1 - set2 = #<strset_t "two" "one" "three">

Subset
set1 <= set1: true
set2 <= set2: true
set1 <= set2: false
set2 <= set1: false
(set1 ^ set2) <= set1: true
(set1 ^ set2) <= set2: true

Equal
set1 = set1: true
set2 = set2: true
set1 = set2: false
set2 = set1: false
(set1 ^ set2) = (set2 ^ set1): true
(set1 ^ set2) = set1: false
(set1 ^ set2) = set2: false```

## AutoHotkey

```test(Set,element){
for i, val in Set
if (val=element)
return true
return false
}

Union(SetA,SetB){
SetC:=[], Temp:=[]
for i, val in SetA
SetC.Insert(val), Temp[val] := true
for i, val in SetB
if !Temp[val]
SetC.Insert(val)
return SetC
}

intersection(SetA,SetB){
SetC:=[], Temp:=[]
for i, val in SetA
Temp[val] := true
for i, val in SetB
if Temp[val]
SetC.Insert(val)
return SetC
}

difference(SetA,SetB){
SetC:=[], Temp:=[]
for i, val in SetB
Temp[val] := true
for i, val in SetA
if !Temp[val]
SetC.Insert(val)
return SetC
}

subset(SetA,SetB){
Temp:=[], A:=B:=0
for i, val in SetA
Temp[val] := true , A++
for i, val in SetB
if Temp[val]{
B++
IfEqual, A, %B%, return 1
} return 0
}

equal(SetA,SetB){
return (SetA.MaxIndex() = SetB.MaxIndex() && subset(SetA,SetB)) ? 1: 0
}
```

Examples:

```A:= ["apple", "cherry", "elderberry", "grape"]
B:= ["banana", "cherry", "date", "elderberry", "fig"]
C:= ["apple", "cherry", "elderberry", "grape", "orange"]
D:= ["apple", "cherry", "elderberry", "grape"]
E:= ["apple", "cherry", "elderberry"]
M:= "banana"

Res =
(
A:= ["apple", "cherry", "elderberry", "grape"]
B:= ["banana", "cherry", "date", "elderberry", "fig"]
C:= ["apple", "cherry", "elderberry", "grape", "orange"]
D:= ["apple", "cherry", "elderberry", "grape"]
E:= ["apple", "cherry", "elderberry"]
M:= "banana"

)

Res .= "`nM is " (test(A,M)?"":"not ") "an element of Set A"
Res .= "`nM is " (test(B,M)?"":"not ") "an element of Set B"

Res .= "`nUnion(A,B) = "
for i, val in Union(A,B)
Res.= (A_Index=1?"`t":", ") val

Res .= "`nintersection(A,B) = "
for i, val in intersection(A,B)
Res.= (A_Index=1?"`t":", ") val

Res .= "`ndifference(A,B) = "
for i, val in difference(A,B)
Res.= (A_Index=1?"`t":", ") val

Res .= "`n`nA is " (subset(A,C)?"":"not ") "a subset of Set C"
Res .= "`nA is " (subset(A,D)?"":"not ") "a subset of Set D"
Res .= "`nA is " (subset(A,E)?"":"not ") "a subset of Set E"

Res .= "`n`nA is " (equal(A,C)?"":"not ") "a equal to Set C"
Res .= "`nA is " (equal(A,D)?"":"not ") "a equal to Set D"
Res .= "`nA is " (equal(A,E)?"":"not ") "a equal to Set E"

MsgBox % Res
```
Output:
```A:= ["apple", "cherry", "elderberry", "grape"]
B:= ["banana", "cherry", "date", "elderberry", "fig"]
C:= ["apple", "cherry", "elderberry", "grape", "orange"]
D:= ["apple", "cherry", "elderberry", "grape"]
E:= ["apple", "cherry", "elderberry"]
M:= "banana"

M is not an element of Set A
M is an element of Set B
Union(A,B) = 	apple, cherry, elderberry, grape, banana, date, fig
intersection(A,B) = 	cherry, elderberry
difference(A,B) = 	apple, grape

A is a subset of Set C
A is a subset of Set D
A is not a subset of Set E

A is not a equal to Set C
A is a equal to Set D
A is not a equal to Set E```

## BASIC

### BBC BASIC

The sets are represented as 32-bit integers, which means that the maximum number of elements is 32.

```      DIM list\$(6)
list\$() = "apple", "banana", "cherry", "date", "elderberry", "fig", "grape"

setA% = %1010101
PRINT "Set A: " FNlistset(list\$(), setA%)
setB% = %0111110
PRINT "Set B: " FNlistset(list\$(), setB%)
elementM% = %0000010
PRINT "Element M: " FNlistset(list\$(), elementM%) '

IF elementM% AND setA% THEN
PRINT "M is an element of set A"
ELSE
PRINT "M is not an element of set A"
ENDIF
IF elementM% AND setB% THEN
PRINT "M is an element of set B"
ELSE
PRINT "M is not an element of set B"
ENDIF

PRINT '"The union of A and B is " FNlistset(list\$(), setA% OR setB%)
PRINT "The intersection of A and B is " FNlistset(list\$(), setA% AND setB%)
PRINT "The difference of A and B is " FNlistset(list\$(), setA% AND NOT setB%)

IF (setA% AND setB%) = setA% THEN
PRINT '"Set A is a subset of set B"
ELSE
PRINT '"Set A is not a subset of set B"
ENDIF
IF setA% = setB% THEN
PRINT "Set A is equal to set B"
ELSE
PRINT "Set A is not equal to set B"
ENDIF
END

DEF FNlistset(list\$(), set%)
LOCAL i%, o\$
FOR i% = 0 TO 31
IF set% AND 1 << i% o\$ += list\$(i%) + ", "
NEXT
= LEFT\$(LEFT\$(o\$))
```
Output:
```Set A: apple, cherry, elderberry, grape
Set B: banana, cherry, date, elderberry, fig
Element M: banana

M is not an element of set A
M is an element of set B

The union of A and B is apple, banana, cherry, date, elderberry, fig, grape
The intersection of A and B is cherry, elderberry
The difference of A and B is apple, grape

Set A is not a subset of set B
Set A is not equal to set B
```

## BQN

BQN can use lists with only unique elements to represent sets. The following implement all the basic set relations.

```Union ← ⍷∾
Inter ← ∊/⊣
Diff ← ¬∘∊/⊣
Subset ← ∧´∊
Eq ← ≡○∧
CreateSet ← ⍷

•Show 2‿4‿6‿8 Union 2‿3‿5‿7
•Show 2‿4‿6‿8 Inter 2‿3‿5‿7
•Show 2‿4‿6‿8 Diff 2‿3‿5‿7
•Show 2‿4‿6‿8 Subset 2‿3‿5‿7
•Show 2‿4‿6‿8 Eq 2‿3‿5‿7
•Show CreateSet 2‿2‿3‿5‿7‿2
```
```⟨ 2 4 6 8 3 5 7 ⟩
⟨ 2 ⟩
⟨ 4 6 8 ⟩
0
0
⟨ 2 3 5 7 ⟩
```

## C

Building a set highly depends on the datatype and use case. For example, a set of string could be implemented by hash table, sort tree or trie, but if all sets are known to have very few number of elements, it might be best to use just flat arrays. There isn't, and shouldn't be, an all-purpose set type for C.

A frequent use of set is that of small, non-negative integers, implemented as a bit field as shown below.

```#include <stdio.h>

typedef unsigned int set_t; /* probably 32 bits; change according to need */

void show_set(set_t x, const char *name)
{
int i;
printf("%s is:", name);
for (i = 0; (1U << i) <= x; i++)
if (x & (1U << i))
printf(" %d", i);
putchar('\n');
}

int main(void)
{
int i;
set_t a, b, c;

a = 0; /* empty set */
for (i = 0; i < 10; i += 3) /* add 0 3 6 9 to set a */
a |= (1U << i);
show_set(a, "a");

for (i = 0; i < 5; i++)
printf("\t%d%s in set a\n", i, (a & (1U << i)) ? "":" not");

b = a;
b |= (1U << 5); b |= (1U << 10); /* b is a plus 5, 10 */
b &= ~(1U << 0);	/* sans 0 */
show_set(b, "b");

show_set(a | b, "union(a, b)");
show_set(c = a & b, "c = common(a, b)");
show_set(a & ~b, "a - b"); /* diff, not arithmetic minus */
show_set(b & ~a, "b - a");
printf("b is%s a subset of a\n", !(b & ~a) ? "" : " not");
printf("c is%s a subset of a\n", !(c & ~a) ? "" : " not");

printf("union(a, b) - common(a, b) %s union(a - b, b - a)\n",
((a | b) & ~(a & b)) == ((a & ~b) | (b & ~a))
? "equals" : "does not equal");

return 0;
}
```

## C#

```using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

class Program
{
static void PrintCollection(IEnumerable<int> x)
{
Console.WriteLine(string.Join(" ", x));
}
static void Main(string[] args)
{
Console.OutputEncoding = Encoding.UTF8;
Console.WriteLine("Set creation");
var A = new HashSet<int> { 4, 12, 14, 17, 18, 19, 20 };
var B = new HashSet<int> { 2, 5, 8, 11, 12, 13, 17, 18, 20 };

PrintCollection(A);
PrintCollection(B);

Console.WriteLine("Test m ∈ S -- \"m is an element in set S\"");
Console.WriteLine("14 is an element in set A: {0}", A.Contains(14));
Console.WriteLine("15 is an element in set A: {0}", A.Contains(15));

Console.WriteLine("A ∪ B -- union; a set of all elements either in set A or in set B.");
var aUb = A.Union(B);
PrintCollection(aUb);

Console.WriteLine("A ∖ B -- difference; a set of all elements in set A, except those in set B.");

Console.WriteLine("A ⊆ B -- subset; true if every element in set A is also in set B.");
Console.WriteLine(A.IsSubsetOf(B));
var C = new HashSet<int> { 14, 17, 18 };
Console.WriteLine(C.IsSubsetOf(A));

Console.WriteLine("A = B -- equality; true if every element of set A is in set B and vice versa.");
Console.WriteLine(A.SetEquals(B));
var D = new HashSet<int> { 4, 12, 14, 17, 18, 19, 20 };
Console.WriteLine(A.SetEquals(D));

Console.WriteLine("If A ⊆ B, but A ≠ B, then A is called a true or proper subset of B, written A ⊂ B or A ⊊ B");
Console.WriteLine(A.IsProperSubsetOf(B));
Console.WriteLine(C.IsProperSubsetOf(A));

Console.WriteLine("Modify a mutable set.  (Add 10 to A; remove 12 from B).");
B.Remove(12);
PrintCollection(A);
PrintCollection(B);

}
}
```
Output:
```Set creation
4 12 14 17 18 19 20
2 5 8 11 12 13 17 18 20
Test m ∈ S -- "m is an element in set S"
14 is an element in set A: True
15 is an element in set A: False
A ∪ B -- union; a set of all elements either in set A or in set B.
4 12 14 17 18 19 20 2 5 8 11 13
A ∖ B -- difference; a set of all elements in set A, except those in set B.
4 14 19
A ⊆ B -- subset; true if every element in set A is also in set B.
False
True
A = B -- equality; true if every element of set A is in set B and vice versa.
False
True
If A ⊆ B, but A ≠ B, then A is called a true or proper subset of B, written A ⊂ B or A ⊊ B
False
True
Modify a mutable set.  (Add 10 to A; remove 12 from B).
4 12 14 17 18 19 20 10
2 5 8 11 13 17 18 20```

## C++

C++ standard library contains a set class, which is a sorted container without duplicates and implemented as a binary tree. Additional set functionality can be implemented in terms of standard library algorithms.

C++11 standard library also contains unordered_set based on a hash table. However, algorithms like std::set_intersection etc take sorted ranges, so set-specific functions should be hand-rolled.

```#include <set>
#include <iostream>
#include <iterator>
#include <algorithm>

namespace set_display {
template <class T>
std::ostream& operator<<(std::ostream& os, const std::set<T>& set)
{
os << '[';
if (!set.empty()) {
std::copy(set.begin(), --set.end(), std::ostream_iterator<T>(os, ", "));
os << *--set.end();
}
return os << ']';
}
}

template <class T>
bool contains(const std::set<T>& set, const T& key)
{
return set.count(key) != 0;
}

template <class T>
std::set<T> set_union(const std::set<T>& a, const std::set<T>& b)
{
std::set<T> result;
std::set_union(a.begin(), a.end(), b.begin(), b.end(), std::inserter(result, result.end()));
return result;
}

template <class T>
std::set<T> set_intersection(const std::set<T>& a, const std::set<T>& b)
{
std::set<T> result;
std::set_intersection(a.begin(), a.end(), b.begin(), b.end(), std::inserter(result, result.end()));
return result;
}

template <class T>
std::set<T> set_difference(const std::set<T>& a, const std::set<T>& b)
{
std::set<T> result;
std::set_difference(a.begin(), a.end(), b.begin(), b.end(), std::inserter(result, result.end()));
return result;
}

template <class T>
bool is_subset(const std::set<T>& set, const std::set<T>& subset)
{
return std::includes(set.begin(), set.end(), subset.begin(), subset.end());
}

int main()
{
using namespace set_display;
std::set<int> a{2, 5, 7, 5, 9, 2}; //C++11 initialization syntax
std::set<int> b{1, 5, 9, 7, 4 };
std::cout << "a = " << a << '\n';
std::cout << "b = " << b << '\n';

int value1 = 8, value2 = 5;
std::cout << "Set a " << (contains(a, value1) ? "contains " : "does not contain ") << value1 << '\n';
std::cout << "Set a " << (contains(a, value2) ? "contains " : "does not contain ") << value2 << '\n';

std::cout << "Union of a and b: " << set_union(a, b) << '\n';
std::cout << "Intersection of a and b: " << set_intersection(a, b) << '\n';
std::cout << "Difference of a and b: " << set_difference(a, b) << '\n';

std::set<int> sub{5, 9};
std::cout << "Set b " << (is_subset(a, b) ? "is" : "is not") << " a subset of a\n";
std::cout << "Set " << sub << ' ' << (is_subset(a, sub) ? "is" : "is not") << " a subset of a\n";

std::set<int> copy = a;
std::cout << "a " << (a == copy ? "equals " : "does not equal ") << copy << '\n';

return 0;
}
```

## Ceylon

```shared void run() {
value a = set {1, 2, 3};
value b = set {3, 4, 5};
value union = a | b;
value intersection = a & b;
value difference = a ~ b;
value subset = a.subset(b);
value equality = a == b;

print("set a:         ``a``
set b:         ``b``
1 in a?        ``1 in a``
a | b:         ``union``
a & b:         ``intersection``
a ~ b:         ``difference``
a subset of b? ``subset``
a == b?        ``equality``");
}
```

## Clojure

```(require 'clojure.set)

; sets can be created using the set method or set literal syntax
(def a (set [1 2 3 4]))
(def b #{4 5 6 7})

(a 10) ; returns the element if it's contained in the set, otherwise nil

(clojure.set/union a b)

(clojure.set/intersection a b)

(clojure.set/difference a b)

(clojure.set/subset? a b)
```

## CoffeeScript

This implements functions from the task, along with an iteration helper called "each".

```# For ad-hoc set features, it sometimes makes sense to use hashes directly,
# rather than abstract to this level, but I'm showing a somewhat heavy
# solution to show off CoffeeScript class syntax.
class Set
constructor: (elems...) ->
@hash = {}
for elem in elems
@hash[elem] = true

@hash[elem] = true

remove: (elem) ->
delete @hash[elem]

has: (elem) ->
@hash[elem]?

union: (set2) ->
set = new Set()
for elem of @hash
for elem in set2.to_array()
set

intersection: (set2) ->
set = new Set()
for elem of @hash
set

minus: (set2) ->
set = new Set()
for elem of @hash
set

is_subset_of: (set2) ->
for elem of @hash
return false if !set2.has elem
true

equals: (set2) ->
this.is_subset_of(set2) and set2.is_subset_of this

to_array: ->
(elem for elem of @hash)

each: (f) ->
for elem of @hash
f(elem)

to_string: ->
@to_array()

run_tests = ->
set1 = new Set("apple", "banana") # creation
console.log set1.has "apple" # true (membership)
console.log set1.has "worms" # false (membership)

set2 = new Set("banana", "carrots")
console.log set1.union(set2).to_string() # [ 'apple', 'banana', 'carrots' ] (union)
console.log set1.intersection(set2).to_string() # [ 'banana' ] (intersection)
console.log set1.minus(set2).to_string() # [ 'apple' ] (difference)

set3 = new Set("apple")
console.log set3.is_subset_of set1 # true
console.log set3.is_subset_of set2 # false

set4 = new Set("apple", "banana")
console.log set4.equals set1 # true
console.log set4.equals set2 # false

set5 = new Set("foo")
console.log set5.to_string() # [ 'foo', 'bar' ]
set5.remove "bar" # remove
console.log set5.to_string() # [ 'foo' ]

# iteration, prints apple then banana (order not guaranteed)
set1.each (elem) ->
console.log elem

run_tests()
```

## Common Lisp

Common Lisp provides some set operations on lists.

```(setf a '(1 2 3 4))
(setf b '(2 3 4 5))

(format t "sets: ~a ~a~%" a b)

;;; element
(loop for x from 1 to 6 do
(format t (if (member x a)
"~d ∈ A~%"
"~d ∉ A~%") x))

(format t "A ∪ B: ~a~%" (union a b))
(format t "A ∩ B: ~a~%" (intersection a b))
(format t "A \\ B: ~a~%" (set-difference a b))
(format t (if (subsetp a b)
"~a ⊆ ~a~%"
"~a ⊈ ~a~%") a b)

(format t (if (and (subsetp a b)
(subsetp b a))
"~a = ~a~%"
"~a ≠ ~a~%") a b)
```

## D

```void main() {
import std.stdio, std.algorithm, std.range;

// Not true sets, items can be repeated, but must be sorted.
auto s1 = [1, 2, 3, 4, 5, 6].assumeSorted;
auto s2 = [2, 5, 6, 3, 4, 8].sort(); // [2,3,4,5,6,8].
auto s3 = [1, 2, 5].assumeSorted;

assert(s1.canFind(4)); // Linear search.
assert(s1.contains(4)); // Binary search.
assert(s1.setUnion(s2).equal([1,2,2,3,3,4,4,5,5,6,6,8]));
assert(s1.setIntersection(s2).equal([2, 3, 4, 5, 6]));
assert(s1.setDifference(s2).equal([1]));
assert(s1.setSymmetricDifference(s2).equal([1, 8]));
assert(s3.setDifference(s1).empty); // It's a subset.
assert(!s1.equal(s2));

auto s4 = [[1, 4, 7, 8], [1, 7], [1, 7, 8], [4], [7]];
const s5 = [1, 1, 1, 4, 4, 7, 7, 7, 7, 8, 8];
assert(s4.nWayUnion.equal(s5));
}
```

==D==

```module set;
import std.typecons : Tuple, tuple;
struct Set(V) { // Limited set of V-type elements                                        // here 'this' is named A, s is B, v V-type item

protected V[] array;

this(const Set s) {                                                              // construct A by copy of B
array = s.array.dup;
}

this(V[] arg...){                                                                // construct A with items
foreach(v; arg) if (v.isNotIn(array)) array ~= v;
}

enum : Set { empty = Set() }                                                     // ∅

ref Set opAssign()(const Set s) {                                                // A = B
array = s.array.dup;
return this;
}

bool opBinaryRight(string op : "in")(const V v) const {                          // v ∈ A
return v.isIn(array);
}

ref Set opOpAssign(string op)(const V v) if (op == "+" || op == "|") {           // A += {v}          // + = ∪ = |
if (v.isIn(array)) return this;
array ~= v;
return this;
}

ref Set opOpAssign(string op)(const Set s) if (op == "+" || op == "|") {         // A += B
foreach(x; s.array) if (x.isNotIn(array)) array ~= x;
return this;
}

Set opBinary(string op)(const V v) const if (op == "+" || op == "|"){            // A + {v}
Set result = this;
result += v;
return result;
}

Set opBinaryRight(string op)(const V v) const if (op == "+" || op == "|") {      // {v} + A
Set result = this;
result += v;
return result;
}

Set opBinary(string op)(const Set s) const if (op == "+" || op == "|") {         // A + B
Set result = this;
result += s;
return result;
}

Set opBinary(string op : "&")(const Set s) const{                                // A ∩ B               // ∩ = &
Set result;
foreach(x; array) if(x.isIn(s.array)) result += x;
return result;
}

ref Set opOpAssign(string op : "&")(const Set s) {                               // A ∩= B
return this(this & s);
}

Set opBinary(string op : "^")(const Set s) const {                               // (A ∪ B) - (A ∩ B)    //  = A ^ B
Set result;
foreach(x; array) if (x.isNotIn(s.array)) result += x;
foreach(x; s.array) if(x.isNotIn(array)) result += x;
return result;
}

ref opOpAssign(string op : "^")(const Set s) {
return this = this ^ s;
}

Set opBinary(string op : "-")(const Set s) const {                                // A - B
Set r;
foreach(x; array) if(x.isNot(s.array)) r += x;
return r;
}

ref Set opOpAssign(string op : "-")(const Set s) {                                // A -= B
return this = this - s;
}

Set!(Tuple!(V,U)) opBinary(U, string op : "*")(const Set!U s) const {             // A × B = { (x, y) | ∀x ∈ A ∧ ∀y ∈ B }
Set!(Tuple!(V, U)) r;
foreach(x; array) foreach(y; s.array) r += tuple(x, y);
return r;
}

bool isEmpty() const { return !array.length;}                                     // A ≟ ∅

bool opBinary(string op : "in")(const Set s) const {                              // A ⊂ s
foreach(v; array) if(v.isNotIn(s.array)) return false;
return true;
}

bool opEquals(const Set s) const {                                                // A ≟ B
if (array.length != s.array.length) return false;
return this in s;
}

T[] array() const @property { return array.dup;}

}

Set!(Tuple!(T, T)) sqr(T)(const Set!T s) { return s * s; }                                 // A²

auto pow(T, uint n : 0)(const Set!T s) {                                                   // A ^ 0
return Set!T.empty;
}

auto pow(T, uint n : 1)(const Set!T s) {                                                   // A ^ 1 = A
return s;
}

auto pow(T, uint n : 2)(const Set!T s) {                                                   // A ^ 2 (=A²)
return sqr!T(s);
}

auto pow(T, uint n)(const Set!T s) if(n % 2) {                                             // if n Odd,  A^n = A * (A^(n/2))²
return s * sqr!T(pow!(T, n/2)(s));
}

auto pow(T, uint n)(const Set!T s) if(!(n % 2)) {                                           // if n Even, A^n = (A^(n/2))²
return sqr!T(pow!(T, n/2)(s));
}

size_t Card(T)(const Set!T s) {return s.length; }                                           // Card(A)

Set!(Set!T) power(T)(Set!T s) {                                                             // ∀B ∈ P(A) ⇒ B ⊂ A
Set!(Set!T) ret;
foreach(e; s.array) {
Set!(Set!T) rs;
foreach(x; ret.array) {
x += e;
rs += x;
}
ret += rs;
}
return ret;
}

bool isIn(T)(T x, T[] array){
foreach(a; array) if(a == x) return true;
return false;
}
bool isNotIn(T)(T x, T[] array){
foreachj(a; array) if(a == x) return false;
return true;
}
```

## Dart

```void main(){
//Set Creation
Set A = new Set.from([1,2,3]);
Set B = new Set.from([1,2,3,4,5]);
Set C = new Set.from([1,2,4,5]);

print('Set A = \$A');
print('Set B = \$B');
print('Set C = \$C');
print('');
//Test if element is in set
int m = 3;
print('m = 5');
print('m in A = \${A.contains(m)}');
print('m in B = \${B.contains(m)}');
print('m in C = \${C.contains(m)}');
print('');
//Union of two sets
Set AC = A.union(C);
print('Set AC = Union of A and C = \$AC');
print('');
//Intersection of two sets
Set A_C = A.intersection(C);
print('Set A_C = Intersection of A and C = \$A_C');
print('');
//Difference of two sets
Set A_diff_C = A.difference(C);
print('Set A_diff_C = Difference between A and C = \$A_diff_C');
print('');
//Test if set is subset of another set
print('A is a subset of B = \${B.containsAll(A)}');
print('C is a subset of B = \${B.containsAll(C)}');
print('A is a subset of C = \${C.containsAll(A)}');
print('');
//Test if two sets are equal
print('A is equal to B  = \${B.containsAll(A) && A.containsAll(B)}');
print('B is equal to AC = \${B.containsAll(AC) && AC.containsAll(B)}');
}
```
Output:
```Set A = {1, 2, 3}
Set B = {1, 2, 3, 4, 5}
Set C = {1, 2, 4, 5}

m = 5
m in A = true
m in B = true
m in C = false

Set AC = Union of A and C = {1, 2, 3, 4, 5}

Set A_C = Intersection of A and C = {1, 2}

Set A_diff_C = Difference between A and C = {3}

A is a subset of B = true
C is a subset of B = true
A is a subset of C = false

A is equal to B  = false
B is equal to AC = true```

## Delphi

The library Boost.Generics.Collection.

```program Set_task;

{\$APPTYPE CONSOLE}

uses
System.SysUtils,
Boost.Generics.Collection;

begin
var s1 := TSet<Integer>.Create([1, 2, 3, 4, 5, 6]);
var s2 := TSet<Integer>.Create([2, 5, 6, 3, 4, 8]);
var s3 := TSet<Integer>.Create([1, 2, 5]);

Writeln('S1 ', s1.ToString);
Writeln('S2 ', s2.ToString);
Writeln('S3 ', s3.ToString, #10);

Writeln('4 is in S1? ', s1.Has(4));
Writeln('S1 union S2 ', (s1 + S2).ToString);
Writeln('S1 intersection S2 ', (s1 * S2).ToString);
Writeln('S1 difference S2 ', (s1 - S2).ToString);
Writeln('S3 is subset S2 ', s1.IsSubSet(s3));
Writeln('S1 equality S2? ', s1 = s2);
end.
```
Output:
```S1 { 1, 2, 3, 4, 5, 6 }
S2 { 2, 3, 4, 5, 6, 8 }
S3 { 1, 2, 5 }

4 is in S1? TRUE
S1 union S2 { 1, 2, 3, 4, 5, 6, 8 }
S1 intersection S2 { 2, 3, 4, 5, 6 }
S1 difference S2 { 1 }
S3 is subset S2 TRUE
S1 equality S2? FALSE```

## Diego

```use_namespace(rosettacode)_me();

// Set creation

// Membership
ms_msg()_calc([M]∈[B])
? with_msg()_msg(set M is an element in set B);
: with_msg()_msg(set M is not an element in set B);
;

ms_msg()_calc(🐖∈[A])
? with_msg()_msg(🐖 is an element in set A);
: with_msg()_msg(🐖 is not an element in set A);
;

// Union
ms_msg()_msg(A∪B=[])_calc([A]∪[B]);

// Intersection
ms_msg()_msg(A∩B=[])_calc([A]∩[B]);

// Difference
ms_msg()_msg(A∖B=[])_calc([A]∖[B]);   // U+2216 is used not U+005c (\)
ms_msg()_msg(A\\B=[])_calc([A]\\[B]); // U+005c (\) has to be escaped

// Subset
ms_msg()_calc([C]⊆[A])
? with_msg()_msg(set C is a subset of set A);
: with_msg()_msg(set C is not a subset of set A);
;

ms_msg()_calc([C]⊆[B])
? with_msg()_msg(set C is a subset of set B);
: with_msg()_msg(set C is not a subset of set B);
;

// Equality
ms_msg()_calc([A]=[B])
? with_msg()_msg(set A is equal to set B);
: with_msg()_msg(set A is not equal to set B);
;

// Test
ms_msg()_calc([A]⊂[B])_or()_calc([A]⊊[B])
? with_msg()_msg(set A is a proper subset of set B);
: with_msg()_msg(set A is not a proper subset of set B);
;

ms_msg()_calc([C]⊂[B]||[C]⊊[B])     // alternative syntax
? with_msg()_msg(set C is a proper subset of set B);
: with_msg()_msg(set C is not a proper subset of set B);
;

// Modify a mutable set (all sets are mutable)
with_set(M)_push(🦬,🦘,🦫,🦭);
ms_msg()_calc([M]=[A])
? with_msg()_msg(set M is equal to set A);
: with_msg()_msg(set M is not equal to set A);
;

reset_namespace[];```
Output:
```set M is an element in set B
🐖 is an element in set A
A∪B=🐖,🦬,🦘,🦫,🦭,🐈‍⬛,🦤,🐐
A∩B=🦬,🦫
A∖B=🐖,🦘,🦭
A\B=🐖,🦘,🦭
set C is not a subset of set A
set C is a subset of set B
set A is not equal to set B
set A is not a proper subset of set B
set C is a proper subset of set B
set M is equal to set A```

## EchoLisp

EchoLisp sets are lists, i.e the set of all sets is a proper subset of the set of all lists. Sets elements may be any object, including sets.

The set operations are: ∩ ∪ ⊆ / ∈ = ∆ ×

```; use { } to read a set
(define A { 1 2 3 4 3 5 5}) → { 1 2 3 4 5 } ; duplicates are removed from a set
; or use make-set to make a set from a list
(define B (make-set ' ( 3 4 5 6 7 8 8))) → { 3 4 5 6 7 8 }
(set-intersect A B) → { 3 4 5 }
(set-intersect? A B) → #t ; predicate
(set-union A B) → { 1 2 3 4 5 6 7 8 }
(set-substract A B) → { 1 2 }
(set-sym-diff A B) → { 1 2 6 7 8 } ; ∆ symmetric difference
(set-equal? A B) →  #f
(set-equal? { a b c} { c b a}) → #t ; order is unimportant
(set-subset? A B) → #f ; B in A or B = A
(set-subset? A { 3 4 }) → #t
(member 4 A) → (4 5) ; same as #t : true
(member 9 A) → #f

; check basic equalities
(set-equal? A (set-union (set-intersect A B) (set-substract A B))) → #t
(set-equal? (set-union A B) (set-union (set-sym-diff A B) (set-intersect A B))) → #t

; × : cartesian product of two sets : all pairs (a . b) , a in A, b in B
; returns a list (not a set)
(define A { albert simon})
(define B { antoinette ornella marylin})

(set-product A B)
→ ((albert . antoinette) (albert . marylin) (albert . ornella) (simon . antoinette) (simon . marylin) (simon . ornella))

; sets elements may be sets
{ { a b c} {c b a } { a b d}} → { { a b c } { a b d } } ; duplicate removed

; A few functions return sets :
(primes 10) → { 2 3 5 7 11 13 17 19 23 29 }
```

## Elixir

Works with: Elixir version 1.1
```iex(1)> s = MapSet.new
#MapSet<[]>
iex(2)> sa = MapSet.put(s, :a)
#MapSet<[:a]>
iex(3)> sab = MapSet.put(sa, :b)
#MapSet<[:a, :b]>
iex(4)> sbc = Enum.into([:b, :c], MapSet.new)
#MapSet<[:b, :c]>
iex(5)> MapSet.member?(sab, :a)
true
iex(6)> MapSet.member?(sab, :c)
false
iex(7)> :a in sab
true
iex(8)> MapSet.union(sab, sbc)
#MapSet<[:a, :b, :c]>
iex(9)> MapSet.intersection(sab, sbc)
#MapSet<[:b]>
iex(10)> MapSet.difference(sab, sbc)
#MapSet<[:a]>
iex(11)> MapSet.disjoint?(sab, sbc)
false
iex(12)> MapSet.subset?(sa, sab)
true
iex(13)> MapSet.subset?(sab, sa)
false
iex(14)> sa == sab
false
```

## Erlang

Built in.

```2> S = sets:new().
4> Sab = sets:from_list([a, b]).
5> sets:is_element(a, Sa).
true
6> Union = sets:union(Sa, Sab).
7> sets:to_list(Union).
[a,b]
8> Intersection = sets:intersection(Sa, Sab).
9> sets:to_list(Intersection).
[a]
10> Subtract = sets:subtract(Sab, Sa).
11> sets:to_list(Subtract).
[b]
12> sets:is_subset(Sa, Sab).
true
13> Sa =:= Sab.
false
```

## F#

The Collections.Set<'T> class implements "Immutable sets based on binary trees, where comparison is the F# structural comparison function, potentially using implementations of the IComparable interface on key values." (http://msdn.microsoft.com/en-us/library/ee353619.aspx)

```[<EntryPoint>]
let main args =
// Create some sets (of int):
let s1 = Set.ofList [1;2;3;4;3]
let s2 = Set.ofArray [|3;4;5;6|]

printfn "Some sets (of int):"
printfn "s1 = %A" s1
printfn "s2 = %A" s2
printfn "Set operations:"
printfn "2 ∈ s1? %A" (s1.Contains 2)
printfn "10 ∈ s1? %A" (s1.Contains 10)
printfn "s1 ∪ s2 = %A" (Set.union s1 s2)
printfn "s1 ∩ s2 = %A" (Set.intersect s1 s2)
printfn "s1 ∖ s2 = %A" (Set.difference s1 s2)
printfn "s1 ⊆ s2? %A" (Set.isSubset s1 s1)
printfn "{3, 1} ⊆ s1? %A" (Set.isSubset (Set.ofList [3;1]) s1)
printfn "{3, 2, 4, 1} = s1? %A" ((Set.ofList [3;2;4;1]) = s1)
printfn "s1 = s2? %A" (s1 = s2)
printfn "More set operations:"
printfn "#s1 = %A" s1.Count
printfn "s1 ∪ {99} = %A" (s1.Add 99)
printfn "s1 ∖ {3} = %A" (s1.Remove 3)
printfn "s1 ⊂ s1? %A" (Set.isProperSubset s1 s1)
printfn "s1 ⊂ s2? %A" (Set.isProperSubset s1 s2)
0
```
Output:
```Some sets (of int):
s1 = set [1; 2; 3; 4]
s2 = set [3; 4; 5; 6]
Set operations:
2 ∈ s1? true
10 ∈ s1? false
s1 ∪ s2 = set [1; 2; 3; 4; 5; 6]
s1 ∩ s2 = set [3; 4]
s1 ∖ s2 = set [1; 2]
s1 ⊆ s2? true
{3, 1} ⊆ s1? true
{3, 2, 4, 1} = s1? true
s1 = s2? false
More set operations:
#s1 = 4
s1 ∪ {99} = set [1; 2; 3; 4; 99]
s1 ∖ {3} = set [1; 2; 4]
s1 ⊂ s1? false
s1 ⊂ s2? false```

## Factor

We will use Factor's hash-sets for this task. A hash-set is created with `HS{ ... }`.

```( scratchpad ) USE: sets
( scratchpad ) HS{ 2 5 4 3 } HS{ 5 6 7 } union .
HS{ 2 3 4 5 6 7 }
( scratchpad ) HS{ 2 5 4 3 } HS{ 5 6 7 } intersect .
HS{ 5 }
( scratchpad ) HS{ 2 5 4 3 } HS{ 5 6 7 } diff .
HS{ 2 3 4 }
( scratchpad ) HS{ 2 5 4 3 } HS{ 5 6 7 } subset? .
f
( scratchpad ) HS{ 5 6 } HS{ 5 6 7 } subset? .
t
( scratchpad ) HS{ 5 6 } HS{ 5 6 7 } set= .
f
( scratchpad ) HS{ 6 5 7 } HS{ 5 6 7 } set= .
t
```

## Forth

Works with: Forth

Works with any ANS Forth. Needs the FMS-SI (single inheritance) library code located here: http://soton.mpeforth.com/flag/fms/index.html

```include FMS-SI.f
include FMS-SILib.f

: union {: a b -- c :}
begin
b each:
while dup
a indexOf: if 2drop else a add: then
repeat b <free a dup sort: ; ok

i{ 2 5 4 3 } i{ 5 6 7 } union p: i{ 2 3 4 5 6 7 } ok

: free2 ( a b -- ) <free <free ;
: intersect {: a b | c -- c :}
heap> 1-array2 to c
begin
b each:
while dup
a indexOf: if drop c add: else drop then
repeat a b free2 c dup sort: ;

i{ 2 5 4 3 } i{ 5 6 7 } intersect p: i{ 5 } ok

: diff {: a b | c -- c :}
heap> 1-array2 to c
begin
a each:
while dup
b indexOf: if 2drop else c add: then
repeat a b free2 c dup sort: ;

i{ 2 5 4 3 } i{ 5 6 7 } diff p: i{ 2 3 4 } ok

: subset {: a b -- flag :}
begin
a each:
while
b indexOf: if drop else false exit then
repeat a b free2 true ;

i{ 2 5 4 3 } i{ 5 6 7 } subset . 0 ok
i{ 5 6 } i{ 5 6 7 } subset .  -1 ok

: set= {: a b -- flag :}
a size: b size: <> if a b free2 false exit then
a sort: b sort:
begin
a each: drop b each:
while
<> if a b free2 false exit then
repeat a b free2 true ;

i{ 5 6 } i{ 5 6 7 } set= .  0 ok
i{ 6 5 7 } i{ 5 6 7 } set= .  -1 ok
```

## FreeBASIC

```function is_in( N as integer, S() as integer ) as boolean
'test if the value N is in the set S
for i as integer = 0 to ubound(S)
if N=S(i) then return true
next i
return false
end function

sub add_to_set( N as integer, S() as integer )
'adds the element N to the set S
if is_in( N, S() ) then return
dim as integer k = ubound(S)
redim preserve S(0 to k+1)
S(k+1)=N
end sub

sub setunion( S() as integer, T() as integer, U() as integer )
'makes U() the union of the sets S and T
dim as integer k = ubound(S)
redim U(-1)
for i as integer = 0 to k
next i
k = ubound(T)
for i as integer = 0 to k
if not is_in( T(i), U() ) then
end if
next i
end sub

sub setintersect( S() as integer, T() as integer, U() as integer )
'makes U() the intersection of the sets S and T
dim as integer k = ubound(S)
redim U(-1)
for i as integer = 0 to k
if is_in(S(i), T()) then add_to_set( S(i), U() )
next i
end sub

sub setsubtract( S() as integer, T() as integer, U() as integer )
'makes U() the difference of the sets S and T
dim as integer k = ubound(S)
redim U(-1)
for i as integer = 0 to k
if not is_in(S(i), T()) then add_to_set( S(i), U() )
next i
end sub

function is_subset( S() as integer, T() as integer ) as boolean
for i as integer = 0 to ubound(S)
if not is_in( S(i), T() ) then return false
next i
return true
end function

function is_equal( S() as integer, T() as integer ) as boolean
if not is_subset( S(), T() ) then return false
if not is_subset( T(), S() ) then return false
return true
end function

function is_proper_subset( S() as integer, T() as integer ) as boolean
if not is_subset( S(), T() ) then return false
if is_equal( S(), T() ) then return false
return true
end function

sub show_set( L() as integer )
'display a set
dim as integer num = ubound(L)
if num=-1 then
print "[]"
return
end if
print "[";
for i as integer = 0 to num-1
print str(L(i))+", ";
next i
print str(L(num))+"]"
end sub

'sets are created by making an empty array
redim as integer S1(-1), S2(-1), S3(-1), S4(-1), S5(-1)
'and populated by adding elements one-by-one
print "S1    ",
show_set S1()
print "S2    ",
show_set S2()
print "S3    ",
show_set S3()
print "S4    ",
show_set S4()
print "S5    ",
show_set S5()
print "----"
redim as integer S_U(-1)
setunion S1(), S2(), S_U()
print "S1 U S2    ",
show_set S_U()
redim as integer S_U(-1)
setintersect S1(), S2(), S_U()
print "S1 n S2    ",
show_set S_U()
redim as integer S_U(-1)
setsubtract S1(), S2(), S_U()
print "S1 \ S2    ",
show_set S_U()
redim as integer S_U(-1)
setsubtract S3(), S1(), S_U()
print "S3 \ S1    ",
show_set S_U()
print "S4 in S3? ", is_subset(S4(), S3())
print "S3 in S4? ", is_subset(S3(), S4())
print "S5 in S3? ", is_subset(S5(), S3())  'empty set is a subset of every set
print "S2 = S3?  ", is_equal(S2(), S3())
print "S4 proper subset of S3?   ", is_proper_subset( S4(), S3() )
print "S2 proper subset of S3?   ", is_proper_subset( S2(), S3() )```
Output:
```S1            [20, 30, 40, 50]
S2            [19, 20, 21, 22]
S3            [22, 21, 19, 20]
S4            [21]
S5            []
----
S1 U S2       [20, 30, 40, 50, 19, 21, 22]
S1 n S2       [20]
S1 \ S2       [30, 40, 50]
S3 \ S1       [22, 21, 19]
S4 in S3?     true
S3 in S4?     false
S5 in S3?     true
S2 = S3?      true
S4 proper subset of S3?     true
S2 proper subset of S3?     false```

## Frink

```a = new set[1, 2]
b = toSet[[2,3]]   // Construct a set from an array

a.contains[2]  // Element test (returns true)
union[a,b]
intersection[a,b]
setDifference[a,b]
isSubset[a,b]  // Returns true if a is a subset of b
a==b           // set equality test```

## FunL

```A = {1, 2, 3}
B = {3, 4, 5}
C = {1, 2, 3, 4, 5}
D = {2, 1, 3}

println( '2 is in A: ' + (2 in A) )
println( '4 is in A: ' + (4 in A) )
println( 'A union B: ' + A.union(B) )
println( 'A intersect B: ' + A.intersect(B) )
println( 'A difference B: ' + A.diff(B) )
println( 'A subset of B: ' + A.subsetOf(B) )
println( 'A subset of B: ' + A.subsetOf(C) )
println( 'A equal B: ' + (A == B) )
println( 'A equal D: ' + (A == D) )

S = set( A )

println( 'S (mutable version of A): ' + S )
println( 'S with 4 added: ' + S )
println( 'S subset of C: ' + S.subsetOf(C) )
S.remove( 1 )
println( 'S after 1 removed: ' + S )```
Output:
```2 is in A: true
4 is in A: false
A union B: {4, 5, 1, 2, 3}
A intersect B: {3}
A difference B: {1, 2}
A subset of B: false
A subset of B: true
A equal B: false
A equal D: true
S (mutable version of A): {1, 2, 3}
S with 4 added: {1, 2, 3, 4}
S subset of C: true
S after 1 removed: {2, 3, 4}
```

## FutureBasic

```include "NSLog.incl"

local fn DoIt
// create
CFSetRef s1 = fn SetWithArray( @[@"a",@"b",@"c",@"d",@"e"] )
CFSetRef s2 = fn SetWithArray( @[@"b",@"c",@"d",@"e",@"f",@"h"] )
CFSetRef s3 = fn SetWithArray( @[@"b",@"c",@"d"] )
CFSetRef s4 = fn SetWithArray( @[@"b",@"c",@"d"] )
NSLog(@"s1: %@",s1)
NSLog(@"s2: %@",s2)
NSLog(@"s3: %@",s3)
NSLog(@"s4: %@\n",s4)

// membership
NSLog(@"\"b\" in s1: %d", fn SetContainsObject( s1, @"b" ))
NSLog(@"\"f\" in s1: %d\n", fn SetContainsObject( s1, @"f" ))

// union
CFMutableSetRef s12 = fn MutableSetWithSet( s1 )
MutableSetUnionSet( s12, s2 )
NSLog(@"s1 union s2: %@\n", s12)

// intersection
CFMutableSetRef s1i2 = fn MutableSetWithSet( s1 )
MutableSetIntersectSet( s1i2, s2 )
NSLog(@"s1 intersect s2: %@\n", s1i2)

// difference
CFMutableSetRef s1d2 = fn MutableSetWithSet( s1 )
MutableSetMinusSet( s1d2, s2 )
NSLog(@"s1 - s2: %@\n", s1d2)

// subsetof
NSLog(@"s3 subset of s1: %d\n", fn SetIsSubsetOfSet( s3, s1 ))

// equality
NSLog(@"s3 == s4: %d\n", fn SetIsEqual( s3, s4 ))

// cardinality
NSLog(@"size of s1: %lu\n", fn SetCount(s1))

// has intersection (not disjoint)
NSLog(@"s1 intersects s2: %d\n", fn SetIntersectsSet( s1, s2 ))

// adding and removing elements from mutable set
CFMutableSetRef s1mut = fn MutableSetWithSet( s1 )
NSLog(@"s1mut after adding \"g\": %@\n", s1mut)
NSLog(@"s1mut after adding \"b\" again: %@\n", s1mut)
MutableSetRemoveObject( s1mut, @"c" )
NSLog(@"s1mut after removing \"c\": %@\n", s1mut)
end fn

fn DoIt

HandleEvents```
Output:
```s1: {(
d,
b,
e,
c,
a
)}
s2: {(
d,
b,
e,
c,
h,
f
)}
s3: {(
b,
c,
d
)}
s4: {(
b,
c,
d
)}

"b" in s1: 1
"f" in s1: 0

s1 union s2: {(
d,
b,
f,
e,
c,
a,
h
)}

s1 intersect s2: {(
d,
b,
e,
c
)}

s1 - s2: {(
a
)}

s3 subset of s1: 1

s3 == s4: 1

size of s1: 5

s1 intersects s2: 1

d,
b,
g,
e,
c,
a
)}

s1mut after adding "b" again: {(
d,
b,
g,
e,
c,
a
)}

s1mut after removing "c": {(
d,
b,
g,
e,
a
)}
```

## Go

A common complaint is that Go has no native set type and so there are a number of third-party libraries offering to fill this perceived gap. Yet Go has good native support for most applications for sets.

### Maps

Go maps meet the task description in that they do not require orderable elements. To demonstrate that, a set of complex numbers is shown here. Complex numbers can be compared for equality but are not ordered.

```package main

import "fmt"

// Define set as a type to hold a set of complex numbers.  A type
// could be defined similarly to hold other types of elements.  A common
// variation is to make a map of interface{} to represent a set of
// mixed types.  Also here the map value is a bool.  By always storing
// true, the code is nicely readable.  A variation to use less memory
// is to make the map value an empty struct.  The relative advantages
// can be debated.
type set map[complex128]bool

func main() {
s0 := make(set)             // create empty set
s1 := set{3: true}          // create set with one element
s2 := set{3: true, 1: true} // create set with two elements

// option: another way to create a set
s3 := newSet(3, 1, 4, 1, 5, 9)

// option: output!
fmt.Println("s0:", s0)
fmt.Println("s1:", s1)
fmt.Println("s2:", s2)
fmt.Println("s3:", s3)

fmt.Printf("%v ∈ s0: %t\n", 3, s0.hasElement(3))
fmt.Printf("%v ∈ s3: %t\n", 3, s3.hasElement(3))
fmt.Printf("%v ∈ s3: %t\n", 2, s3.hasElement(2))

b := set{4: true, 2: true}
fmt.Printf("s3 ∪ %v: %v\n", b, union(s3, b))

fmt.Printf("s3 ∩ %v: %v\n", b, intersection(s3, b))

fmt.Printf("s3 \\ %v: %v\n", b, difference(s3, b))

fmt.Printf("%v ⊆ s3: %t\n", b, subset(b, s3))
fmt.Printf("%v ⊆ s3: %t\n", s2, subset(s2, s3))
fmt.Printf("%v ⊆ s3: %t\n", s0, subset(s0, s3))

s2Same := set{1: true, 3: true}
fmt.Printf("%v = s2: %t\n", s2Same, equal(s2Same, s2))

// option: proper subset
fmt.Printf("%v ⊂ s2: %t\n", s2Same, properSubset(s2Same, s2))
fmt.Printf("%v ⊂ s3: %t\n", s2Same, properSubset(s2Same, s3))

// option: delete.  it's built in.
delete(s3, 3)
fmt.Println("s3, 3 deleted:", s3)
}

func newSet(ms ...complex128) set {
s := make(set)
for _, m := range ms {
s[m] = true
}
return s
}

func (s set) String() string {
if len(s) == 0 {
return "∅"
}
r := "{"
for e := range s {
r = fmt.Sprintf("%s%v, ", r, e)
}
return r[:len(r)-2] + "}"
}

func (s set) hasElement(m complex128) bool {
return s[m]
}

func union(a, b set) set {
s := make(set)
for e := range a {
s[e] = true
}
for e := range b {
s[e] = true
}
return s
}

func intersection(a, b set) set {
s := make(set)
for e := range a {
if b[e] {
s[e] = true
}
}
return s
}

func difference(a, b set) set {
s := make(set)
for e := range a {
if !b[e] {
s[e] = true
}
}
return s
}

func subset(a, b set) bool {
for e := range a {
if !b[e] {
return false
}
}
return true
}

func equal(a, b set) bool {
return len(a) == len(b) && subset(a, b)
}

func properSubset(a, b set) bool {
return len(a) < len(b) && subset(a, b)
}
```
Output:
```s0: ∅
s1: {(3+0i)}
s2: {(3+0i), (1+0i)}
s3: {(3+0i), (1+0i), (4+0i), (5+0i), (9+0i)}
3 ∈ s0: false
3 ∈ s3: true
2 ∈ s3: false
s3 ∪ {(4+0i), (2+0i)}: {(5+0i), (9+0i), (2+0i), (3+0i), (1+0i), (4+0i)}
s3 ∩ {(2+0i), (4+0i)}: {(4+0i)}
s3 \ {(4+0i), (2+0i)}: {(5+0i), (9+0i), (3+0i), (1+0i)}
{(2+0i), (4+0i)} ⊆ s3: false
{(3+0i), (1+0i)} ⊆ s3: true
∅ ⊆ s3: true
{(1+0i), (3+0i)} = s2: true
{(1+0i), (3+0i)} ⊂ s2: false
{(1+0i), (3+0i)} ⊂ s3: true
s3, 3 deleted: {(5+0i), (9+0i), (1+0i), (4+0i)}
```

### Big.Int

If elements of your set are integers or can be indexed by integers, are zero based and relatively "dense", then the big.Int type in the standard library can serve efficiently as a set. The solution here doesn't even bother to define a set type, it just defines functions that use big.Ints directly as sets.

Note that the elements here, being integers, are of course ordered and so might not meet a strict reading of the task requirements.

```package main

import (
"fmt"
"math/big"
)

func main() {
// create an empty set
var s0 big.Int

// create sets with elements
s1 := newSet(3)
s2 := newSet(3, 1)
s3 := newSet(3, 1, 4, 1, 5, 9)

// output
fmt.Println("s0:", format(s0))
fmt.Println("s1:", format(s1))
fmt.Println("s2:", format(s2))
fmt.Println("s3:", format(s3))

// element predicate
fmt.Printf("%v ∈ s0: %t\n", 3, hasElement(s0, 3))
fmt.Printf("%v ∈ s3: %t\n", 3, hasElement(s3, 3))
fmt.Printf("%v ∈ s3: %t\n", 2, hasElement(s3, 2))

// union
b := newSet(4, 2)
fmt.Printf("s3 ∪ %v: %v\n", format(b), format(union(s3, b)))

// intersection
fmt.Printf("s3 ∩ %v: %v\n", format(b), format(intersection(s3, b)))

// difference
fmt.Printf("s3 \\ %v: %v\n", format(b), format(difference(s3, b)))

// subset predicate
fmt.Printf("%v ⊆ s3: %t\n", format(b), subset(b, s3))
fmt.Printf("%v ⊆ s3: %t\n", format(s2), subset(s2, s3))
fmt.Printf("%v ⊆ s3: %t\n", format(s0), subset(s0, s3))

// equality
s2Same := newSet(1, 3)
fmt.Printf("%v = s2: %t\n", format(s2Same), equal(s2Same, s2))

// proper subset
fmt.Printf("%v ⊂ s2: %t\n", format(s2Same), properSubset(s2Same, s2))
fmt.Printf("%v ⊂ s3: %t\n", format(s2Same), properSubset(s2Same, s3))

// delete
remove(&s3, 3)
fmt.Println("s3, 3 removed:", format(s3))
}

func newSet(ms ...int) (set big.Int) {
for _, m := range ms {
set.SetBit(&set, m, 1)
}
return
}

func remove(set *big.Int, m int) {
set.SetBit(set, m, 0)
}

func format(set big.Int) string {
if len(set.Bits()) == 0 {
return "∅"
}
r := "{"
for e, l := 0, set.BitLen(); e < l; e++ {
if set.Bit(e) == 1 {
r = fmt.Sprintf("%s%v, ", r, e)
}
}
return r[:len(r)-2] + "}"
}

func hasElement(set big.Int, m int) bool {
return set.Bit(m) == 1
}

func union(a, b big.Int) (set big.Int) {
set.Or(&a, &b)
return
}

func intersection(a, b big.Int) (set big.Int) {
set.And(&a, &b)
return
}

func difference(a, b big.Int) (set big.Int) {
set.AndNot(&a, &b)
return
}

func subset(a, b big.Int) bool {
ab := a.Bits()
bb := b.Bits()
if len(ab) > len(bb) {
return false
}
for i, aw := range ab {
if aw&^bb[i] != 0 {
return false
}
}
return true
}

func equal(a, b big.Int) bool {
return a.Cmp(&b) == 0
}

func properSubset(a, b big.Int) (p bool) {
ab := a.Bits()
bb := b.Bits()
if len(ab) > len(bb) {
return false
}
for i, aw := range ab {
bw := bb[i]
if aw&^bw != 0 {
return false
}
if aw != bw {
p = true
}
}
return
}
```
Output:
```s0: ∅
s1: {3}
s2: {1, 3}
s3: {1, 3, 4, 5, 9}
3 ∈ s0: false
3 ∈ s3: true
2 ∈ s3: false
s3 ∪ {2, 4}: {1, 2, 3, 4, 5, 9}
s3 ∩ {2, 4}: {4}
s3 \ {2, 4}: {1, 3, 5, 9}
{2, 4} ⊆ s3: false
{1, 3} ⊆ s3: true
∅ ⊆ s3: true
{1, 3} = s2: true
{1, 3} ⊂ s2: false
{1, 3} ⊂ s3: true
s3, 3 removed: {1, 4, 5, 9}
```

### Intsets

Not quite in the stanard library but still in the official "sub repository", intsets are a sparse bit set. Like big.Int they use a single bit to represent a possible element, but the sparse representation efficiently allows for large "holes" in the element sequence. Also the intsets API provides a more set-like terminology so the RC task can be coded more directly.

```package main

import (
"fmt"

"golang.org/x/tools/container/intsets"
)

func main() {
var s0, s1 intsets.Sparse // create some empty sets
s1.Insert(3)              // insert an element
s2 := newSet(3, 1)        // create sets with elements
s3 := newSet(3, 1, 4, 1, 5, 9)

// output
fmt.Println("s0:", &s0)
fmt.Println("s1:", &s1)
fmt.Println("s2:", s2)
fmt.Println("s3:", s3)

// element predicate
fmt.Printf("%v ∈ s0: %t\n", 3, s0.Has(3))
fmt.Printf("%v ∈ s3: %t\n", 3, s3.Has(3))
fmt.Printf("%v ∈ s3: %t\n", 2, s3.Has(2))

// union
b := newSet(4, 2)
var s intsets.Sparse
s.Union(s3, b)
fmt.Printf("s3 ∪ %v: %v\n", b, &s)

// intersection
s.Intersection(s3, b)
fmt.Printf("s3 ∩ %v: %v\n", b, &s)

// difference
s.Difference(s3, b)
fmt.Printf("s3 \\ %v: %v\n", b, &s)

// subset predicate
fmt.Printf("%v ⊆ s3: %t\n", b, b.SubsetOf(s3))
fmt.Printf("%v ⊆ s3: %t\n", s2, s2.SubsetOf(s3))
fmt.Printf("%v ⊆ s3: %t\n", &s0, s0.SubsetOf(s3))

// equality
s2Same := newSet(1, 3)
fmt.Printf("%v = s2: %t\n", s2Same, s2Same.Equals(s2))

// delete
s3.Remove(3)
fmt.Println("s3, 3 removed:", s3)
}

func newSet(ms ...int) *intsets.Sparse {
var set intsets.Sparse
for _, m := range ms {
set.Insert(m)
}
return &set
}
```
Output:
```s0: {}
s1: {3}
s2: {1 3}
s3: {1 3 4 5 9}
3 ∈ s0: false
3 ∈ s3: true
2 ∈ s3: false
s3 ∪ {2 4}: {1 2 3 4 5 9}
s3 ∩ {2 4}: {4}
s3 \ {2 4}: {1 3 5 9}
{2 4} ⊆ s3: false
{1 3} ⊆ s3: true
{} ⊆ s3: true
{1 3} = s2: true
s3, 3 removed: {1 4 5 9}
```

## Groovy

```def s1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] as Set
def m1 = 6
def m2 = 7
def s2 = [0, 2, 4, 6, 8] as Set
assert m1 in s1                                        : 'member'
assert ! (m2 in s2)                                    : 'not a member'
def su = s1 + s2
assert su == [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] as Set : 'union'
def si = s1.intersect(s2)
assert si == [8, 6, 4, 2] as Set                       : 'intersection'
def sd = s1 - s2
assert sd == [1, 3, 5, 7, 9, 10] as Set                : 'difference'
assert s1.containsAll(si)                              : 'subset'
assert ! s1.containsAll(s2)                            : 'not a subset'
assert (si + sd) == s1                                 : 'equality'
assert (s2 + sd) != s1                                 : 'inequality'
assert s1 != su && su.containsAll(s1)                  : 'proper subset'
s1 << 0
assert s1 == su                                        : 'added element 0 to s1'
```

Works with: GHC

GHC offers a functional, persistent set data structure in its `Data.Set` module. It is implemented using a binary search tree. Elements must be of an orderable type (instance of `Ord`).

```Prelude> import Data.Set
Prelude Data.Set> empty :: Set Integer -- Empty set
fromList []
Prelude Data.Set> let s1 = fromList [1,2,3,4,3] -- Convert list into set
Prelude Data.Set> s1
fromList [1,2,3,4]
Prelude Data.Set> let s2 = fromList [3,4,5,6]
Prelude Data.Set> union s1 s2 -- Union
fromList [1,2,3,4,5,6]
Prelude Data.Set> intersection s1 s2 -- Intersection
fromList [3,4]
Prelude Data.Set> s1 \\ s2 -- Difference
fromList [1,2]
Prelude Data.Set> s1 `isSubsetOf` s1 -- Subset
True
Prelude Data.Set> fromList [3,1] `isSubsetOf` s1
True
Prelude Data.Set> s1 `isProperSubsetOf` s1 -- Proper subset
False
Prelude Data.Set> fromList [3,1] `isProperSubsetOf` s1
True
Prelude Data.Set> fromList [3,2,4,1] == s1 -- Equality
True
Prelude Data.Set> s1 == s2
False
Prelude Data.Set> 2 `member` s1 -- Membership
True
Prelude Data.Set> 10 `notMember` s1
True
Prelude Data.Set> size s1 -- Cardinality
4
Prelude Data.Set> insert 99 s1 -- Create a new set by inserting
fromList [1,2,3,4,99]
Prelude Data.Set> delete 3 s1 -- Create a new set by deleting
fromList [1,2,4]
```

Regular lists can also be used as sets. Haskell has some functions to help with using lists as sets. No requirement is made of element type. However, these are not very efficient because they require linear time to find an element.

```Prelude> import Data.List
Prelude Data.List> let s3 = nub [1,2,3,4,3] -- Remove duplicates from list
Prelude Data.List> s3
[1,2,3,4]
Prelude Data.List> let s4 = [3,4,5,6]
Prelude Data.List> union s3 s4 -- Union
[1,2,3,4,5,6]
Prelude Data.List> intersect s3 s4 -- Intersection
[3,4]
Prelude Data.List> s3 \\ s4 -- Difference
[1,2]
Prelude Data.List> 42 : s3 -- Return new list with element inserted at the beginning
[42,1,2,3,4]
Prelude Data.List> delete 3 s3 -- Return new list with first occurrence of element removed
[1,2,4]
```

## Icon and Unicon

The set is a basic datatype (structure) in Icon and Unicon, which supports 'member', 'union', 'intersection' and 'difference' operations. Subset and equality must be implemented separately, or use the routines in the 'sets' library.

Implemented directly:

```procedure display_set (s)
writes ("[")
every writes (!s || " ")
write ("]")
end

# fail unless s1 and s2 contain the same elements
procedure set_equals (s1, s2)
return subset(s1, s2) & subset(s2, s1)
end

# fail if every element in s2 is not contained in s1
procedure subset (s1, s2)
every (a := !s2) do {
if not(member(s1,a)) then fail
}
return s2
end

procedure main ()
a := set(1, 1, 2, 3, 4)
b := set(2, 3, 5)
writes ("a: ")
display_set (a)
writes ("b: ")
display_set (b)
# basic set operations
writes ("Intersection: ")
display_set (a ** b)
writes ("Union: ")
display_set (a ++ b)
writes ("Difference: ")
display_set (a -- b)
# membership
if member(a, 2) then
write ("2 is a member of a")
else
write ("2 is not a member of a")
if member(a, 5) then
write ("5 is a member of a")
else
write ("5 is not a member of a")
# equality
if set_equals(a, set(1,2,3,4,4)) then
write ("a equals set(1,2,3,4,4)")
else
write ("a does not equal set(1,2,3,4,4)")
if set_equals(a, b) then
write ("a equals b")
else
write ("a does not equal b")
# subset
if subset(a, set(1,2)) then
write ("(1,2) is included in a")
else
write ("(1,2) is not included in a")
if subset(a, set(1,2,5)) then
write ("(1,2,5) is included in a")
else
write ("(1,2,5) is not included in a")
end
```
Output:
```a: [2 4 1 3 ]
b: [5 2 3 ]
Intersection: [2 3 ]
Union: [5 2 4 1 3 ]
Difference: [4 1 ]
2 is a member of a
5 is not a member of a
a equals set(1,2,3,4,4)
a does not equal b
(1,2) is included in a
(1,2,5) is not included in a
```

Using library:

```link sets

procedure main ()
a := set(1, 1, 2, 3, 4)
b := set(2, 3, 5)
write ("a: ", simage(a))
write ("b: ", simage(b))
# basic set operations
write ("Intersection: ", simage (a**b))
write ("Union: ", simage        (a++b))
write ("Difference: ", simage   (a--b))
# membership
if member(a, 2) then
write ("2 is a member of a")
else
write ("2 is not a member of a")
if member(a, 5) then
write ("5 is a member of a")
else
write ("5 is not a member of a")
# equality
if seteq(a, set(1,2,3,4,4)) then
write ("a equals set(1,2,3,4,4)")
else
write ("a does not equal set(1,2,3,4,4)")
if seteq(a, b) then
write ("a equals b")
else
write ("a does not equal b")
# check subset
if setlt(set(1,2), a) then
write ("(1,2) is included in a")
else
write ("(1,2) is not included in a")
if setlt(a, set(1,2,5), a) then
write ("(1,2,5) is included in a")
else
write ("(1,2,5) is not included in a")
end
```
Output:
```a: { 2, 4, 1, 3 }
b: { 5, 2, 3 }
Intersection: { 2, 3 }
Union: { 5, 2, 4, 1, 3 }
Difference: { 4, 1 }
2 is a member of a
5 is not a member of a
a equals set(1,2,3,4,4)
a does not equal b
(1,2) is included in a
(1,2,5) is not included in a
```

## J

In J, we use a sequence to represent a set. This actually winds up being a family of set implementations. In this example, we chose to ignore order and specify that duplicate elements are not allowed.

Here are definitions for the required operations:

```union=: ~.@,
intersection=: [ -. -.
difference=: -.
subset=: *./@e.
equality=: -:&(/:~)
```

Examples:

```  2 4 6 8 ~.@, 2 3 5 7
2 4 6 8 3 5 7
2 4 6 8 ([ -. -.) 2 3 5 7
2
2 4 6 8 -. 2 3 5 7
4 6 8
2 4 6 8 *./@e. 2 3 5 7
0
'' *./@e. 2 3 5 7
1
2 4 6 8 3 5 7 -:&(/:~) 8 7 6 5 4 3 2
1
```

Examples again, using names rather than code:

```   2 4 6 8 union 2 3 5 7
2 4 6 8 3 5 7
2 4 6 8 intersection 2 3 5 7
2
2 4 6 8 difference 2 3 5 7
4 6 8
2 4 6 8 subset 2 3 5 7
0
'' subset 2 3 5 7
1
2 4 6 8 3 5 7 equality 8 7 6 5 4 3 2
1
```

Note that J uses 1 for true and 0 for false. Mathematical revisionists object to this, but this is consistent with the original (and revised) formulations of boolean algebra. (And there are deep ties to Bayes' rule.)

Note that these operations can be combined in sentences with other operations. For example we could define

```properSubset=: subset * 1 - equality
```

## Java

Works with: Java version 7+

To use this in Java 5 replace all "<>" with "<Integer>".

```import java.util.Arrays;
import java.util.Collections;
import java.util.Set;
import java.util.TreeSet;

public class Sets {
public static void main(String[] args){
Set<Integer> a = new TreeSet<>();
//TreeSet sorts on natural ordering (or an optional comparator)
//other options: HashSet (hashcode)
//               EnumSet (optimized for enum values)
Set<Integer> b = new TreeSet<>();
Set<Integer> c = new TreeSet<>();
Set<Integer> d = new TreeSet<>();

b.addAll(Arrays.asList(2, 3, 4, 5, 6, 8));
System.out.println("a: " + a);
System.out.println("b: " + b);
System.out.println("c: " + c);
System.out.println("d: " + d);

System.out.println("2 in a: " + a.contains(2));
System.out.println("6 in a: " + a.contains(6));

Set<Integer> ab = new TreeSet<>();
System.out.println("a union b: " + ab);

Set<Integer> a_b = new TreeSet<>();
a_b.removeAll(b);
System.out.println("a - b: " + a_b);

System.out.println("c subset of a: " + a.containsAll(c));
//use a.conatins() for single elements

System.out.println("c = d: " + c.equals(d));
System.out.println("d = c: " + d.equals(c));

Set<Integer> aib = new TreeSet<>();
aib.retainAll(b);
System.out.println("a intersect b: " + aib);

//other noteworthy things related to sets:
Set<Integer> empty = Collections.EMPTY_SET; //immutable empty set
empty.isEmpty(); //test if a set is empty
empty.size();
Collections.disjoint(a, b); //returns true if the sets have no common elems (based on their .equals() methods)
Collections.unmodifiableSet(a); //returns an immutable copy of a
}
}```
Output:
```a: [1, 2, 3, 4, 5]
b: [2, 3, 4, 5, 6, 8]
c: [2, 3, 4]
d: [2, 3, 4]
2 in a: true
6 in a: false
a union b: [1, 2, 3, 4, 5, 6, 8]
a - b: [1]
c subset of a: true
c = d: true
d = c: true
a intersect b: [2, 3, 4, 5]
add 2 to a again: false```

## JavaScript

JavaScript does not support native sets before ECMAScript 6.

```var set = new Set();

set.has(0); //=> true
set.has(3); //=> false
set.has('two'); // true
set.has(Math.sqrt(4)); //=> false
set.has('TWO'.toLowerCase()); //=> true

set.size; //=> 4

set.delete('two');
set.has('two'); //==> false
set.size; //=> 3

//iterating set using ES6 for..of
//Set order is preserved in order items are added.
for (var item of set) {
console.log('item is ' + item);
}
```

## jq

Works with: jq version 1.4

Neither JSON nor jq has a "set" type, but as explained below in the first part of this entry, finite sets of Unicode strings can be directly represented in JSON and thus jq.

The second part of this entry focuses on a jq library of set-theoretic functions that support finite sets of arbitrary JSON entities.

### Finite Sets of Unicode Strings

There is an obvious 1-1 mapping between the collection of finite sets of Unicode strings and the collection of JSON objects with distinct keys the values of which all have the boolean value "true". For example, the set of strings {"a", "b"} corresponds to the JSON object {"a": true, "b": true }.

When restricted to such JSON objects, jq's equality operator ("==") yields set-theoretic semantics, and similarly, jq's + operator yields set-theoretic union.

For example:

```{"a":true, "b":true } == {"b":true, "a":true}.
{"a":true} + {"b":true } == { "a":true, "b":true}```

Thus, it can be seen that jq has built-in support for sets of finite-length Unicode strings.

For simplicity and to avoid confusion, we shall refer to JSON objects all of whose keys are distinct and all values of which have the boolean value "true" as "string sets".

String-set test

Here is a jq filter for determining whether a JSON object is a "string set":

```def is_stringset:
. as \$in | type == "object" and reduce keys[] as \$key (true; . and \$in[\$key] == true);```

String-set membership:

The test for set membership, m ∈ S, where m is a string and S is a set of strings, corresponds exactly to the jq test:

`T | has(m)`

where T is the JSON object corresponding to S. This test is also efficient.

String-set intersection

```# Set-intersection: A ∩ B
def stringset_intersection(A;B):
reduce (A|keys)[] as \$k
({}; if (B|has(\$k)) then . + {(\$k):true} else . end);```

String-set difference

```# stringset_difference: A \ B
def stringset_difference(A;B):
reduce (A|keys)[] as \$k
({}; if (B|has(\$k)) then . else . + {(\$k):true} end);```

Subset

```# A ⊆ B iff string_subset(A;B)
def stringset_subset(A;B):
reduce (A|keys)[] as \$k
(true; . and (B|has(\$k)));```

### Finite Sets of JSON Entities

Finite sets of arbitrary JSON entities can be represented by sets of strings using an invertible serialization of JSON entities, but in the remainder of this entry, we provide a more straightforward and probably more efficient implementation of finite sets using JSON arrays.

Specifically, the empty set is represented by [] and a non-empty set of JSON entities with distinct members m1, m2, ... mN is represented by the JSON array [s1, s2, ... sN] where:

[s1, s2, ... sN] is the result of ([m1, m2, ... mN] | sort)

When confined to sorted arrays, jq's equality operator (==) yields set-theoretic semantics, and therefore, for the remainder of this entry, we shall refer to sorted arrays simply as sets.

To convert an arbitrary jq or JSON array to a set, we can simply use the built-in jq operator "unique". To test whether an arbitrary JSON entity is a set without sorting:

```def is_set:
. as \$in
| type == "array" and
reduce range(0;length-1) as \$i
(true; if . then \$in[\$i] < \$in[\$i+1] else false end);```

The following library of set-theoretic functions is intended for use with jq version 1.4 or later. However, as noted below, if used with a version of jq that does not have bsearch, then it is assumed that a definition of bsearch equivalent to that given in Binary search is available.

Set creation

• [] is the empty set;
• if m1 <= m2 <= ... mN then [m1, m2, ... mN] is the set containing the listed elements;
• The set of elements in an array, a, can be constructed by writing: a | unique
• The set of strings in the string-set SS is: SS|keys

m ∈ S

If m is a JSON entity and S a set, then the jq expression S[m] can be used to test whether m is an element of S, but for large sets, this is inefficient. A generally more efficient test membership of m in S would use bsearch as defined at Binary search or as provided in recent versions of jq:

`def is_member(m):  bsearch(m) > -1;`

Intersection

```# If A and B are sets, then intersection(A;B) emits their intersection:
def intersection(\$A;\$B):
def pop:
.[0] as \$i
| .[1] as \$j
| if \$i == (\$A|length) or \$j == (\$B|length) then empty
elif \$A[\$i] == \$B[\$j] then \$A[\$i], ([\$i+1, \$j+1] | pop)
elif \$A[\$i] <  \$B[\$j] then [\$i+1, \$j] | pop
else [\$i, \$j+1] | pop
end;
[[0,0] | pop];```

Difference

```# If A and B are sets, then A-B is emitted
def difference(A;B):
(A|length) as \$al
| (B|length) as \$bl
| if \$al == 0 then [] elif \$bl == 0 then A
else
reduce range(0; \$al + \$bl) as \$k
( [0, 0, []];
.[0] as \$i | .[1] as \$j
| if \$i < \$al and \$j < \$bl then
if A[\$i] == B[\$j] then [ \$i+1, \$j+1,  .[2] ]
elif  A[\$i] < B[\$j] then [ \$i+1, \$j, .[2] + [A[\$i]] ]
else [ \$i , \$j+1, .[2] ]
end
elif \$i < \$al then [ \$i+1, \$j,  .[2] + [A[\$i]] ]
else .
end
) | .[2]
end ;```

Union

A simple but inefficient implementation would use: (A + B) | unique

To compute the union of two sets efficiently, it is helpful to define a function for merging sorted arrays.

```# merge input array with array x by comparing the heads of the arrays in turn;
# if both arrays are sorted, the result will be sorted:
def merge(x):
length as \$length
| (x|length) as \$xl
| if \$length == 0 then x
elif \$xl == 0 then .
else
. as \$in
| reduce range(0; \$xl + \$length) as \$z
# state [ix, xix, ans]
( [0, 0, []];
if .[0] < \$length and ((.[1] < \$xl and \$in[.[0]] <= x[.[1]]) or .[1] == \$xl)
then [(.[0] + 1), .[1], (.[2] + [\$in[.[0]]]) ]
else [.[0], (.[1] + 1), (.[2] + [x[.[1]]]) ]
end
) | .[2]
end ;

def union(A;B):
A|merge(B)
| reduce .[] as \$m ([]; if length == 0 or .[length-1] != \$m then . + [\$m] else . end);```

A ⊆ B

```def subset(A;B):
# TCO
def _subset:
if .[0]|length == 0 then true
elif .[1]|length == 0 then false
elif .[0][0] == .[1][0] then [.[0][1:], .[1][1:]] | _subset
elif .[0][0] < .[1][0] then false
else [ .[0], .[1][1:] ] | _subset
end;
[A,B] | _subset;```

Test whether two sets intersect

The following implementation assumes a version of jq with bsearch/1.

If A and B are sets (i.e. A == (A|unique) and B == (B|unique)), then [A,B] | intersect emits true if A and B have at least one element in common:

```def intersect:
.[0] as \$A  | .[1] as \$B
| (\$A|length) as \$al
| (\$B|length) as \$bl
| if \$al == 0 or \$bl == 0 then false
else
(\$B | bsearch(\$A[0])) as \$b
| if \$b >= 0 then true
else [\$A[1:], \$B[- (1 + \$b) :]] | intersect
end
end;```

## Julia

```julia> S1 = Set(1:4) ; S2 = Set(3:6) ; println(S1,"\n",S2)
Set{Int64}({4,2,3,1})
Set{Int64}({5,4,6,3})

julia> 5 in S1 , 5 in S2
(false,true)

julia> intersect(S1,S2)
Set{Int64}({4,3})

julia> union(S1,S2)
Set{Int64}({5,4,6,2,3,1})

julia> setdiff(S1,S2)
Set{Int64}({2,1})

julia> issubset(S1,S2)
false

julia> isequal(S1,S2)
false

julia> symdiff(S1,S2)
Set{Int64}({5,6,2,1})```

## Kotlin

```// version 1.0.6

fun main(args: Array<String>) {
val fruits  = setOf("apple", "pear", "orange", "banana")
println("fruits  : \$fruits")
val fruits2 = setOf("melon", "orange", "lemon", "gooseberry")
println("fruits2 : \$fruits2\n")

println("fruits  contains 'banana'     : \${"banana" in fruits}")
println("fruits2 contains 'elderberry' : \${"elderbury" in fruits2}\n")

println("Union        : \${fruits.union(fruits2)}")
println("Intersection : \${fruits.intersect(fruits2)}")
println("Difference   : \${fruits.minus(fruits2)}\n")

println("fruits2 is a subset of fruits : \${fruits.containsAll(fruits2)}\n")
val fruits3 = fruits
println("fruits3 : \$fruits3\n")
var areEqual = fruits.containsAll(fruits2) && fruits3.containsAll(fruits)
println("fruits2 and fruits are equal  : \$areEqual")
areEqual = fruits.containsAll(fruits3) && fruits3.containsAll(fruits)
println("fruits3 and fruits are equal  : \$areEqual\n")

val fruits4 = setOf("apple", "orange")
println("fruits4 : \$fruits4\n")
var isProperSubset = fruits.containsAll(fruits3) && !fruits3.containsAll(fruits)
println("fruits3 is a proper subset of fruits : \$isProperSubset")
isProperSubset = fruits.containsAll(fruits4) && !fruits4.containsAll(fruits)
println("fruits4 is a proper subset of fruits : \$isProperSubset\n")

val fruits5 = mutableSetOf("cherry", "blueberry", "raspberry")
println("fruits5 : \$fruits5\n")
fruits5 += "guava"
println("fruits5 + 'guava'  : \$fruits5")
println("fruits5 - 'cherry' : \${fruits5 - "cherry"}")
}
```
Output:
```fruits  : [apple, pear, orange, banana]
fruits2 : [melon, orange, lemon, gooseberry]

fruits  contains 'banana'     : true
fruits2 contains 'elderberry' : false

Union        : [apple, pear, orange, banana, melon, lemon, gooseberry]
Intersection : [orange]
Difference   : [apple, pear, banana]

fruits2 is a subset of fruits : false

fruits3 : [apple, pear, orange, banana]

fruits2 and fruits are equal  : false
fruits3 and fruits are equal  : true

fruits4 : [apple, orange]

fruits3 is a proper subset of fruits : false
fruits4 is a proper subset of fruits : true

fruits5 : [cherry, blueberry, raspberry]

fruits5 + 'guava'  : [cherry, blueberry, raspberry, guava]
fruits5 - 'cherry' : [blueberry, raspberry, guava]
```

## Lasso

```// Extend set type
define set->issubsetof(p::set) => .intersection(#p)->size == .size
define set->oncompare(p::set) => .intersection(#p)->size - .size

//	Set creation
local(set1) = set('j','k','l','m','n')
local(set2) = set('m','n','o','p','q')

//Test m ∈ S -- "m is an element in set S"
#set1 >> 'm'

// A ∪ B -- union; a set of all elements either in set A or in set B.
#set1->union(#set2)

//A ∩ B -- intersection; a set of all elements in both set A and set B.
#set1->intersection(#set2)

//A ∖ B -- difference; a set of all elements in set A, except those in set B.
#set1->difference(#set2)

//A ⊆ B -- subset; true if every element in set A is also in set B.
#set1->issubsetof(#set2)

//A = B -- equality; true if every element of set A is in set B and vice-versa.
#set1 == #set2
```
Output:
```true
set(j, k, l, m, n, o, p, q)
set(m, n)
set(j, k, l)
false
false```

## LFE

Translation of: Erlang
```> (set set-1 (sets:new))
#(set 0 16 16 8 80 48 ...)
> (set set-2 (sets:add_element 'a set-1))
#(set 1 16 16 8 80 48 ...)
> (set set-3 (sets:from_list '(a b)))
#(set 2 16 16 8 80 48 ...)
> (sets:is_element 'a set-2)
true
> (set union (sets:union set-2 set-3))
#(set 2 16 16 8 80 48 ...)
> (sets:to_list union)
(a b)
> (set intersect (sets:intersection set-2 set-3))
#(set 1 16 16 8 80 48 ...)
> (sets:to_list intersect)
(a)
> (set subtr (sets:subtract set-3 set-2))
#(set 1 16 16 8 80 48 ...)
> (sets:to_list subtr)
(b)
> (sets:is_subset set-2 set-3)
true
> (=:= set-2 set-3)
false
> (set set-4 (sets:add_element 'b set-2))
#(set 2 16 16 8 80 48 ...)
> (=:= set-3 set-4)
true
```

## Liberty BASIC

Sets are not natively available- implemented here in string form so no need to dim/redim or pass number of elements.

```A\$ ="red hot chili peppers rule OK"

print " New set, in space-separated form. Extra spaces and duplicates will be removed. "
input newSet\$
newSet\$  =trim\$(           newSet\$)
newSet\$  =stripBigSpaces\$( newSet\$)
newSet\$  =removeDupes\$(    newSet\$)
print " Set stored as the string '"; newSet\$; "'"

print
print " 'red'  is an element of '"; A\$; "' is "; isAnElementOf\$( "red",  A\$)
print " 'blue' is an element of '"; A\$; "' is "; isAnElementOf\$( "blue",  A\$)
print " 'red'  is an element of '"; B\$; "' is "; isAnElementOf\$( "red",  B\$)
print
print " Union        of '"; A\$; "' & '"; B\$; "' is '"; unionOf\$( A\$, B\$); "'."
print
print " Intersection of '"; A\$; "' & '"; B\$; "' is '"; intersectionOf\$( A\$, B\$); "'."
print
print " Difference   of '"; A\$; "' & '"; B\$; "' is '"; differenceOf\$( A\$, B\$); "'."
print
print " '"; A\$; "' equals '";        A\$; "' is "; equalSets\$( A\$, A\$)
print " '"; A\$; "' equals '";        B\$; "' is "; equalSets\$( A\$, B\$)
print
print  " '"; A\$; "' is a subset of '"; B\$; "' is "; isSubsetOf\$( A\$, B\$)
print  " 'red peppers' is a subset of 'red hot chili peppers rule OK' is "; isSubsetOf\$( "red peppers", "red hot chili peppers rule OK")

end

function removeDupes\$( a\$)
numElements =countElements( a\$)
redim elArray\$( numElements)         '   ie 4 elements are array entries 1 to 4 and 0 is spare =""
for m =0 to numElements
el\$ =word\$( a\$, m, " ")
elArray\$( m) =el\$
next m
sort elArray\$(), 0, numElements
b\$           =""
penultimate\$ ="999"
for jk =0 to numElements    '   do not use "" ( nuls) or elementsalready seen
if elArray\$( jk) ="" then [on]
if elArray\$( jk) <>penultimate\$ then b\$ =b\$ +elArray\$( jk) +" ": penultimate\$ =elArray\$( jk)
[on]
next jk
b\$ =trim\$( b\$)
removeDupes\$ =b\$
end function

function stripBigSpaces\$( a\$)   '   copy byte by byte, but id=f a space had a preceding space, ignore it.
lenA =len( a\$)
penul\$ =""
for i =1 to len( a\$)
c\$ =mid\$( a\$, i, 1)
if c\$ <>" " then
if penul\$ <>" " then
b\$ =b\$ +c\$
else
b\$ =b\$ +" " +c\$
end if
end if
penul\$ =c\$
next i
stripBigSpaces\$ =b\$
end function

function countElements( a\$) '   count elements repr'd by space-separated words in string rep'n.
if isNul\$( a\$) ="True" then countElements =0: exit function
i  =0
do
el\$ =word\$( a\$, i +1, " ")
i =i +1
loop until el\$ =""
countElements =i -1
end function

function isNul\$( a\$)    '   a nul set implies its string rep'n is length zero.
if a\$ ="" then isNul\$ ="True" else isNul\$ ="False"
end function

function isAnElementOf\$( a\$, b\$)    '   check element a\$ exists in set b\$.
isAnElementOf\$ ="False"
i  =0
do
el\$ =word\$( b\$, i +1, " ")
if a\$ =el\$ then isAnElementOf\$ ="True"
i =i +1
loop until el\$ =""
end function

function unionOf\$( a\$, b\$)
i  =1
o\$ =a\$
do
w\$ =word\$( b\$, i, " ")
if w\$ ="" then exit do
if isAnElementOf\$( w\$, a\$) ="False" then o\$ =o\$ +" " +w\$
i =i +1
loop until w\$ =""
unionOf\$ =o\$
end function

function intersectionOf\$( a\$, b\$)
i  =1
o\$ =""
do
el\$ =word\$( a\$, i, " ")
if el\$ ="" then exit do
if ( isAnElementOf\$( el\$, b\$) ="True") and ( o\$ ="")  then o\$ =el\$
if ( isAnElementOf\$( el\$, b\$) ="True") and ( o\$ <>el\$) then o\$ =o\$ +" " +el\$
i =i +1
loop until el\$ =""
intersectionOf\$ =o\$
end function

function equalSets\$( a\$, b\$)
if len( a\$) <>len( b\$) then equalSets\$ ="False": exit function
i =1
do
el\$ =word\$( a\$, i, " ")
if isAnElementOf\$( el\$, b\$) ="False" then equalSets\$ ="False": exit function
i =i +1
loop until w\$ =""
equalSets\$ ="True"
end function

function differenceOf\$( a\$, b\$)
i  =1
o\$ =""
do
el\$ =word\$( a\$, i, " ")
if el\$ ="" then exit do
if ( isAnElementOf\$( el\$, b\$) ="False") and ( o\$ ="")   then o\$ =el\$
if ( isAnElementOf\$( el\$, b\$) ="False") and ( o\$ <>el\$) then o\$ =o\$ +" " +el\$
i =i +1
loop until el\$ =""
differenceOf\$ =o\$
end function

function isSubsetOf\$( a\$, b\$)
isSubsetOf\$ ="True"
i  =1
do
el\$ =word\$( a\$, i, " ")
if el\$ ="" then exit do
if ( isAnElementOf\$( el\$, b\$) ="False") then isSubsetOf\$ ="False": exit function
i =i +1
loop until el\$ =""
end function```

```New set, in space-separated form. Extra spaces and duplicates will be removed.
? now is the the time for all good all men
Set stored as the string 'all for good is men now the time'
'red' is an element of 'red hot chili peppers rule OK' is True
'blue' is an element of 'red hot chili peppers rule OK' is False
'red' is an element of 'lady in red' is True
Union of 'red hot chili peppers rule OK' & 'lady in red' is 'red hot chili peppers rule OK lady in'.
Intersection of 'red hot chili peppers rule OK' & 'lady in red' is 'red'.
Difference of 'red hot chili peppers rule OK' & 'lady in red' is 'hot chili peppers rule OK'.
'red hot chili peppers rule OK' equals 'red hot chili peppers rule OK' is True
'red hot chili peppers rule OK' equals 'lady in red' is False
'red hot chili peppers rule OK' is a subset of 'lady in red' is False
'red peppers' is a subset of 'red hot chili peppers rule OK' is True
```

## Lua

Works with: lua version 5.1
```function emptySet()         return { }  end
function insert(set, item)  set[item] = true  end
function remove(set, item)  set[item] = nil  end
function member(set, item)  return set[item]  end
function size(set)
local result = 0
for _ in pairs(set) do result = result + 1 end
return result
end
function fromTable(tbl) -- ignore the keys of tbl
local result = { }
for _, val in pairs(tbl) do
result[val] = true
end
return result
end
function toArray(set)
local result = { }
for key in pairs(set) do
table.insert(result, key)
end
return result
end
function printSet(set)
print(table.concat(toArray(set), ", "))
end
function union(setA, setB)
local result = { }
for key, _ in pairs(setA) do
result[key] = true
end
for key, _ in pairs(setB) do
result[key] = true
end
return result
end
function intersection(setA, setB)
local result = { }
for key, _ in pairs(setA) do
if setB[key] then
result[key] = true
end
end
return result
end
function difference(setA, setB)
local result = { }
for key, _ in pairs(setA) do
if not setB[key] then
result[key] = true
end
end
return result
end
function subset(setA, setB)
for key, _ in pairs(setA) do
if not setB[key] then
return false
end
end
return true
end
function properSubset(setA, setB)
return subset(setA, setB) and (size(setA) ~= size(setB))
end
function equals(setA, setB)
return subset(setA, setB) and (size(setA) == size(setB))
end
```
Works with: lua version 5.3

(May work with earlier versions but not tested on those.)

This implementation creates, in effect, a set type with operators for comparisons (subset, equality, true subset), and set operations like unions, differences, and intersections. It is a mutable set type, so primitives exist for insertion and removal of elements. Elements can be tested for presence O(1) with the has() method or can be iterated over as an array-flavoured table since this is what the type presents as. (All of its functionality is buried in metatables.)

The code is intended to be placed into a file and accessed as a module. E.g. if placed into the file "set.lua" it would be accessed with `set = require 'set'`.

```local function new(_, ...)
local r = {}
local s = setmetatable({}, {
-- API operations
__index = {

-- single value insertion
insert = function(s, v)
if not r[v] then
table.insert(s, v)
r[v] = #s
end
return s
end,

-- single value removal
remove = function(s, v)
local i = r[v]
if i then
r[v] = nil
local t = table.remove(s)
if t ~= v then
r[t] = i
s[i] = t
end
end
return s
end,

-- multi-value insertion
batch_insert = function(s, ...)
for _,v in pairs {...} do
s:insert(v)
end
return s
end,

-- multi-value removal
batch_remove = function(s, ...)
for _,v in pairs {...} do
s:remove(v)
end
return s
end,

-- membership test
has = function(s, e)
return r[e] ~= nil
end
},

-- set manipulation operators

-- union
r = set()
r:batch_insert(table.unpack(s1))
r:batch_insert(table.unpack(s2))
return r
end,

-- subtraction
__sub = function(s1, s2)
r = set()
r:batch_insert(table.unpack(s1))
r:batch_remove(table.unpack(s2))
return r
end,

-- intersection
__mul = function(s1, s2)
r = set()
for _,v in ipairs(s1) do
if s2:has(v) then
r:insert(v)
end
end
return r
end,

-- equality
__eq = function(s1, s2)
if #s1 ~= #s2 then return false end
for _,v in ipairs(s1) do
if not s2:has(v) then return false end
end
return true
end,

-- proper subset
__lt = function(s1, s2)
if s1 == s2 then return false end
for _,v in ipairs(s1) do
if not s2:has(v) then return false end
end
return true
end,

-- subset
__lte = function(s1, s2)
return (s1 == s2) or (s1 < s2)
end,

-- metatable type tag
__type__ = 'set'
})
s:batch_insert(...)
return s
end

return setmetatable({}, { __call = new })
```

## M2000 Interpreter

A tuple is a referenced type of an array, with variant type for each item (may also be a reference to another tuple). The empty tuple is this (,), and the one item is this (1,), and two items (1,2) For search in a tuple we have O(N).

```Module Sets {
setA=("apple", "cherry", "grape")
setB=("banana","cherry", "date")

Print Len(setA)=3 'true
Print setA#pos("apple")>=0=true   ' exist
Print setA#pos("banana")>=0=False  ' not exist

intersection=lambda  SetB (x\$)-> SetB#pos(x\$)>=0
SetC=SetA#filter(intersection,(,))
Print SetC

Difference= lambda (aSet)->{
=lambda  aSet (x\$)-> aSet#pos(x\$)<0
}
IsetC=SetB#filter(Difference(setA),(,))
Print SetC
SetC=SetA#filter(Difference(setB),(,))
Print SetC

k=each(setB)
SetC=cons(setA)
while k
if setA#pos(SetB#val\$(k^))<0 then Append SetC, (SetB#val\$(k^),)
end while
Print SetC
\\ subset if items exists in same order
Print SetA#pos("cherry","grape")>=0 ' true ' is a subset of SetA
Print SetA#pos(("apple", "cherry"))>=0 ' true ' is a subset of SetA
Print SetA#pos(("apple","grape"))>=0 ' false ' is not a subset of SetA in that order
\\ subset in any position
fold1=lambda (aSet)-> {
=lambda aSet (x\$, cond) ->{
push cond and aSet#pos(x\$)>=0
}
}
SetC=("banana", "date")
print SetC#Fold(fold1(SetA), True)  ' False
print SetC#Fold(fold1(SetB), True)  ' True
SetC=("cherry",)
print SetC#Fold(fold1(SetA), True)  ' True
print SetC#Fold(fold1(SetB), True)  ' True
\\ Mutation
\\ change value at position 0
return SetC, 0:="banana"
print SetC#Fold(fold1(SetA), True)  ' False
print SetC#Fold(fold1(SetB), True)  ' True

\\ equality
SetC=Cons(SetA)  ' we get a copy of one or more tuple
\\ SetC is subset of SetA and SetA is subset of  SetC
Print SetC#Fold(fold1(SetA), True)=SetA#Fold(fold1(SetC), True)  ' True
\\ another way
Print Len(SetC#filter(Difference(setA),(,)))=0   ' true   \\ difference is an empty tuple
append SetC, SetB
Print Len(SetC)=6 ' true
print SetC#pos(0 ->"cherry")=1 ' true
print SetC#pos(2 -> "cherry")=4 ' true
print SetC#pos(5 -> "cherry")=-1 ' true
print SetC#pos(0 -> "banana","cherry")=3 ' true
print SetC#pos( "banana","cherry")=3 ' true
mapU=lambda ->{
push ucase\$(letter\$)
}
fold2=lambda (x\$, k\$)->{
push replace\$(")(", ", ",k\$+"("+quote\$(x\$)+")")
}
Print SetC#map(mapU)#fold\$(fold2, "") ' ("APPLE", "CHERRY", "GRAPE", "BANANA", "CHERRY", "DATE")
Print SetC#map(mapU)  ' APPLE CHERRY GRAPE BANANA CHERRY DATE
Print SetC#fold\$(fold2, "")  ' ("apple", "cherry", "grape", "banana", "cherry", "date")

}
Sets```

## Maple

Sets in Maple are built-in, native data structures, and are immutable. Sets are formed by enclosing a sequence of objects between braces. You can get something essentially equivalent to set comprehensions by using {seq}, which applies the set constructor ("{}") to the sequencing operation "seq".

```> S := { 2, 3, 5, 7, 11, Pi, "foo", { 2/3, 3/4, 4/5 } };
S := {2, 3, 5, 7, 11, "foo", Pi, {2/3, 3/4, 4/5}}

> type( S, set );
true

> Pi in S;
Pi  in  {2, 3, 5, 7, 11, "foo", Pi, {2/3, 3/4, 4/5}}

> if Pi in S then print( yes ) else print( no ) end:
yes

> member( Pi, S );
true

> if 4 in S then print( yes ) else print( no ) end:
no

> evalb( { 2/3, 3/4, 4/5 } in S );
true

> { a, b, c } union { 1, 2, 3 };
{1, 2, 3, a, b, c}

> { a, b, c } intersect { b, c, d };
{b, c}

> { a, b, c } minus { b, c, d };
{a}

> { a, b } subset { a, b, c };
true

> { a, d } subset { a, b, c };
false

> evalb( { 1, 2, 3 } = { 1, 2, 3 } );
true

> evalb( { 1, 2, 3 } = { 1, 2, 4 } );
false```

## Mathematica/Wolfram Language

```set1 = {"a", "b", "c", "d", "e"}; set2 = {"a", "b", "c", "d", "e", "f", "g"};
MemberQ[set1, "a"]
Union[set1 , set2]
Intersection[set1 , set2]
Complement[set2, set1](*Set Difference*)
MemberQ[Subsets[set2], set1](*Subset*)
set1 == set2(*Equality*)
set1 == set1(*Equality*)
```
Output:
```True
{"a", "b", "c", "d", "e", "f", "g"}
{"a", "b", "c", "d", "e"}
{"f", "g"}
True
False
True```

## MATLAB / Octave

There are two types of sets supported, sets with numeric values are stored in a vector, sets with string elements are stored in a cell-array.

```    % Set creation
s = [1, 2, 4];     % numeric values
t = {'a','bb','ccc'}; % cell array of strings
u = unique([1,2,3,3,2,3,2,4,1]);   % set consists only of unique elements
% Test m ∈ S -- "m is an element in set S"
ismember(m, S)
% A ∪ B -- union; a set of all elements either in set A or in set B.
union(A, B)
% A ∩ B -- intersection; a set of all elements in both set A and set B.
intersect(A, B)
% A ∖ B -- difference; a set of all elements in set A, except those in set B.
setdiff(A, B)
% A ⊆ B -- subset; true if every element in set A is also in set B.
all(ismember(A, B))
% A = B -- equality; true if every element of set A is in set B and vice-versa.
isempty(setxor(A, B))
```

## Maxima

```/* illustrating some functions on sets; names are self-explanatory */

a: {1, 2, 3, 4};
{1, 2, 3, 4}

b: {2, 4, 6, 8};
{2, 4, 6, 8}

intersection(a, b);
{2, 4}

union(a, b);
{1, 2, 3, 4, 6, 8}

powerset(a);
{{}, {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 4}, {1, 3}, {1, 3, 4}, {1, 4}, {2}, {2, 3}, {2, 3, 4}, {2, 4}, {3}, {3, 4}, {4}}

set_partitions(a);
{{{1}, {2}, {3}, {4}}, {{1}, {2}, {3, 4}}, {{1}, {2, 3}, {4}}, {{1}, {2, 3, 4}}, {{1}, {2, 4}, {3}}, {{1, 2}, {3}, {4}},
{{1, 2}, {3, 4}}, {{1, 2, 3}, {4}}, {{1, 2, 3, 4}}, {{1, 2, 4}, {3}}, {{1, 3}, {2}, {4}}, {{1, 3}, {2, 4}}, {{1, 3, 4}, {2}},
{{1, 4}, {2}, {3}}, {{1, 4}, {2, 3}}}

setdifference(a, b);
{1, 3}

emptyp(a);
false

elementp(2, a);
true

cardinality(a);
4

cartesian_product(a, b);
{[1, 2], [1, 4], [1, 6], [1, 8], [2, 2], [2, 4], [2, 6], [2, 8], [3, 2], [3, 4], [3, 6], [3, 8], [4, 2], [4, 4], [4, 6], [4, 8]}

subsetp(a, b);
false

symmdifference(a, b);
{1, 3, 6, 8}

partition_set(union(a, b), evenp);
[{1, 3}, {2, 4, 6, 8}]

c: setify(makelist(fib(n), n, 1, 20));
{1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765}

equiv_classes(c, lambda([m, n], mod(m - n, 3) = 0));
{{1, 13, 34, 55, 610, 1597, 2584}, {2, 5, 8, 89, 233, 377, 4181}, {3, 21, 144, 987, 6765}}

disjointp(a, b);
false

{1, 2, 3, 4, 7}

a;
{1, 2, 3, 4}

disjoin(1, a);
{2, 3, 4}

a;
{1, 2, 3, 4}

subset(c, primep);
{2, 3, 5, 13, 89, 233, 1597}

permutations(a);
{[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 3, 4, 2], [1, 4, 2, 3], [1, 4, 3, 2],
[2, 1, 3, 4], [2, 1, 4, 3], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1],
[3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1],
[4, 1, 2, 3], [4, 1, 3, 2], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]}

setequalp(a, b);
false
```

## Nanoquery

This is a full implementation of a set class.

```class set
declare internal_list

def set()
internal_list = list()
end
def set(list)
internal_list = list
end

def append(value)
if not value in internal_list
internal_list.append(value)
end
return this
end
def contains(value)
return value in internal_list
end
def difference(other)
diff = list()
for value in this.internal_list
diff.append(value)
end

for i in range(len(diff) - 1, 0)
if diff[i] in other.internal_list
diff.remove(i)
end
end

return new(set, diff)
end
def operator=(other)
for value in other.internal_list
if not value in this.internal_list
return false
end
end
return true
end
def intersection(other)
intersect = list()
for value in this.internal_list
if other.contains(value)
intersect.append(value)
end
end
return new(set, intersect)
end
def subset(other)
for value in this.internal_list
if not value in other.internal_list
return false
end
end
return true
end
def union(other)
u = list()

for value in this.internal_list
u.append(value)
end

for value in other.internal_list
if not value in u
u.append(value)
end
end

return new(set, u)
end

def toString()
return str(this.internal_list)
end
end```

Testing this implementation:

```import "rosetta-code/set.nq"

a = new(set, {1, 2, 3, 4, 5})
b = new(set, {2, 3, 4, 5, 6, 8})
c = new(set, {2, 3, 4})
d = new(set, {2, 3, 4})

println "a: " + a
println "b: " + b
println "c: " + c
println "d: " + d

println "2 in a: " + a.contains(2)
println "6 in a: " + a.contains(6)

println "a union b: "        + a.union(b)
println "a - b: "            + a.difference(b)
println "c subset of a: "    + c.subset(a)

println "c = d: "            + (c = d)
println "d = c: "            + (d = c)

println "a intersect b: "    + a.intersection(b)
println "add 7 to a: "       + a.append(7)
println "add 2 to a again: " + a.append(2)```
Output:
```a: [1, 2, 3, 4, 5]
b: [2, 3, 4, 5, 6, 8]
c: [2, 3, 4]
d: [2, 3, 4]
2 in a: true
6 in a: false
a union b: [1, 2, 3, 4, 5, 6, 8]
a - b: [1]
c subset of a: true
c = d: true
d = c: true
a intersect b: [2, 3, 4, 5]
add 7 to a: [1, 2, 3, 4, 5, 7]
add 2 to a again: [1, 2, 3, 4, 5, 7]```

## Nemerle

The Nemerle.Collections namespace provides an implementation of a Set.

```using System.Console;
using Nemerle.Collections;

module RCSet
{
HasSubset[T](this super : Set[T], sub : Set[T]) : bool
{
super.ForAll(x => sub.Contains(x))
}

Main() : void
{
def names1 = Set(["Bob", "Billy", "Tom", "Dick", "Harry"]);
def names2 = Set(["Bob", "Mary", "Alice", "Louisa"]);
//def names3 = Set(["Bob", "Bob"]);        // unfortunately, duplicated elements are not well handled by the stock
// implementation, this statement would throw an ArgumentException
def elem = names1.Contains("Bob");         // element test
def names1u2 = names1.Sum(names2);         // union
def names1d2 = names1.Subtract(names2);    // difference
def names1i2 = names1.Intersect(names2);   // intersection
def same = names1.Equals(names2);          // equality
def sub12 = names1.HasSubset(names2);      // subset

WriteLine(\$"\$names1u2\n\$names1d2\n\$names1i2");
WriteLine(\$"\$same\t\$sub12");
}
}
```

## Nim

Nim provides a set constructor which accepts elements of an ordinal type with at most 65536 values. Each element is represented by a bit. For non-ordinal elements or if the possible number of elements is too large (for instance for a set of strings or a set of 32 or 64 bits integers), the standard library provides a set constructor named HashSet.

Here is an example of usage of a set.

```var # creation
s = {0, 3, 5, 10}
t = {3..20, 50..55}

if 5 in s: echo "5 is in!" # element test

var
c = s + t # union
d = s * t # intersection
e = s - t # difference

if s <= t: echo "s ⊆ t" # subset

if s <  t: echo "s ⊂ t" # strong subset

if s == t: echo "s = s" # equality

s.incl(4) # add 4 to set
s.excl(5) # remove 5 from set
```

HashSets are not very different, less efficient but more versatile. Here is the same program with HashSets. The only difference is the way to create a HashSet: this is done by converting an array or sequence to the HashSet. As there is no way to specify a range in an array, we use a conversion of range to sequence with “toSeq” and a union.

```import sequtils, sets

var # creation
s = [0, 3, 5, 10].toHashSet
t = toSeq(3..20).toHashSet + toSeq(50..55).toHashSet

if 5 in s: echo "5 is in!" # element test

var
c = s + t # union
d = s * t # intersection
e = s - t # difference

if s <= t: echo "s ⊆ t" # subset

if s <  t: echo "s ⊂ t" # strong subset

if s == t: echo "s = s" # equality

s.incl(4) # add 4 to set
s.excl(5) # remove 5 from set
```

Note that there exists also an Ordered constructor which remembers the insertion order. And for sets of integers, the module “intsets” provides another constructor better suited for sparse integer sets.

## Objective-C

```#import <Foundation/Foundation.h>

int main (int argc, const char *argv[]) {
@autoreleasepool {

NSSet *s1 = [NSSet setWithObjects:@"a", @"b", @"c", @"d", @"e", nil];
NSSet *s2 = [NSSet setWithObjects:@"b", @"c", @"d", @"e", @"f", @"h", nil];
NSSet *s3 = [NSSet setWithObjects:@"b", @"c", @"d", nil];
NSSet *s4 = [NSSet setWithObjects:@"b", @"c", @"d", nil];
NSLog(@"s1: %@", s1);
NSLog(@"s2: %@", s2);
NSLog(@"s3: %@", s3);
NSLog(@"s4: %@", s4);

// Membership
NSLog(@"b in s1: %d", [s1 containsObject:@"b"]);
NSLog(@"f in s1: %d", [s1 containsObject:@"f"]);

// Union
NSMutableSet *s12 = [NSMutableSet setWithSet:s1];
[s12 unionSet:s2];
NSLog(@"s1 union s2: %@", s12);

// Intersection
NSMutableSet *s1i2 = [NSMutableSet setWithSet:s1];
[s1i2 intersectSet:s2];
NSLog(@"s1 intersect s2: %@", s1i2);

// Difference
NSMutableSet *s1_2 = [NSMutableSet setWithSet:s1];
[s1_2 minusSet:s2];
NSLog(@"s1 - s2: %@", s1_2);

// Subset of
NSLog(@"s3 subset of s1: %d", [s3 isSubsetOfSet:s1]);

// Equality
NSLog(@"s3 = s4: %d", [s3 isEqualToSet:s4]);

// Cardinality
NSLog(@"size of s1: %lu", [s1 count]);

// Has intersection (not disjoint)
NSLog(@"does s1 intersect s2? %d", [s1 intersectsSet:s2]);

// Adding and removing elements from a mutable set
NSMutableSet *mut_s1 = [NSMutableSet setWithSet:s1];
NSLog(@"mut_s1 after adding g: %@", mut_s1);
NSLog(@"mut_s1 after adding b again: %@", mut_s1);
[mut_s1 removeObject:@"c"];
NSLog(@"mut_s1 after removing c: %@", mut_s1);

}
return 0;
}
```
Output:
```s1: {(
d,
b,
e,
c,
a
)}
s2: {(
d,
b,
e,
c,
h,
f
)}
s3: {(
b,
c,
d
)}
s4: {(
b,
c,
d
)}
b in s1: 1
f in s1: 0
s1 union s2: {(
c,
h,
d,
e,
a,
f,
b
)}
s1 intersect s2: {(
d,
b,
e,
c
)}
s1 - s2: {(
a
)}
s3 subset of s1: 1
s3 = s4: 1
size of s1: 5
does s1 intersect s2? 1
d,
b,
g,
e,
c,
a
)}
mut_s1 after adding b again: {(
d,
b,
g,
e,
c,
a
)}
mut_s1 after removing c: {(
d,
b,
g,
e,
a
)}
```

## OCaml

OCaml offers a functional, persistent set data structure in its `Set` module. It is implemented using a binary search tree. `Set` works in the functor model, which means you need to first use the `Set.Make` functor to create a module for your kind of set that you can use. You must give the functor an argument that is a module containing the type and ordering function.

In the interactive toplevel:

Works with: OCaml version 4.02+
```# module IntSet = Set.Make(struct type t = int let compare = compare end);; (* Create a module for our type of set *)
module IntSet :
sig
type elt = int
type t
val empty : t
val is_empty : t -> bool
val mem : elt -> t -> bool
val add : elt -> t -> t
val singleton : elt -> t
val remove : elt -> t -> t
val union : t -> t -> t
val inter : t -> t -> t
val diff : t -> t -> t
val compare : t -> t -> int
val equal : t -> t -> bool
val subset : t -> t -> bool
val iter : (elt -> unit) -> t -> unit
val fold : (elt -> 'a -> 'a) -> t -> 'a -> 'a
val for_all : (elt -> bool) -> t -> bool
val exists : (elt -> bool) -> t -> bool
val filter : (elt -> bool) -> t -> t
val partition : (elt -> bool) -> t -> t * t
val cardinal : t -> int
val elements : t -> elt list
val min_elt : t -> elt
val max_elt : t -> elt
val choose : t -> elt
val split : elt -> t -> t * bool * t
val find : elt -> t -> elt
val of_list : elt list -> t
end
# IntSet.empty;; (* Empty set. A set is an abstract type that will not display in the interpreter *)
- : IntSet.t = <abstr>
# IntSet.elements (IntSet.empty);; (* Get the previous set into a list *)
- : IntSet.elt list = []
# let s1 = IntSet.of_list [1;2;3;4;3];;
val s1 : IntSet.t = <abstr>
# IntSet.elements s1;;
- : IntSet.elt list = [1; 2; 3; 4]
# let s2 = IntSet.of_list [3;4;5;6];;
val s2 : IntSet.t = <abstr>
# IntSet.elements s2;;
- : IntSet.elt list = [3; 4; 5; 6]
# IntSet.elements (IntSet.union s1 s2);; (* Union *)
- : IntSet.elt list = [1; 2; 3; 4; 5; 6]
# IntSet.elements (IntSet.inter s1 s2);; (* Intersection *)
- : IntSet.elt list = [3; 4]
# IntSet.elements (IntSet.diff s1 s2);; (* Difference *)
- : IntSet.elt list = [1; 2]
# IntSet.subset s1 s1;; (* Subset *)
- : bool = true
# IntSet.subset (IntSet.of_list [3;1]) s1;;
- : bool = true
# IntSet.equal (IntSet.of_list [3;2;4;1]) s1;; (* Equality *)
- : bool = true
# IntSet.equal s1 s2;;
- : bool = false
# IntSet.mem 2 s1;; (* Membership *)
- : bool = true
# IntSet.mem 10 s1;;
- : bool = false
# IntSet.cardinal s1;; (* Cardinality *)
- : int = 4
# IntSet.elements (IntSet.add 99 s1);; (* Create a new set by inserting *)
- : IntSet.elt list = [1; 2; 3; 4; 99]
# IntSet.elements (IntSet.remove 3 s1);; (* Create a new set by deleting *)
- : IntSet.elt list = [1; 2; 4]
```

(Note: `of_list` is only available in OCaml 4.02+. In earlier versions, you can implement one yourself like `let of_list lst = List.fold_right IntSet.add lst IntSet.empty;;`)

Regular lists can also be used as sets.

In addition, you can use imperative hash tables from the `Hashtbl` module as a hash table-based set, using the unit type as the "value" for each key.

## Ol

```; test set
(define set1 '(1 2 3 4 5 6 7 8 9))
(define set2 '(3 4 5 11 12 13 14))
(define set3 '(4 5 6 7))
(define set4 '(1 2 3 4 5 6 7 8 9))

; union
(print (union set1 set2))
; ==> (1 2 6 7 8 9 3 4 5 11 12 13 14)

; intersection
(print (intersect set1 set2))
; ==> (3 4 5)

; difference
(print (diff set1 set2))
; ==> (1 2 6 7 8 9)

; subset (no predefined function)
(define (subset? a b)
(all (lambda (i) (has? b i)) a))
(print (subset? set3 set1))
; ==> #true
(print (subset? set3 set2))
; ==> #false

; equality
(print (equal? set1 set2))
; ==> #false
(print (equal? set1 set4))
; ==> #true
```

## ooRexx

```-- Set creation
-- Using the OF method
s1 = .set~of(1, 2, 3, 4, 5, 6)
-- Explicit addition of individual items
s2 = .set~new
s2~put(2)
s2~put(4)
s2~put(6)
s3 = .set~new
s3~putall(.array~of(1, 3, 5))
-- Test m ? S -- "m is an element in set S"
say s1~hasindex(1) s3~hasindex(2)  -- "1 0", which is "true" and "false"
--    A ? B -- union; a set of all elements either in set A or in set B.
s4 = s2~union(s3)   -- {1, 2, 3, 4, 5, 6}
Call show 's4',s4
--    A ? B -- intersection; a set of all elements in both set A and set B.
s5 = s1~intersection(s2)   -- {2, 4, 6}
Call show 's5',s5
--    A ? B -- difference; a set of all elements in set A, except those in set B.
s6 = s1~difference(s2)   -- {1, 3, 5}
Call show 's6',s6
--    A ? B -- subset; true if every element in set A is also in set B.
say s1~subset(s2) s2~subset(s1) --  "0 1"
--    A = B -- equality; true if every element of set A is in set B and vice-versa.
-- No direct equivalence method, but the XOR method can be used to determine this
say s1~xor(s4)~isempty   -- true
Exit
show: Procedure
Use Arg set_name,set
Say set_name':' set~makearray~makestring((LINE),',')
return```

The set operators don't come out too well :-(

Output:
```1 0
s4: 1,2,3,4,5,6
s5: 2,4,6
s6: 1,3,5
0 1
1```

## PARI/GP

Aside from ⊆, all operations are already a part of GP.

```setsubset(s,t)={
for(i=1,#s,
if(!setsearch(t,s[i]), return(0))
);
1
};
s=Set([1,2,2])
t=Set([4,2,4])
setsearch(s,1)
setunion(s,t)
setintersect(s,t)
setminus(s,t)
setsubset(s,t)
s==t```
Output:
```%1 = [1, 2]
%2 = [2, 4]
%3 = 1
%4 = [1, 2, 4]
%5 = [2]
%6 = [1]
%7 = 0
%8 = 0```

## Pascal

Works with: Free Pascal

Freepascal/object pascal handles sets very well. --Guionardo 22:03, 7 January 2012 (UTC)

```program Rosetta_Set;

{\$mode objfpc}{\$H+}

Classes;

{\$R *.res}
type
CharSet = set of char;

var
A, B, C, S: CharSet;
M: char;

function SetToString(const ASet: CharSet): string;
var
J: char;
begin
Result := '';
// Test all chars
for J in char do
// If the char is in set, add to result
if J in ASet then
Result := Result + J + ', ';
// Clear the result
if Result > '' then
Delete(Result, Length(Result) - 1, 2);
end;

procedure PrintSet(const ASet: CharSet; const ASetName: string;
const ATitle: string = '');
begin
if ATitle > '' then
WriteLn(ATitle);
WriteLn(ASetName, ' = [', SetToString(ASet), ']', #10);
end;

procedure ShowEqual(const ASetA, ASetB: CharSet; const ASetNameA, ASetNameB: string);
begin
WriteLn(ASetNameA, ' = [', SetToString(ASetA), ']');
WriteLn(ASetNameB, ' = [', SetToString(ASetB), ']');
if ASetA = ASetB then
WriteLn(ASetNameA, ' = ', ASetNameB)
else
WriteLn(ASetNameA, ' <> ', ASetNameB);
end;

begin
// Set Creation
A := ['A', 'B', 'C', 'D', 'E', 'F'];
B := ['E', 'F', 'G', 'H', 'I', 'J'];
PrintSet(A, 'A', 'Set Creation');
PrintSet(B, 'B');

// Test m ∈ S -- "m is an element in set S"
M := 'A';
if M in A then
WriteLn('"A" is in set A');

// A ∪ B -- union; a set of all elements either in set A or in set B.
S := A + B;
PrintSet(S, 'S', 'S = A U B -- union; a set of all elements either in set A or in set B.');

// A ∩ B -- intersection; a set of all elements in both set A and set B.
S := A * B;
PrintSet(S, 'S',
'S = A ∩ B -- intersection; a set of all elements in both set A and set B.');

// A \ B -- difference; a set of all elements in set A, except those in set B.
S := A - B;
PrintSet(S, 'S',
'S = A \ B -- difference; a set of all elements in set A, except those in set B.');

// A ⊆ B -- subset; true if every element in set A is also in set B.
Writeln('A ⊆ B -- subset; true if every element in set A is also in set B.');
if A <= B then
WriteLn('A in B')
else
Writeln('A is not in B');
Writeln;
//A = B -- equality; true if every element of set A is in set B and vice-versa.
Writeln('A = B -- equality; true if every element of set A is in set B and vice-versa.');

ShowEqual(A, B, 'A', 'B');
S := A * B;
C := ['E', 'F'];
ShowEqual(S, C, 'S', 'C');

end.
```

## Perl

For real code, try Set::Object from CPAN. Here we provide a primitive implementation using hashes.

```use strict;

package Set; # likely will conflict with stuff on CPAN
'""'	=> \&str,
'bool'	=> \&count,
'-='	=> \&del,
'-'	=> \&diff,
'=='	=> \&eq,
'&'	=> \&intersection,
'|'	=> \&union,
'^'	=> \&xdiff;

sub str {
my \$set = shift;
# This has drawbacks: stringification is used as set key
# if the set is added to another set as an element, which
# may cause inconsistencies if the element set is modified
# later.  In general, a hash key loses its object identity
# anyway, so it's not unique to us.
"Set{ ".  join(", " => sort map("\$_", values %\$set)) . " }"
}

sub new {
my \$pkg = shift;
my \$h = bless {};
\$h
}

my (\$set, \$elem) = @_;
\$set->{\$elem} = \$elem;
\$set
}

sub del {
my (\$set, \$elem) = @_;
delete \$set->{\$elem};
\$set
}

sub has { # set has element
my (\$set, \$elem) = @_;
exists \$set->{\$elem}
}

sub union {
my (\$this, \$that) = @_;
bless { %\$this, %\$that }
}

sub intersection {
my (\$this, \$that) = @_;
my \$s = new Set;
for (keys %\$this) {
\$s->{\$_} = \$this->{\$_} if exists \$that->{\$_}
}
\$s
}

sub diff {
my (\$this, \$that) = @_;
my \$s = Set->new;
for (keys %\$this) {
\$s += \$this->{\$_} unless exists \$that->{\$_}
}
\$s
}

sub xdiff { # xor, symmetric diff
my (\$this, \$that) = @_;
my \$s = new Set;
bless { %{ (\$this - \$that) | (\$that - \$this) } }
}

sub count { scalar(keys %{+shift}) }

sub eq {
my (\$this, \$that) = @_;
!(\$this - \$that) && !(\$that - \$this);
}

sub contains { # this is a superset of that
my (\$this, \$that) = @_;
for (keys %\$that) {
return 0 unless \$this->has(\$_)
}
return 1
}

package main;
my (\$x, \$y, \$z, \$w);

\$x = Set->new(1, 2, 3);
\$x += \$_ for (5 .. 7);
\$y = Set->new(1, 2, 4, \$x); # not the brightest idea

print "set x is: \$x\nset y is: \$y\n";
for (1 .. 4, \$x) {
print "\$_ is", \$y->has(\$_) ? "" : " not", " in y\n";
}

print "union: ", \$x | \$y, "\n";
print "intersect: ", \$x & \$y, "\n";
print "z = x - y = ", \$z = \$x - \$y, "\n";
print "y is ", \$x->contains(\$y) ? "" : "not ", "a subset of x\n";
print "z is ", \$x->contains(\$z) ? "" : "not ", "a subset of x\n";
print "z = (x | y) - (x & y) = ", \$z = (\$x | \$y) - (\$x & \$y), "\n";
print "w = x ^ y = ", \$w = (\$x ^ \$y), "\n";
print "w is ", (\$w == \$z) ? "" : "not ", "equal to z\n";
print "w is ", (\$w == \$x) ? "" : "not ", "equal to x\n";
```

## Phix

First, a simple implementaion using native sequences:

```with javascript_semantics
function is_element(object x, sequence set)
return find(x,set)!=0
end function

function set_union(sequence set1, set2)
for i=1 to length(set2) do
if not is_element(set2[i],set1) then
set1 = append(set1,set2[i])
end if
end for
return set1
end function

function set_intersection(sequence set1, set2)
sequence res = {}
for i=1 to length(set1) do
if is_element(set1[i],set2) then
res = append(res,set1[i])
end if
end for
return res
end function

function set_difference(sequence set1, set2)
sequence res = {}
for i=1 to length(set1) do
if not is_element(set1[i],set2) then
res = append(res,set1[i])
end if
end for
return res
end function

function set_subset(sequence set1, set2)
for i=1 to length(set1) do
if not is_element(set1[i],set2) then
return false
end if
end for
return true
end function

function set_equality(sequence set1, set2)
if length(set1)!=length(set2) then
return false
end if
return set_subset(set1,set2)
end function

--test code:
?is_element(3,{1,2,3})              -- 1
?is_element(4,{1,2,3})              -- 0
?set_union({1,2,3},{3,4,5})         -- {1,2,3,4,5}
?set_intersection({1,2,3},{3,4,5})  -- {3}
?set_difference({1,2,3},{3,4,5})    -- {1,2}
?set_subset({1,2,3},{3,4,5})        -- 0
?set_subset({1,2},{1,2,3})          -- 1
?set_equality({1,2,3},{3,4,5})      -- 0
?set_equality({1,2,3},{3,1,2})      -- 1
```
Output:

Note that you get true/false instead of 1/0 under pwa/p2js - use printf(1,"%t\n",...) in place of ? to get an exact match.

```1
0
{1,2,3,4,5}
{3}
{1,2}
0
1
0
1
```

Alternative using dictionaries, which needs several additional visitor routines (at a pinch they could be merged), but performance is better on larger sets:

```with javascript_semantics
function create_set(sequence s={})
integer res = new_dict()
for i=1 to length(s) do
setd(s[i],0,res)
end for
return res
end function

function element(object x, integer set)
return getd_index(x,set)!=0
end function

function u_visitor(object key, object data, object user_data)
integer {union_set,set2} = user_data
if set2=0
or not element(key,union_set) then
setd(key,data,union_set)
end if
return 1
end function

function set_union(integer set1, integer set2)
integer union_set = new_dict()
traverse_dict(u_visitor,{union_set,0},set1)
traverse_dict(u_visitor,{union_set,set2},set2)
return union_set
end function

function i_visitor(object key, object data, object user_data)
integer {inter_sect,set2} = user_data
if element(key,set2) then
setd(key,data,inter_sect)
end if
return 1
end function

function set_intersection(integer set1, integer set2)
integer inter_sect = new_dict()
traverse_dict(i_visitor,{inter_sect,set2},set1)
return inter_sect
end function

function d_visitor(object key, object data, object user_data)
integer {diff_set,set2} = user_data
if not element(key,set2) then
setd(key,data,diff_set)
end if
return 1
end function

function set_difference(integer set1, integer set2)
integer diff_set = new_dict()
traverse_dict(d_visitor,{diff_set,set2},set1)
return diff_set
end function

bool subset_res
function s_visitor(object key, object data, object user_data)
integer set2 = user_data
if not element(key,set2) then
subset_res = false
return 0 -- cease traversal
end if
return 1
end function

function subset(integer set1, integer set2)
subset_res = true
traverse_dict(s_visitor,set2,set1)
return subset_res
end function

function equality(integer set1, integer set2)
if dict_size(set1)!=dict_size(set2) then
return false
end if
return subset(set1,set2)
end function

include builtins/map.e -- for keys()

-- matching test code:
integer set1 = create_set({3,1,2}),
set2 = create_set({5,3,4}),
set3 = create_set({2,1})

?element(3,set1)        -- 1
?element(4,set1)        -- 0
?keys(set_union(set1,set2)) -- {1,2,3,4,5}
?keys(set_intersection(set1,set2)) -- {3}
?keys(set_difference(set1,set2)) -- {1,2}
?subset(set1,set2)      -- 0
?subset(set3,set1)      -- 1
?equality(set1,set2)    -- 0
setd(3,0,set3)
?equality(set1,set3)    -- 1
```

Same output as above. I have written a builtins/sets.e, and tried another approach with builtins/sets2.e, but neither really hit the spot.

## Phixmonti

```include ..\Utilitys.pmt

def isElement find enddef

def setUnion dup >ps remove ps> chain enddef

def setIntersection over >ps remove ps> swap remove enddef

def setDifference remove enddef

def setSubset swap remove len not nip enddef

def setEquality sort swap sort == enddef

( 1 2 3 ) 1 isElement ?
4 isElement ?
( 3 4 5 ) setUnion ?
( 1 2 3 ) ( 3 4 5 ) setIntersection ?
( 1 2 3 ) ( 3 4 5 ) setDifference ?
( 1 2 3 ) ( 3 4 5 ) setSubset ?
( 1 2 3 ) ( 1 2 ) setSubset ?
( 1 2 3 ) ( 3 4 5 ) setEquality ?
( 1 2 3 ) ( 3 1 2 ) setEquality ?```
Output:
```1
0
[1, 2, 3, 4, 5]
[3]
[1, 2]
0
1
0
1

=== Press any key to exit ===```

## PicoLisp

We may use plain lists, or 'idx' structures for sets. A set may contain any type of data.

### Using lists

```(setq
Set1 (1 2 3 7 abc "def" (u v w))
Set2 (2 3 5 hello (x y z))
Set3 (3 hello (x y z)) )

# Element tests (any non-NIL value means "yes")
: (member "def" Set1)
-> ("def" (u v w))

: (member "def" Set2)
-> NIL

: (member '(x y z) Set2)
-> ((x y z))

# Union
: (uniq (append Set1 Set2))
-> (1 2 3 7 abc "def" (u v w) 5 hello (x y z))

# Intersection
: (sect Set1 Set2)
-> (2 3)

# Difference
: (diff Set1 Set2)
-> (1 7 abc "def" (u v w))

# Test for subset
: (not (diff Set1 Set2))
-> NIL  # Set1 is not a subset of Set2

: (not (diff Set3 Set2))
-> T  # Set3 is a subset of Set2

# Test for equality
: (= (sort (copy Set1)) (sort (copy Set2)))
-> NIL

: (= (sort (copy Set2)) (sort (copy Set2)))
-> T```

### Using 'idx' structures

```# Create three test-sets
(balance 'Set1 (1 2 3 7 abc "def" (u v w)))
(balance 'Set2 (2 3 5 hello (x y z)))
(balance 'Set3 (3 hello (x y z)))

# Get contents
: (idx 'Set1)
-> (1 2 3 7 abc "def" (u v w))

: (idx 'Set2)
-> (2 3 5 hello (x y z))

# Element tests (any non-NIL value means "yes")
: (idx 'Set1 "def")
-> ("def" (abc) (u v w))

: (idx 'Set2 "def")
-> NIL

: (idx 'Set2 '(x y z))
-> ((x y z))

# Union
: (use S
(balance 'S (idx 'Set1))
(balance 'S (idx 'Set2) T)
(idx 'S) )
-> (1 2 3 5 7 abc "def" hello (u v w) (x y z))

# Intersection
: (sect (idx 'Set1) (idx 'Set2))
-> (2 3)

# Difference
: (diff (idx 'Set1) (idx 'Set2))
-> (1 7 abc "def" (u v w))

# Test for subset
: (not (diff (idx 'Set1) (idx 'Set2)))
-> NIL  # Set1 is not a subset of Set2

: (not (diff (idx 'Set3) (idx 'Set2)))
-> T  # Set3 is a subset of Set2

# Test for equality
: (= (idx 'Set1) (idx 'Set2))
-> NIL

: (= (idx 'Set2) (idx 'Set2))
-> T```

## PowerShell

.NET offers the HashSet type which seems to act in most ways like a set.

When used in PowerShell, the syntax is clumsy. In addition, the "reference" set (`\$set1`) is modified in place to become the result. (All examples assume the variable `\$set1` contains the value `@(1,2,3,4)`)

```[System.Collections.Generic.HashSet[object]]\$set1 = 1..4
[System.Collections.Generic.HashSet[object]]\$set2 = 3..6

#            Operation           +     Definition      +          Result
#--------------------------------+---------------------+-------------------------
\$set1.UnionWith(\$set2)           # Union                 \$set1 = 1, 2, 3, 4, 5, 6
\$set1.IntersectWith(\$set2)       # Intersection          \$set1 = 3, 4
\$set1.ExceptWith(\$set2)          # Difference            \$set1 = 1, 2
\$set1.SymmetricExceptWith(\$set2) # Symmetric difference  \$set1 = 1, 2, 6, 5
\$set1.IsSupersetOf(\$set2)        # Test superset         False
\$set1.IsSubsetOf(\$set2)          # Test subset           False
\$set1.Equals(\$set2)              # Test equality         False
\$set1.IsProperSupersetOf(\$set2)  # Test proper superset  False
\$set1.IsProperSubsetOf(\$set2)    # Test proper subset    False

5 -in \$set1                      # Test membership       False
7 -notin \$set1                   # Test non-membership   True
```

## Prolog

Works with SWI-Prolog, library(lists).

```:- use_module(library(lists)).

set :-
A = [2, 4, 1, 3],
B = [5, 2, 3, 2],
(   is_set(A) -> format('~w is a set~n', [A])
;   format('~w is not a set~n', [A])),
(   is_set(B) -> format('~w is a set~n', [B])
;   format('~w is not a set~n', [B])),

% create a set from a list

list_to_set(B, BS),
(   is_set(BS) -> format('~nCreate a set from a list~n~w is a set~n', [BS])
;   format('~w is not a set~n', [BS])),

intersection(A, BS, I),
format('~n~w intersection ~w => ~w~n', [A, BS, I]),
union(A, BS, U),
format('~w union ~w => ~w~n', [A, BS, U]),
difference(A, BS, D),
format('~w difference ~w => ~w~n', [A, BS, D]),

X = [1,2],
(   subset(X, A) -> format('~n~w is a subset of ~w~n', [X, A])
;   format('~w is not a subset of ~w~n', [X, A])),
Y = [1,5],
(   subset(Y, A) -> format('~w is a subset of ~w~n', [Y, A])
;   format('~w is not a subset of ~w~n', [Y, A])),
Z = [1, 2, 3, 4],
(  equal(Z, A) -> format('~n~w is equal to ~w~n', [Z, A])
;   format('~w is not equal to ~w~n', [Z, A])),
T = [1, 2, 3],
(  equal(T, A) -> format('~w is equal to ~w~n', [T, A])
;   format('~w is not equal to ~w~n', [T, A])).

% compute difference of sets
difference(A, B, D) :-
exclude(member_(B), A, D).

member_(L, X) :-
member(X, L).

equal([], []).
equal([H1 | T1], B) :-
select(H1, B, B1),
equal(T1, B1).
```
Output:
``` ?- set.
[2,4,1,3] is a set
[5,2,3,2] is not a set

Create a set from a list
[5,2,3] is a set

[2,4,1,3] intersection [5,2,3] => [2,3]
[2,4,1,3] union [5,2,3] => [4,1,5,2,3]
[2,4,1,3] difference [5,2,3] => [4,1]

[1,2] is a subset of [2,4,1,3]
[1,5] is not a subset of [2,4,1,3]

[1,2,3,4] is equal to [2,4,1,3]
[1,2,3] is not equal to [2,4,1,3]
true.
```

Works with: SWI-Prolog version library(ordset)

SWI-Prolog provides a standard library(ordsets). It is loaded by default. I demonstrate almost all of these predicates in the interactive top-level (`?-` is the prompt). Variables prefixed with `\$` refer back to the value of the last instantiation. It treats sets as ordered lists of unique elements:

```%%  Set creation

?- list_to_ord_set([1,2,3,4], A), list_to_ord_set([2,4,6,8], B).
A = [1, 2, 3, 4],
B = [2, 4, 6, 8].

%% Test m ∈ S -- "m is an element in set S"

?- ord_memberchk(2, \$A).
true.

%% A ∪ B -- union; a set of all elements either in set A or in set B.

?- ord_union(\$A, \$B, Union).
Union = [1, 2, 3, 4, 6, 8].

%% A ∩ B -- intersection; a set of all elements in both set A and set B.

?- ord_intersection(\$A, \$B, Intersection).
Intersection = [2, 4].

%% A ∖ B -- difference; a set of all elements in set A, except those in set B.

?- ord_subtract(\$A, \$B, Diff).
Diff = [1, 3].

%% A ⊆ B -- subset; true if every element in set A is also in set B.

?- ord_subset(\$A, \$B).
false.

?- ord_subset([2,4], \$B).
true.

%% A = B -- equality; true if every element of set A is in set B and vice-versa.

?- \$A == \$B.
false.

?- \$A == [1,2,3,4].
true.

%% Definition of a proper subset:

ord_propsubset(A, B) :-
ord_subset(A, B),
\+(A == B).

NewA = [1, 2, 3, 4, 19].

NewerA = [1, 2, 4, 19].
```

## PureBasic

This solution uses PureBasic's maps (hash tables).

```Procedure.s booleanText(b) ;returns 'True' or 'False' for a boolean input
If b: ProcedureReturn "True": EndIf
ProcedureReturn "False"
EndProcedure

Procedure.s listSetElements(Map a(), delimeter.s = " ") ;format elements for display
Protected output\$

ForEach a()
output\$ + MapKey(a()) + delimeter
Next

ProcedureReturn "(" + RTrim(output\$, delimeter) + ")"
EndProcedure

Procedure.s listSortedSetElements(Map a(), delimeter.s = " ") ;format elements for display as sorted for easy comparison
Protected output\$
NewList b.s()

ForEach a()
Next
SortList(b(), #PB_Sort_Ascending | #PB_Sort_NoCase)
ForEach b()
output\$ + b() + delimeter
Next

ProcedureReturn "(" + RTrim(output\$, delimeter) + ")"
EndProcedure

Procedure cardinalityOf(Map a())
ProcedureReturn MapSize(a())
EndProcedure

Procedure createSet(elements.s, Map o(), delimeter.s = " ", clearSet = 1)
Protected i, elementCount

If clearSet: ClearMap(o()): EndIf
elementCount = CountString(elements, delimeter) + 1 ;add one for the last element which won't have a delimeter
For i = 1 To elementCount
Next

ProcedureReturn MapSize(o())
EndProcedure

Procedure adjoinTo(elements.s, Map o(), delimeter.s = " ")
ProcedureReturn createSet(elements, o(), delimeter, 0)
EndProcedure

Procedure disjoinFrom(elements.s, Map o(), delimeter.s = " ")
Protected i, elementCount

elementCount = CountString(elements, delimeter) + 1 ;add one for the last element which won't have a delimeter
For i = 1 To elementCount
DeleteMapElement(o(), StringField(elements, i, delimeter))
Next

ProcedureReturn MapSize(o())
EndProcedure

Procedure isElementOf(element.s, Map a())
ProcedureReturn FindMapElement(a(), element)
EndProcedure

Procedure unionOf(Map a(), Map b(), Map o())
CopyMap(a(), o())
ForEach b()
Next

ProcedureReturn MapSize(o())
EndProcedure

Procedure intersectionOf(Map a(), Map b(), Map o())
ClearMap(o())
ForEach a()
If FindMapElement(b(), MapKey(a()))
EndIf
Next

ProcedureReturn MapSize(o())
EndProcedure

Procedure differenceOf(Map a(), Map b(), Map o())
CopyMap(a(), o())
ForEach b()
If FindMapElement(o(), MapKey(b()))
DeleteMapElement(o())
Else
EndIf
Next

ProcedureReturn MapSize(o())
EndProcedure

Procedure isSubsetOf(Map a(), Map b()) ;boolean
ForEach a()
If Not FindMapElement(b(), MapKey(a()))
ProcedureReturn 0
EndIf
Next
ProcedureReturn 1
EndProcedure

Procedure isProperSubsetOf(Map a(), Map b()) ;boolean
If MapSize(a()) = MapSize(b())
ProcedureReturn 0
EndIf
ProcedureReturn isSubsetOf(a(), b())
EndProcedure

Procedure isEqualTo(Map a(), Map b())
If MapSize(a()) = MapSize(b())
ProcedureReturn isSubsetOf(a(), b())
EndIf
ProcedureReturn 0
EndProcedure

Procedure isEmpty(Map a()) ;boolean
If MapSize(a())
ProcedureReturn 0
EndIf
ProcedureReturn 1
EndProcedure

If OpenConsole()
NewMap a()
NewMap b()
NewMap o() ;for output sets
NewMap c()

createSet("red blue green orange yellow", a())
PrintN("Set A = " + listSortedSetElements(a()) + " of cardinality " + Str(cardinalityOf(a())) + ".")
PrintN("Set B = " + listSortedSetElements(b()) + " of cardinality " + Str(cardinalityOf(b())) + ".")
PrintN("'red' is an element of A is " + booleanText(isElementOf("red", a())) + ".")
PrintN("'red' is an element of B is " + booleanText(isElementOf("red", b())) + ".")
PrintN("'blue' is an element of B is " + booleanText(isElementOf("blue", b())) + ".")

unionOf(a(), b(), o())
PrintN(#crlf\$ + "Union of A & B is " + listSortedSetElements(o()) + ".")
intersectionOf(a(), b(), o())
PrintN("Intersection of  A & B is " + listSortedSetElements(o()) + ".")
differenceOf(a(), b(), o())
PrintN("Difference of  A & B is " + listSortedSetElements(o()) + ".")

PrintN(listSortedSetElements(a()) + " equals " + listSortedSetElements(a()) + " is " + booleanText(isEqualTo(a(), a())) + ".")
PrintN(listSortedSetElements(a()) + " equals " + listSortedSetElements(b()) + " is " + booleanText(isEqualTo(a(), b())) + ".")

createSet("red green", c())
PrintN(#crlf\$ + listSortedSetElements(c()) + " is a subset of " + listSortedSetElements(a()) + " is "+ booleanText(isSubsetOf(c(), a())) + ".")
PrintN(listSortedSetElements(c()) + " is a proper subset of " + listSortedSetElements(b()) + " is "+ booleanText(isProperSubsetOf(c(), b())) + ".")
PrintN(listSortedSetElements(c()) + " is a proper subset of " + listSortedSetElements(a()) + " is "+ booleanText(isProperSubsetOf(c(), a())) + ".")
PrintN(listSortedSetElements(b()) + " is a proper subset of " + listSortedSetElements(b()) + " is "+ booleanText(isProperSubsetOf(b(), b())) + ".")

PrintN(#crlf\$ + "Set C = " + listSortedSetElements(c()) + " of cardinality " + Str(cardinalityOf(c())) + ".")
PrintN("Add 'dog cat mouse' to C to get " + listSortedSetElements(c()) + " of cardinality " + Str(cardinalityOf(c())) + ".")
disjoinFrom("red green dog", c())
PrintN("Take away 'red green dog' from C to get " + listSortedSetElements(c()) + " of cardinality " + Str(cardinalityOf(c())) + ".")

Print(#crlf\$ + #crlf\$ + "Press ENTER to exit"): Input()
CloseConsole()
EndIf```
Output:
```Set A = (blue green orange red yellow) of cardinality 5.
Set B = (green lady red) of cardinality 3.
'red' is an element of A is True.
'red' is an element of B is True.
'blue' is an element of B is False.

Union of A & B is (blue green lady orange red yellow).
Intersection of  A & B is (green red).
Difference of  A & B is (blue lady orange yellow).
(blue green orange red yellow) equals (blue green orange red yellow) is True.
(blue green orange red yellow) equals (green lady red) is False.

(green red) is a subset of (blue green orange red yellow) is True.
(green red) is a proper subset of (green lady red) is True.
(green red) is a proper subset of (blue green orange red yellow) is True.
(green lady red) is a proper subset of (green lady red) is False.

Set C = (green red) of cardinality 2.
Add 'dog cat mouse' to C to get (cat dog green mouse red) of cardinality 5.
Take away 'red green dog' from C to get (cat mouse) of cardinality 2.```

## Python

In Python, `set` is a standard type since Python 2.4. There is also `frozenset` which is an immutable version of `set`. (In Python 2.3, they were provided as `Set` and `ImmutableSet` types in the module `sets`.)

Language syntax for set literals is supported starting in Python 3.0 and 2.7. (For versions prior to 2.7, use `set([1, 2, 3, 4])` instead of `{1, 2, 3, 4}`. Even in Python 2.7+ and 3.0+, it is necessary to write `set()` to express the empty set.)

Works with: Python version 2.7+ and 3.0+
```>>> s1, s2 = {1, 2, 3, 4}, {3, 4, 5, 6}
>>> s1 | s2 # Union
{1, 2, 3, 4, 5, 6}
>>> s1 & s2 # Intersection
{3, 4}
>>> s1 - s2 # Difference
{1, 2}
>>> s1 < s1 # True subset
False
>>> {3, 1} < s1 # True subset
True
>>> s1 <= s1 # Subset
True
>>> {3, 1} <= s1 # Subset
True
>>> {3, 2, 4, 1} == s1 # Equality
True
>>> s1 == s2 # Equality
False
>>> 2 in s1 # Membership
True
>>> 10 not in s1 # Non-membership
True
>>> {1, 2, 3, 4, 5} > s1 # True superset
True
>>> {1, 2, 3, 4} > s1 # True superset
False
>>> {1, 2, 3, 4} >= s1 # Superset
True
>>> s1 ^ s2 # Symmetric difference
{1, 2, 5, 6}
>>> len(s1) # Cardinality
4
>>> s1
{99, 1, 2, 3, 4}
>>> s1
{1, 2, 3, 4}
>>> s1 |= s2 # Mutability
>>> s1
{1, 2, 3, 4, 5, 6}
>>> s1 -= s2 # Mutability
>>> s1
{1, 2}
>>> s1 ^= s2 # Mutability
>>> s1
{1, 2, 3, 4, 5, 6}
>>>
```

## Quackery

Sets are not implemented in Quackery, so we start with an implementation of sets as sorted nests of strings without duplicates.

This was the first pass at coding sets. I felt that bitmap sets were the way to go, but I wanted to get my head around the problem space and fill in a gap in my understanding, specifically pertaining to bignum bitmaps and the universal set. By the time I had coded an inefficient version that avoided the issue, I realised the solution - we don't need to accommodate everything, just everything we know about so far. So in the second version the ancillary stack `elements` holds a nest of the names of every set element we know of so far, and the word `elementid` returns the position of an element in that nest, if it has been encountered before, and adds it to the nest if it is previously unknown, then returns its position.

In both versions, a set element is any string of non-whitespace characters apart from "}set". If you desperately need set elements with spaces or carriage returns, or called "}set" there are workarounds.

Use the second version, it's a lot more efficient and the set of everything so far encountered is available if wanted. It's `[ elements share size bit 1 - ]`.

The set of everything is -1, just don't try to enumerate it; your program will crash when it gets to naming things it hasn't heard of.

### Sorted Nests of Strings

```  [ [] \$ "" rot
sort\$ witheach
[ tuck != if
[ dup dip
[ nested join ] ] ]
drop ]                             is -duplicates  (   { --> {   )

[ [] \$ "" rot
sort\$ witheach
[ tuck = if
[ nested join
\$ "" ] ]
drop -duplicates ]                 is duplicates   (   { --> {   )

[ [] \$ "" rot
sort\$ witheach
[ tuck != iff
[ dup dip [ nested join ] ]
else
[ dip [ -1 pluck ]
over != if
[ nested join \$ "" ] ] ]
drop ]                             is --duplicates (   { --> {   )

[ [] swap
[ trim
dup \$ "" = if
[ \$ '"set{" without "}set"'
message put bail ]
nextword
dup \$ "}set" != while
nested rot join swap
again ]
drop swap
-duplicates
' [ ' ] swap nested join
swap dip [ nested join ] ]     builds set{         ( [ \$ --> [ \$ )

[ -duplicates
say "{ "
witheach [ echo\$ sp ]
say "}" ]                          is echoset      (   { --> {   )

[ join duplicates ]                  is intersection ( { { --> {   )

[ join -duplicates ]                 is union        ( { { --> {   )

[ join --duplicates ]                is symmdiff     ( { { --> {   )

[ over intersection symmdiff ]       is difference   ( { { --> {   )

[ over intersection = ]              is subset       ( { { --> b   )

[ dip nested subset ]                is element      ( \$ { --> b   )

[ 2dup = iff
[ 2drop false ]
else subset ]                      is propersubset ( { { --> b   )

( ------------------------------ demo ------------------------------ )

set{ apple peach pear melon
apricot banana orange }set        is fruits       (     --> {   )

set{ red orange green blue
purple apricot peach }set         is colours      (     --> {   )

fruits  dup echoset say " are fruits" cr

colours dup echoset say " are colours" cr

2dup intersection echoset say " are both fruits and colours" cr

2dup union echoset say " are fruits or colours" cr

2dup symmdiff echoset say " are fruits or colours but not both" cr

difference echoset say " are fruits that are not colours" cr

set{ red green blue }set dup echoset say " are"
colours subset not if [ say " not" ] say " all colours"  cr

say "fruits and colours are" fruits colours = not if [ say " not" ]
say " exactly the same" cr

\$ "orange" dup echo\$ say " is"
fruits element not if [ say " not" ] say " a fruit" cr

set{ orange }set dup echoset say " is"
fruits propersubset dup if [ say " not" ] say " the only fruit"
not if [ say " or not a fruit" ] cr```
Output:
```{ apple apricot banana melon orange peach pear } are fruits
{ apricot blue green orange peach purple red } are colours
{ apricot orange peach } are both fruits and colours
{ apple apricot banana blue green melon orange peach pear purple red } are fruits or colours
{ apple banana blue green melon pear purple red } are fruits or colours but not both
{ apple banana melon pear } are fruits that are not colours
{ blue green red } are all colours
fruits and colours are not exactly the same
orange is a fruit
{ orange } is not the only fruit```

### Indexed Bitmaps

```  [ stack [ ] ]                     is elements     (     --> s   )

[ elements share 2dup find
dup rot found iff nip done
swap elements take
swap nested join
elements put ]                  is elementid    (   \$ --> n   )

[ 0 temp put
[ trim
dup \$ "" = if
[ \$ '"set{" without "}set"'
message put bail ]
nextword
dup \$ "}set" = iff drop done
elementid bit
temp take | temp put
again ]
temp take
swap dip
[ nested nested join ] ]   builds set{         ( [ \$ --> [ \$ )

[ [] 0 rot
[ dup while
dup 1 & if
[ over elements share
swap peek nested
swap dip
[ rot join swap ] ]
dip 1+
1 >>
again ]
2drop ]                         is set->nest    (   { --> [   )

[ say "{ "
set->nest witheach [ echo\$ sp ]
say "}" ]                       is echoset      (   { -->     )

[ & ]                             is intersection ( { { --> {   )

[ | ]                             is union        ( { { --> {   )

[ ^ ]                             is symmdiff     ( { { --> {   )

[ over intersection symmdiff ]    is difference   ( { { --> {   )

[ over intersection = ]           is subset       ( { { --> b   )

[ dip [ elementid bit ] subset ]  is element      ( \$ { --> b   )

[ 2dup = iff
[ 2drop false ]
else subset ]                   is propersubset ( { { --> b   )

( ----------------------------- demo ---------------------------- )

set{ apple peach pear melon
apricot banana orange }set   is fruits       (     --> {   )

set{ red orange green blue
purple apricot peach }set    is colours      (     --> {   )

fruits dup echoset say " are fruits" cr

colours dup echoset say " are colours" cr

2dup intersection echoset say " are both fruits and colours" cr

2dup union echoset say " are fruits or colours" cr

2dup symmdiff echoset say " are fruits or colours but not both" cr

difference echoset say " are fruits that are not colours" cr

set{ red green blue }set dup echoset say " are"
colours subset not if [ say " not" ] say " all colours"  cr

say "fruits and colours are" fruits colours = not if [ say " not" ]
say " exactly the same" cr

\$ "orange" dup echo\$ say " is"
fruits element not if [ say " not" ] say " a fruit" cr

set{ orange }set dup echoset say " is"
fruits propersubset dup if [ say " not" ] say " the only fruit"
not if [ say " or not a fruit" ] cr```
Output:
```{ orange banana apricot melon pear peach apple } are fruits
{ purple blue green red orange apricot peach } are colours
{ orange apricot peach } are both fruits and colours
{ purple blue green red orange banana apricot melon pear peach apple } are fruits or colours
{ purple blue green red banana melon pear apple } are fruits or colours but not both
{ banana melon pear apple } are fruits that are not colours
{ blue green red } are all colours
fruits and colours are not exactly the same
orange is a fruit
{ orange } is not the only fruit```

## Racket

```#lang racket

(define A (set 1 2 3 4))
(define B (set 3 4 5 6))
(define C (set 4 5))

(set-union A B)     ; gives (set 1 2 3 4 5 6)
(set-intersect A B) ; gives (set 3 4)
(set-subtract A B)  ; gives (set 1 2)
(set=? A B)         ; gives #f
(subset? C A)       ; gives #f
(subset? C B)       ; gives #t
```

## Raku

(formerly Perl 6)

```use Test;

my \$a = set <a b c>;
my \$b = set <b c d>;
my \$c = set <a b c d e>;

ok 'c' ∈ \$a, "c is an element in set A";
nok 'd' ∈ \$a, "d is not an element in set A";

is-deeply \$a ∪ \$b, set(<a b c d>), "union; a set of all elements either in set A or in set B";
is-deeply \$a ∩ \$b, set(<b c>), "intersection; a set of all elements in both set A and set B";
is \$a (-) \$b, set(<a>), "difference; a set of all elements in set A, except those in set B";

ok \$a ⊆ \$c, "subset; true if every element in set A is also in set B";
nok \$c ⊆ \$a, "subset; false if every element in set A is not also in set B";
ok \$a ⊂ \$c, "strict subset; true if every element in set A is also in set B";
nok \$a ⊂ \$a, "strict subset; false for equal sets";
ok \$a === set(<a b c>), "equality; true if every element of set A is in set B and vice-versa";
nok \$a === \$b, "equality; false for differing sets";
```
Output:
```ok 1 - c is an element in set A
ok 2 - d is not an element in set A
ok 3 - union; a set of all elements either in set A or in set B
ok 4 - intersection; a set of all elements in both set A and set B
ok 5 - difference; a set of all elements in set A, except those in set B
ok 6 - subset; true if every element in set A is also in set B
ok 7 - subset; false if every element in set A is not also in set B
ok 8 - strict subset; true if every element in set A is also in set B
ok 9 - strict subset; false for equal sets
ok 10 - equality; true if every element of set A is in set B and vice-versa
ok 11 - equality; false for differing sets```

## REXX

REXX doesn't have native set support, but can be easily coded to handle lists as sets.

```/*REXX program  demonstrates some  common   SET   functions.                            */
truth.0= 'false';            truth.1= "true"    /*two common names for a truth table.   */
set.=                                           /*the  order  of sets isn't important.  */

call setAdd 'prime',2 3 2 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
call setSay 'prime'                             /*a small set of some  prime numbers.   */

call setAdd 'emirp',97 97 89 83 79 73 71 67 61 59 53 47 43 41 37 31 29 23 19 17 13 11 7 5 3 2
call setSay 'emirp'                             /*a small set of  backward  primes.     */

call setAdd 'happy',1 7 10 13 19 23 28 31 32 44 49 68 70 79 82 86 91 100 94 97 97 97 97 97
call setSay 'happy'                             /*a small set of some  happy  numbers.  */

do j=11  to 100  by 10                    /*see if  PRIME  contains some numbers. */
call setHas  'prime', j
say '             prime contains'     j":"     truth.result
end   /*j*/

call setUnion  'prime','happy','eweion';  call setSay 'eweion'                /* (sic). */
call setCommon 'prime','happy','common';  call setSay 'common'
call setDiff   'prime','happy','diff'  ;  call setSay 'diff';        _=left('', 12)
call setSubset 'prime','happy'         ;  say _ 'prime is a subset of happy:' truth.result
call setEqual  'prime','emirp'         ;  say _ 'prime is  equal   to emirp:' truth.result
exit                                                   /*stick a fork in it, we're done.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
setHas:    procedure expose set.; arg _ .,! .; return wordpos(!, set._)\==0
setDiff:   return set\$('diff'   , arg(1), arg(2), arg(3))
setSay:    return set\$('say'    , arg(1), arg(2))
setUnion:  return set\$('union'  , arg(1), arg(2), arg(3))
setCommon: return set\$('common' , arg(1), arg(2), arg(3))
setEqual:  return set\$('equal'  , arg(1), arg(2))
setSubset: return set\$('subSet' , arg(1), arg(2))
/*──────────────────────────────────────────────────────────────────────────────────────*/
set\$: procedure expose set.;   arg \$,_1,_2,_3;   set_=set._1;   t=_3;   s=t;   !=1
if \$=='SAY'    then do;   say "[set."_1']= 'set._1;   return set._1;   end
if \$=='UNION'  then do
return set._3
end
if common | diff | eq | subset  then s=_2
if add  then do;  set_=_2;  t=_1;  s=_1;  end

do j=1  for words(set_);       _=word(set_, j);       has=wordpos(_, set.s)\==0
(common &  has) |,
(diff   & \has)       then set.t=space(set.t _)
if (eq | subset) & \has  then return 0
end    /*j*/

if subset  then return 1
if eq      then  if arg()>3  then return 1
else return set\$('equal', _2, _1, 1)
return set.t
```

output

```[set.PRIME]=2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
[set.EMIRP]=97 89 83 79 73 71 67 61 59 53 47 43 41 37 31 29 23 19 17 13 11 7 5 3 2
[set.HAPPY]=1 7 10 13 19 23 28 31 32 44 49 68 70 79 82 86 91 100 94 97
prime contains 11: true
prime contains 21: false
prime contains 31: true
prime contains 41: true
prime contains 51: false
prime contains 61: true
prime contains 71: true
prime contains 81: false
prime contains 91: false
[set.EWEION]=2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 1 10 28 32 44 49 68 70 82 86 91 100 94
[set.COMMON]=7 13 19 23 31 79 97
[set.DIFF]=2 3 5 11 17 29 37 41 43 47 53 59 61 67 71 73 83 89
prime is a subset of happy: false
prime is  equal   to emirp: true
```

## Ring

```# Project : Set

arr = ["apple", "banana", "cherry", "date", "elderberry", "fig", "grape"]
for n = 1 to 25
next
seta = "1010101"
see "Set A: " + arrset(arr,seta) + nl
setb = "0111110"
see "Set B: " + arrset(arr,setb) + nl
elementm = "0000010"
see "Element M: " + arrset(arr,elementm) + nl

temp = arrsetinsec(elementm,seta)
if len(temp) > 0
see "M is an element of set A" + nl
else
see "M is not an element of set A" + nl
ok
temp = arrsetinsec(elementm,setb)
if len(temp) > 0
see "M is an element of set B" + nl
else
see "M is not an element of set B" + nl
ok

see "The union of A and B is: "
see arrsetunion(seta,setb) + nl
see "The intersection of A and B is: "
see  arrsetinsec(seta,setb) + nl
see "The difference of A and B is: "
see arrsetnot(seta,setb) + nl

flag = arrsetsub(seta,setb)
if flag = 1
see "Set A is a subset of set B" + nl
else
see "Set A is not a subset of set B" + nl
ok
if seta = setb
see "Set A is equal to set B" + nl
else
see "Set A is not equal to set B" + nl
ok

func arrset(arr,set)
o = ""
for i = 1 to 7
if set[i] = "1"
o = o + arr[i] + ", "
ok
next
return left(o,len(o)-2)

func arrsetunion(seta,setb)
o = ""
union = list(len(seta))
for n = 1 to len(seta)
if seta[n] = "1" or setb[n] = "1"
union[n] = "1"
else
union[n] = "0"
ok
next
for i = 1 to len(union)
if union[i] = "1"
o = o + arr[i] + ", "
ok
next
return o

func arrsetinsec(setc,setd)
o = ""
union = list(len(setc))
for n = 1 to len(setc)
if setc[n] = "1" and setd[n] = "1"
union[n] = "1"
else
union[n] = "0"
ok
next
for i = 1 to len(union)
if union[i] = "1"
o = o + arr[i] + ", "
ok
next
return o

func arrsetnot(seta,setb)
o = ""
union = list(len(seta))
for n = 1 to len(seta)
if seta[n] = "1" and setb[n] = "0"
union[n] = "1"
else
union[n] = "0"
ok
next
for i = 1 to len(union)
if union[i] = "1"
o = o + arr[i] + ", "
ok
next
return o

func arrsetsub(setc,setd)
flag = 1
for n = 1 to len(setc)
if setc[n] = "1" and setd[n] = "0"
flag = 0
ok
next
return flag```

Output:

```Set A: apple, cherry, elderberry, grape
Set B: banana, cherry, date, elderberry, fig
Element M: fig
M is not an element of set A
M is an element of set B
The union of A and B is: apple, banana, cherry, date, elderberry, fig, grape,
The intersection of A and B is: cherry, elderberry,
The difference of A and B is: apple, grape,
Set A is not a subset of set B
Set A is not equal to set B
```

## RPL

RPL can handle lists, which are ordered sets. They can be created either by enumerating them:

```{ "Azerty" #4 3.14 }
```

or by first pushing their components in the stack, then invoking the appropriate function:

```"Azerty" #4 3.14 3 →LIST
```
Works with: Halcyon Calc version 4.2.7

We first need a command to remove an item for a list

```≪ DUP2 1 + OVER SIZE SUB ROT ROT
IF DUP 1 == THEN DROP2 ELSE
1 SWAP 1 - SUB SWAP +
END
≫ 'POPL' STO   @ ( { a1 .. ak .. an } k → { a1 .. an } )
```
RPL code Comment
```≪ POS SIGN ≫ 'IN?' STO

≪ → a b
≪ a 1 b SIZE FOR j
b j GET IF a OVER POS THEN DROP ELSE + END
NEXT
≫ ≫ 'UNION' STO

≪ → a b
≪ { } b
1 a SIZE FOR j
a j GET
IF DUP2 POS THEN
LAST ROT SWAP POPL
ROT ROT + SWAP
ELSE DROP END
NEXT DROP
≫ ≫ ‘INTER’ STO

≪ → a b
≪ { } b
1 a SIZE FOR j
a j GET
IF DUP2 POS NOT THEN
LAST ROT SWAP POPL
ROT ROT + SWAP
ELSE DROP END
NEXT DROP
≫ ≫ ‘DIFFL’ STO

≪ → a b
≪ 1 CF 1 a SIZE FOR j
IF b a j GET POS NOT THEN 1 SF a SIZE 'j' STO END
NEXT 1 FC?
≫ ≫ 'INCL'?' STO

≪ DUP2 INCL? ROT ROT SWAP 'DIFFL' AND
≫ 'SAME?' STO
```
```IN? ( {A} m -- boolean ) // 1 if m ∈ A

UNION ( {A} {B} -- {A ∪ B} )
Scan {B}...
... and add to {A} all {B} items not already in {A}

INTER ( {A} {B} -- {A ∩ B} )
Put a copy of {B} in stack
Scan {A}

if {A} item in copy of {B}
remove it from copy of {B}

DIFFL ( {A} {B} -- {A ∖ B} )
Put a copy of {B} in stack
Scan {A}

if {A} item not in copy of {B}
remove it from copy of {B}

INCL? ( {A} {B} -- boolean ) // true if {A} ⊆ {B}
Scan {A}...
... and break loop if an {A} item is not in {B}
return flag 1, set if loop has been broken

SAME ( {A} {B} -- boolean ) // true if {A} = {B}
{A} = {B} <=> {A} ⊆ {B} and {B} ⊆ {A}
```
Input:
```3 {1 2 3 4} IN?
{1 2 3 4} {3 5 6} UNION
{1 2 3 4} {3 5 6} INTER
{1 2 3 4} {3 5 6} DIFFL
{1 2} {1 2 3 4} INCL?
{1 2 3} {3 1 2} SAME?

```
Output:
```6: 1
5: { 1 2 3 4 5 6 }
4: { 3 }
3: { 1 2 4 }
2: 1
1: 1
```

## Ruby

Ruby's standard library contains a "set" package, which provides `Set` and `SortedSet` classes.

```>> require 'set'
=> true
>> s1, s2 = Set[1, 2, 3, 4], [3, 4, 5, 6].to_set # different ways of creating a set
=> [#<Set: {1, 2, 3, 4}>, #<Set: {5, 6, 3, 4}>]
>> s1 | s2 # Union
=> #<Set: {5, 6, 1, 2, 3, 4}>
>> s1 & s2 # Intersection
=> #<Set: {3, 4}>
>> s1 - s2 # Difference
=> #<Set: {1, 2}>
>> s1.proper_subset?(s1) # Proper subset
=> false
>> Set[3, 1].proper_subset?(s1) # Proper subset
=> true
>> s1.subset?(s1) # Subset
=> true
>> Set[3, 1].subset?(s1) # Subset
=> true
>> Set[3, 2, 4, 1] == s1 # Equality
=> true
>> s1 == s2 # Equality
=> false
>> s1.include?(2) # Membership
=> true
>> Set[1, 2, 3, 4, 5].proper_superset?(s1) # Proper superset
=> true
>> Set[1, 2, 3, 4].proper_superset?(s1) # Proper superset
=> false
>> Set[1, 2, 3, 4].superset?(s1) # Superset
=> true
>> s1 ^ s2 # Symmetric difference
=> #<Set: {5, 6, 1, 2}>
>> s1.size # Cardinality
=> 4
>> s1 << 99 # Mutability (or s1.add(99) )
=> #<Set: {99, 1, 2, 3, 4}>
>> s1.delete(99) # Mutability
=> #<Set: {1, 2, 3, 4}>
>> s1.merge(s2) # Mutability
=> #<Set: {5, 6, 1, 2, 3, 4}>
>> s1.subtract(s2) # Mutability
=> #<Set: {1, 2}>
>>
```

## Run BASIC

```A\$	= "apple cherry elderberry grape"
B\$	= "banana cherry date elderberry fig"
C\$	= "apple cherry elderberry grape orange"
D\$	= "apple cherry elderberry grape"
E\$	= "apple cherry elderberry"
M\$	= "banana"

print "A = ";A\$
print "B = ";B\$
print "C = ";C\$
print "D = ";D\$
print "E = ";E\$
print "M = ";M\$

if instr(A\$,M\$) = 0 then a\$ = "not "
print "M is ";a\$; "an element of Set A"
a\$ = ""
if instr(B\$,M\$) = 0 then a\$ = "not "
print "M is ";a\$; "an element of Set B"

un\$ = A\$ + " "
for i = 1 to 5
if instr(un\$,word\$(B\$,i)) = 0 then un\$ = un\$ + word\$(B\$,i) + " "
next i
print "union(A,B) = ";un\$

for i = 1 to 5
if instr(A\$,word\$(B\$,i)) <> 0 then ins\$ = ins\$ + word\$(B\$,i) + " "
next i
print "Intersection(A,B) = ";ins\$

for i = 1 to 5
if instr(B\$,word\$(A\$,i)) = 0 then dif\$ = dif\$ + word\$(A\$,i) + " "
next i
print "Difference(A,B) = ";dif\$

a = subs(A\$,B\$,"AB")
a = subs(A\$,C\$,"AC")
a = subs(A\$,E\$,"AE")

a = eqs(A\$,B\$,"AB")
a = eqs(A\$,C\$,"AC")
a = eqs(A\$,E\$,"AE")
end

function subs(a\$,b\$,sets\$)
for i = 1 to 5
if instr(b\$,word\$(a\$,i)) <> 0 then subs = subs + 1
next i
if subs = 4 then
print left\$(sets\$,1);" is a subset of ";right\$(sets\$,1)
else
print left\$(sets\$,1);" is not a subset of ";right\$(sets\$,1)
end if
end function

function eqs(a\$,b\$,sets\$)
for i = 1 to 5
if word\$(a\$,i) <> "" then a = a + 1
if word\$(b\$,i) <> "" then b = b + 1
if instr(b\$,word\$(a\$,i)) <> 0 then c = c + 1
next i
if (a = b) and (a = c) then
print left\$(sets\$,1);" is equal ";right\$(sets\$,1)
else
print left\$(sets\$,1);" is not equal ";right\$(sets\$,1)
end if
end function```

Output:

```A = apple cherry elderberry grape
B = banana cherry date elderberry fig
C = apple cherry elderberry grape orange
D = apple cherry elderberry grape
E = apple cherry elderberry
M = banana
M is not an element of Set A
M is an element of Set B
union(A,B) = apple cherry elderberry grape banana date fig
Intersection(A,B) = cherry elderberry
Difference(A,B) = apple grape
A is not a subset of B
A is a subset of C
A is a subset of D
A is not a subset of E
A is not equal B
A is not equal C
A is equal D
A is not equal E```

## Rust

```use std::collections::HashSet;

fn main() {
let a = vec![1, 3, 4].into_iter().collect::<HashSet<i32>>();
let b = vec![3, 5, 6].into_iter().collect::<HashSet<i32>>();

println!("Set A: {:?}", a.iter().collect::<Vec<_>>());
println!("Set B: {:?}", b.iter().collect::<Vec<_>>());
println!("Does A contain 4? {}", a.contains(&4));
println!("Union: {:?}", a.union(&b).collect::<Vec<_>>());
println!("Intersection: {:?}", a.intersection(&b).collect::<Vec<_>>());
println!("Difference: {:?}", a.difference(&b).collect::<Vec<_>>());
println!("Is A a subset of B? {}", a.is_subset(&b));
println!("Is A equal to B? {}", a == b);
}
```

## Scala

```object sets {
val set1 = Set(1,2,3,4,5)
val set2 = Set(3,5,7,9)
println(set1 contains 3)
println(set1 | set2)
println(set1 & set2)
println(set1 diff set2)
println(set1 subsetOf set2)
println(set1 == set2)
}
```

## Scheme

Implemented based on lists. Not efficient on large sets.

```(define (element? a lst)
(and (not (null? lst))
(or (eq? a (car lst))
(element? a (cdr lst)))))

; util, not strictly needed
(define (uniq lst)
(if (null? lst) lst
(let ((a (car lst)) (b (cdr lst)))
(if (element? a b)
(uniq b)
(cons a (uniq b))))))

(define (intersection a b)
(cond ((null? a) '())
((null? b) '())
(else
(append (intersection (cdr a) b)
(if (element? (car a) b)
(list (car a))
'())))))

(define (union a b)
(if (null? a) b
(union (cdr a)
(if (element? (car a) b)
b
(cons (car a) b)))))

(define (diff a b) ; a - b
(if (null? a) '()
(if (element? (car a) b)
(diff (cdr a) b)
(cons (car a) (diff (cdr a) b)))))

(define (subset? a b) ; A ⊆ B
(if (null? a) #t
(and (element? (car a) b)
(subset? (cdr a) b))))

(define (set-eq? a b)
(and (subset? a b)
(subset? b a)))
```

## Seed7

```\$ include "seed7_05.s7i";

const type: charSet is set of char;
enable_output(charSet);

const proc: main is func
local
const charSet: A is {'A', 'B', 'C', 'D', 'E', 'F'};
var charSet: B is charSet.value;
var char: m is 'A';
begin
B := {'E', 'F', 'G', 'H', 'I', 'K'};
incl(B, 'J');        # Add 'J' to set B
excl(B, 'K');        # Remove 'K' from set B
writeln("A: " <& A);
writeln("B: " <& B);
writeln("m: " <& m);
writeln("m in A -- m is an element in A: " <& m in A);
writeln("A | B  -- union:                " <& A | B);
writeln("A & B  -- intersection:         " <& A & B);
writeln("A - B  -- difference:           " <& A - B);
writeln("A >< B -- symmetric difference: " <& A >< B);
writeln("A <= A -- subset:               " <& A <= A);
writeln("A < A  -- proper subset:        " <& A < A);
writeln("A = B  -- equality:             " <& A = B);
end func;```
Output:
```A: {A, B, C, D, E, F}
B: {E, F, G, H, I, J}
m: A
m in A -- m is an element in A: TRUE
A | B  -- union:                {A, B, C, D, E, F, G, H, I, J}
A & B  -- intersection:         {E, F}
A - B  -- difference:           {A, B, C, D}
A >< B -- symmetric difference: {A, B, C, D, G, H, I, J}
A <= A -- subset:               TRUE
A < A  -- proper subset:        FALSE
A = B  -- equality:             FALSE
```

## SETL

```A := {1, 2, 3, 4};
B := {3, 4, 5, 6};
C := {4, 5};

-- Union, Intersection, Difference, Subset, Equality
print(A + B);       -- {1, 2, 3, 4, 5, 6}
print(A * B);       -- {3, 4}
print(A - B);       -- {1, 2}
print(C subset B);  -- #T
print(C = B);       -- #F```

## Sidef

Translation of: Perl
```class MySet(*set) {

method init {
var elems = set
set = Hash()
elems.each { |e| self += e }
}

method +(elem) {
set{elem} = elem
self
}

method del(elem) {
set.delete(elem)
}

method has(elem) {
set.has_key(elem)
}

method ∪(MySet that) {
MySet(set.values..., that.values...)
}

method ∩(MySet that) {
MySet(set.keys.grep{ |k| k ∈ that } \
.map { |k| set{k} }...)
}

method ∖(MySet that) {
MySet(set.keys.grep{|k| !(k ∈ that) } \
.map {|k| set{k} }...)
}

method ^(MySet that) {
var d = ((self ∖ that) ∪ (that ∖ self))
MySet(d.values...)
}

method count { set.len }

method ≡(MySet that) {
(self ∖ that -> count.is_zero) && (that ∖ self -> count.is_zero)
}

method values { set.values }

method ⊆(MySet that) {
that.set.keys.each { |k|
k ∈ self || return false
}
return true
}

method to_s {
"Set{" + set.values.map{|e| "#{e}"}.sort.join(', ') + "}"
}
}

class Object {
method ∈(MySet set) {
set.has(self)
}
}
```

Usage example:

```var x = MySet(1, 2, 3)
5..7 -> each { |i| x += i }

var y = MySet(1, 2, 4, x)

say "set x is: #{x}"
say "set y is: #{y}"

[1,2,3,4,x].each { |elem|
say ("#{elem} is ", elem ∈ y ? '' : 'not', " in y")
}

var (w, z)
say ("union: ", x ∪ y)
say ("intersect: ", x ∩ y)
say ("z = x ∖ y = ", z = (x ∖ y) )
say ("y is ", x ⊆ y ? "" : "not ", "a subset of x")
say ("z is ", x ⊆ z ? "" : "not ", "a subset of x")
say ("z = (x ∪ y) ∖ (x ∩ y) = ", z = ((x ∪ y) ∖ (x ∩ y)))
say ("w = x ^ y = ", w = (x ^ y))
say ("w is ", w ≡ z ? "" : "not ", "equal to z")
say ("w is ", w ≡ x ? "" : "not ", "equal to x")
```
Output:
```set x is: Set{1, 2, 3, 5, 6, 7}
set y is: Set{1, 2, 4, Set{1, 2, 3, 5, 6, 7}}
1 is  in y
2 is  in y
3 is not in y
4 is  in y
Set{1, 2, 3, 5, 6, 7} is  in y
union: Set{1, 2, 3, 4, 5, 6, 7, Set{1, 2, 3, 5, 6, 7}}
intersect: Set{1, 2}
z = x ∖ y = Set{3, 5, 6, 7}
y is not a subset of x
z is a subset of x
z = (x ∪ y) ∖ (x ∩ y) = Set{3, 4, 5, 6, 7, Set{1, 2, 3, 5, 6, 7}}
w = x ^ y = Set{3, 4, 5, 6, 7, Set{1, 2, 3, 5, 6, 7}}
w is equal to z
w is not equal to x
```

## Simula

```SIMSET
BEGIN

! WE DON'T SUBCLASS HEAD BUT USE COMPOSITION FOR CLASS SET ;
CLASS SET;
BEGIN
BEGIN
IF NOT ISIN(E, THIS SET) THEN E.CLONE.INTO(H);

BOOLEAN PROCEDURE EMPTY; EMPTY := H.EMPTY;
REF(LINK) PROCEDURE FIRST; FIRST :- H.FIRST;

END**OF**SET;

! WE SUBCLASS LINK FOR THE ELEMENTS CONTAINED IN THE SET ;
VIRTUAL:
PROCEDURE ISEQUAL IS
BOOLEAN PROCEDURE ISEQUAL(OTHER); REF(ELEMENT) OTHER;;
PROCEDURE REPR IS
TEXT PROCEDURE REPR;;
PROCEDURE REPR IS
REF(ELEMENT) PROCEDURE CLONE;;
BEGIN
END**OF**ELEMENT;

REF(SET) PROCEDURE UNION(S1, S2); REF(SET) S1, S2;
BEGIN REF(SET) SU, S;
SU :- NEW SET;
FOR S :- S1, S2 DO
BEGIN
IF NOT S.EMPTY THEN
BEGIN REF(ELEMENT) E;
E :- S.FIRST;
WHILE E =/= NONE DO
END;
END;
END;
UNION :- SU;
END**OF**UNION;

REF(SET) PROCEDURE INTERSECTION(S1, S2); REF(SET) S1, S2;
BEGIN REF(SET) SI;
SI :- NEW SET;
IF NOT S1.EMPTY THEN
BEGIN REF(ELEMENT) E;
E :- S1.FIRST;
WHILE E =/= NONE DO
BEGIN IF ISIN(E, S2) THEN SI.ADD(E); E :- E.SUC;
END;
END;
INTERSECTION :- SI;
END**OF**INTERSECTION;

REF(SET) PROCEDURE MINUS(S1, S2); REF(SET) S1, S2;
BEGIN REF(SET) SM;
SM :- NEW SET;
IF NOT S1.EMPTY THEN
BEGIN REF(ELEMENT) E;
E :- S1.FIRST;
WHILE E =/= NONE DO
BEGIN IF NOT ISIN(E, S2) THEN SM.ADD(E); E :- E.SUC;
END;
END;
MINUS :- SM;
END**OF**MINUS;

BOOLEAN PROCEDURE ISSUBSET(S1, S2); REF(SET) S1, S2;
BEGIN BOOLEAN B;
B := TRUE;
IF NOT S1.EMPTY THEN
BEGIN REF(ELEMENT) E;
E :- S1.FIRST;
WHILE B AND E =/= NONE DO
BEGIN
B := ISIN(E, S2);
E :- E.SUC;
END;
END;
ISSUBSET := B;
END**OF**ISSUBSET;

BOOLEAN PROCEDURE ISEQUAL(S1, S2); REF(SET) S1, S2;
BEGIN
ISEQUAL := ISSUBSET(S1, S2) AND THEN ISSUBSET(S2, S1)
END**OF**ISEQUAL;

BOOLEAN PROCEDURE ISIN(ELE,S); REF(ELEMENT) ELE; REF(SET) S;
BEGIN
REF(ELEMENT) E; BOOLEAN FOUND;
IF NOT S.EMPTY THEN
BEGIN
E :- S.FIRST;
FOUND := E.ISEQUAL(ELE);
BEGIN FOUND := E.ISEQUAL(ELE); E :- E.SUC;
END;
END;
ISIN := FOUND
END**OF**ISIN;

PROCEDURE OUTSET(S); REF(SET) S;
BEGIN
REF(ELEMENT) E;
OUTCHAR('{');
IF NOT S.EMPTY THEN
BEGIN
E :- S.FIRST; OUTTEXT(E.REPR);
FOR E :- E.SUC WHILE E =/= NONE DO
BEGIN OUTTEXT(", "); OUTTEXT(E.REPR);
END;
END;
OUTCHAR('}');
END**OF**OUTSET;

COMMENT ============== EXAMPLE USING SETS OF NUMBERS ============== ;

ELEMENT CLASS NUMBER(N); INTEGER N;
BEGIN
BOOLEAN PROCEDURE ISEQUAL(OTHER); REF(ELEMENT) OTHER;
ISEQUAL := N = OTHER QUA NUMBER.N;
TEXT PROCEDURE REPR;
BEGIN TEXT T; INTEGER I;
T :- BLANKS(20); T.PUTINT(N);
T.SETPOS(1);
WHILE T.GETCHAR = ' ' DO;
REPR :- T.SUB(T.POS - 1, T.LENGTH - T.POS + 2);
END;
REF(ELEMENT) PROCEDURE CLONE;
CLONE :- NEW NUMBER(N);
END**OF**NUMBER;

PROCEDURE REPORT(S1, MSG1, S2, MSG2, S3); REF(SET) S1, S2, S3; TEXT MSG1, MSG2;
BEGIN
OUTSET(S1);    OUTCHAR(' ');
OUTTEXT(MSG1); OUTCHAR(' ');
OUTSET(S2);    OUTCHAR(' ');
OUTTEXT(MSG2); OUTCHAR(' ');
OUTSET(S3);
OUTIMAGE;
END**OF**REPORT;

PROCEDURE REPORTBOOL(S1, MSG1, S2, MSG2, B); REF(SET) S1, S2; TEXT MSG1, MSG2; BOOLEAN B;
BEGIN
OUTSET(S1);    OUTCHAR(' ');
OUTTEXT(MSG1); OUTCHAR(' ');
OUTSET(S2);    OUTCHAR(' ');
OUTTEXT(MSG2); OUTCHAR(' ');
OUTTEXT(IF B THEN "T" ELSE "F");
OUTIMAGE;
END**OF**REPORTBOOL;

PROCEDURE REPORTNUMBOOL(N1, MSG1, S1, MSG2, B); REF(ELEMENT) N1; REF(SET) S1; TEXT MSG1, MSG2; BOOLEAN B;
BEGIN
OUTTEXT(N1.REPR);    OUTCHAR(' ');
OUTTEXT(MSG1); OUTCHAR(' ');
OUTSET(S1);    OUTCHAR(' ');
OUTTEXT(MSG2); OUTCHAR(' ');
OUTTEXT(IF B THEN "T" ELSE "F");
OUTIMAGE;
END**OF**REPORTNUMBOOL;

REF(SET) S1, S2, S3, S4, S5;
REF(ELEMENT) E;
INTEGER I;

S1 :- NEW SET; FOR I := 1, 2, 3, 4    DO S1.ADD(NEW NUMBER(I));
S2 :- NEW SET; FOR I := 3, 4, 5, 6    DO S2.ADD(NEW NUMBER(I));
S3 :- NEW SET; FOR I := 3, 1          DO S3.ADD(NEW NUMBER(I));
S4 :- NEW SET; FOR I := 1, 2, 3, 4, 5 DO S4.ADD(NEW NUMBER(I));
S5 :- NEW SET; FOR I := 4, 3, 2, 1    DO S5.ADD(NEW NUMBER(I));

REPORT(S1, "UNION", S2, " = ", UNION(S1, S2));

REPORT(S1, "INTERSECTION", S2, " = ", INTERSECTION(S1, S2));

REPORT(S1, "MINUS", S2, " = ", MINUS(S1, S2));

REPORT(S2, "MINUS", S1, " = ", MINUS(S2, S1));

E :- NEW NUMBER(2);
REPORTNUMBOOL(E, "IN", S1, " = ", ISIN(E, S1));

E :- NEW NUMBER(10);
REPORTNUMBOOL(E, "NOT IN", S1, " = ", NOT ISIN(E, S1));

REPORTBOOL(S1, "IS SUBSET OF", S1, " = ", ISSUBSET(S1, S1));
REPORTBOOL(S3, "IS SUBSET OF", S1, " = ", ISSUBSET(S3, S1));
REPORTBOOL(S4, "IS SUPERSET OF", S1, " = ", ISSUBSET(S1, S4));

REPORTBOOL(S1, "IS EQUAL TO", S2, " = ", ISEQUAL(S1, S2));
REPORTBOOL(S2, "IS EQUAL TO", S2, " = ", ISEQUAL(S2, S2));
REPORTBOOL(S1, "IS EQUAL TO", S5, " = ", ISEQUAL(S1, S5));

END.```
Output:
```{1, 2, 3, 4} UNION {3, 4, 5, 6}  =  {1, 2, 3, 4, 5, 6}
{1, 2, 3, 4} INTERSECTION {3, 4, 5, 6}  =  {3, 4}
{1, 2, 3, 4} MINUS {3, 4, 5, 6}  =  {1, 2}
{3, 4, 5, 6} MINUS {1, 2, 3, 4}  =  {5, 6}
2 IN {1, 2, 3, 4}  =  T
10 NOT IN {1, 2, 3, 4}  =  T
{1, 2, 3, 4} IS SUBSET OF {1, 2, 3, 4}  =  T
{3, 1} IS SUBSET OF {1, 2, 3, 4}  =  T
{1, 2, 3, 4, 5} IS SUPERSET OF {1, 2, 3, 4}  =  T
{1, 2, 3, 4} IS EQUAL TO {3, 4, 5, 6}  =  F
{3, 4, 5, 6} IS EQUAL TO {3, 4, 5, 6}  =  T
{1, 2, 3, 4} IS EQUAL TO {4, 3, 2, 1}  =  T

```

## Smalltalk

Works with: Pharo version 1.3-13315
```#(1 2 3) asSet union: #(2 3 4) asSet.
"a Set(1 2 3 4)"

#(1 2 3) asSet intersection: #(2 3 4) asSet.
"a Set(2 3)"

#(1 2 3) asSet difference: #(2 3 4) asSet.
"a Set(1)"

#(1 2 3) asSet includesAllOf: #(1 3) asSet.
"true"

#(1 2 3) asSet includesAllOf: #(1 3 4) asSet.
"false"

#(1 2 3) asSet = #(2 1 3) asSet.
"true"

#(1 2 3) asSet = #(1 2 4) asSet.
"false"
```

## SQL

Works with: Oracle
```-- set of numbers is a table
-- create one set with 3 elements

create table myset1 (element number);

insert into myset1 values (1);
insert into myset1 values (2);
insert into myset1 values (3);

commit;

-- check if 1 is an element

select 'TRUE' BOOL from dual
where 1 in
(select element from myset1);

-- create second set with 3 elements

create table myset2 (element number);

insert into myset2 values (1);
insert into myset2 values (5);
insert into myset2 values (6);

commit;

-- union sets

select element from myset1
union
select element from myset2;

-- intersection

select element from myset1
intersect
select element from myset2;

-- difference

select element from myset1
minus
select element from myset2;

-- subset

-- change myset2 to only have 1 as element

delete from myset2 where not element = 1;

commit;

-- check if myset2 subset of myset1

select 'TRUE' BOOL from dual
where 0 =  (select count(*) from
(select element from myset2
minus
select element from myset1));

-- equality

-- change myset1 to only have 1 as element

delete from myset1 where not element = 1;

commit;

-- check if myset2 subset of myset1 and
-- check if myset1 subset of myset2 and

select 'TRUE' BOOL from dual
where
0 =  (select count(*) from
(select element from myset2
minus
select element from myset1)) and
0 =
(select count(*) from
(select element from myset1
minus
select element from myset2));
```
```SQL>
SQL> -- set of numbers is a table
SQL> -- create one set with 3 elements
SQL>
SQL> create table myset1 (element number);

Table created.

SQL>
SQL> insert into myset1 values (1);

1 row created.

SQL> insert into myset1 values (2);

1 row created.

SQL> insert into myset1 values (3);

1 row created.

SQL>
SQL> commit;

Commit complete.

SQL>
SQL> -- check if 1 is an element
SQL>
SQL> select 'TRUE' BOOL from dual
2  where 1 in
3  (select element from myset1);

BOOL
----
TRUE

SQL>
SQL> -- create second set with 3 elements
SQL>
SQL> create table myset2 (element number);

Table created.

SQL>
SQL> insert into myset2 values (1);

1 row created.

SQL> insert into myset2 values (5);

1 row created.

SQL> insert into myset2 values (6);

1 row created.

SQL>
SQL> commit;

Commit complete.

SQL>
SQL> -- union sets
SQL>
SQL> select element from myset1
2  union
3  select element from myset2;

ELEMENT
----------
1
2
3
5
6

SQL>
SQL> -- intersection
SQL>
SQL> select element from myset1
2  intersect
3  select element from myset2;

ELEMENT
----------
1

SQL>
SQL> -- difference
SQL>
SQL> select element from myset1
2  minus
3  select element from myset2;

ELEMENT
----------
2
3

SQL>
SQL> -- subset
SQL>
SQL> -- change myset2 to only have 1 as element
SQL>
SQL> delete from myset2 where not element = 1;

2 rows deleted.

SQL>
SQL> commit;

Commit complete.

SQL>
SQL> -- check if myset2 subset of myset1
SQL>
SQL> select 'TRUE' BOOL from dual
2  where 0 =  (select count(*) from
3  (select element from myset2
4  minus
5  select element from myset1));

BOOL
----
TRUE

SQL>
SQL> -- equality
SQL>
SQL> -- change myset1 to only have 1 as element
SQL>
SQL> delete from myset1 where not element = 1;

2 rows deleted.

SQL>
SQL> commit;

Commit complete.

SQL>
SQL>  -- check if myset2 subset of myset1 and
SQL>  -- check if myset1 subset of myset2 and
SQL>
SQL> select 'TRUE' BOOL from dual
2  where
3  0 =  (select count(*) from
4  (select element from myset2
5  minus
6  select element from myset1)) and
7  0 =
8  (select count(*) from
9  (select element from myset1
10  minus
11  select element from myset2));

BOOL
----
TRUE
```

## Swift

Works with: Swift version 1.2+
```var s1 : Set<Int> = [1, 2, 3, 4]
let s2 : Set<Int> = [3, 4, 5, 6]
println(s1.union(s2)) // union; prints "[5, 6, 2, 3, 1, 4]"
println(s1.intersect(s2)) // intersection; prints "[3, 4]"
println(s1.subtract(s2)) // difference; prints "[2, 1]"
println(s1.isSubsetOf(s1)) // subset; prints "true"
println(Set<Int>([3, 1]).isSubsetOf(s1)) // subset; prints "true"
println(s1.isStrictSubsetOf(s1)) // proper subset; prints "false"
println(Set<Int>([3, 1]).isStrictSubsetOf(s1)) // proper subset; prints "true"
println(Set<Int>([3, 2, 4, 1]) == s1) // equality; prints "true"
println(s1 == s2) // equality; prints "false"
println(s1.contains(2)) // membership; prints "true"
println(Set<Int>([1, 2, 3, 4]).isSupersetOf(s1)) // superset; prints "true"
println(Set<Int>([1, 2, 3, 4]).isStrictSupersetOf(s1)) // proper superset; prints "false"
println(Set<Int>([1, 2, 3, 4, 5]).isStrictSupersetOf(s1)) // proper superset; prints "true"
println(s1.exclusiveOr(s2)) // symmetric difference; prints "[5, 6, 2, 1]"
println(s1.count) // cardinality; prints "4"
s1.insert(99) // mutability
println(s1) // prints "[99, 2, 3, 1, 4]"
s1.remove(99) // mutability
println(s1) // prints "[2, 3, 1, 4]"
s1.unionInPlace(s2) // mutability
println(s1) // prints "[5, 6, 2, 3, 1, 4]"
s1.subtractInPlace(s2) // mutability
println(s1) // prints "[2, 1]"
s1.exclusiveOrInPlace(s2) // mutability
println(s1) // prints "[5, 6, 2, 3, 1, 4]"
```

## Tcl

Sets in Tcl are modeled as lists of items, with operations that preserve uniqueness of membership.

Library: Tcllib (Package: struct::set)
```package require struct::set

# Many ways to build sets
set s1 [list 1 2 3 4]
set s2 {3 4 5 6}
struct::set add s3 {2 3 4 3 2};   # \$s3 will be proper set...
set item 5

puts "union: [struct::set union \$s1 \$s2]"
puts "intersection: [struct::set intersect \$s1 \$s2]"
puts "difference: [struct::set difference \$s1 \$s2]"
puts "membership predicate: [struct::set contains \$s1 \$item]"
puts "subset predicate: [struct::set subsetof \$s1 \$s2]";   # NB: not strict subset test!
puts "equality predicate: [struct::set equal \$s1 \$s2]"

# Adding an element to a set (note that we pass in the name of the variable holding the set):
struct::set include s3 \$item
# Removing an element from a set:
struct::set exclude s3 \$item
# Getting the cardinality:
puts "cardinality: [struct::set size \$s3]
```

## VBA

```'Implementation of "set" using the built in Collection datatype.
'A collection can hold any object as item. The examples here are only strings.
'A collection stores item, key pairs. With the key you can retrieve the item.
'The keys are hidden and cannot be changed. No duplicate keys are allowed.
'For the "set" implementation item is the same as the key. And keys must
'be a string.
Private Function createSet(t As Variant) As Collection
Dim x As New Collection
For Each elem In t
Next elem
Set createSet = x
End Function
Private Function isElement(s As Variant, x As Collection) As Boolean
Dim errno As Integer, t As Variant
On Error GoTo err
t = x(s)
isElement = True
Exit Function
err:
isElement = False
End Function
Private Function setUnion(A As Collection, B As Collection) As Collection
Dim x As New Collection
For Each elem In A
Next elem
For Each elem In B
On Error Resume Next 'Trying to add a duplicate throws an error
Next elem
Set setUnion = x
End Function
Private Function intersection(A As Collection, B As Collection) As Collection
Dim x As New Collection
For Each elem In A
If isElement(elem, B) Then x.Add elem, elem
Next elem
For Each elem In B
If isElement(elem, A) Then
On Error Resume Next
End If
Next elem
Set intersection = x
End Function
Private Function difference(A As Collection, B As Collection) As Collection
Dim x As New Collection
For Each elem In A
If Not isElement(elem, B) Then x.Add elem, elem
Next elem
Set difference = x
End Function
Private Function subset(A As Collection, B As Collection) As Boolean
Dim flag As Boolean
flag = True
For Each elem In A
If Not isElement(elem, B) Then
flag = False
Exit For
End If
Next elem
subset = flag
End Function
Private Function equality(A As Collection, B As Collection) As Boolean
Dim flag As Boolean
flag = True
If A.Count = B.Count Then
For Each elem In A
If Not isElement(elem, B) Then
flag = False
Exit For
End If
Next elem
Else
flag = False
End If
equality = flag
End Function
Private Function properSubset(A As Collection, B As Collection) As Boolean
Dim flag As Boolean
flag = True
If A.Count < B.Count Then
For Each elem In A
If Not isElement(elem, B) Then
flag = False
Exit For
End If
Next elem
Else
flag = False
End If
properSubset = flag
End Function
Public Sub main()
'Set creation
Dim s As Variant
Dim A As Collection, B As Collection, C As Collection
s = [{"Apple","Banana","Pear","Pineapple"}]
Set A = createSet(s) 'Fills the collection A with the elements of s
'Test m ? S -- "m is an element in set S"
Debug.Print isElement("Apple", A) 'returns True
Debug.Print isElement("Fruit", A) 'returns False
'A ? B -- union; a set of all elements either in set A or in set B.
s = [{"Fruit","Banana","Pear","Orange"}]
Set B = createSet(s)
Set C = setUnion(A, B)
'A n B -- intersection; a set of all elements in both set A and set B.
Set C = intersection(A, B)
'A \ B -- difference; a set of all elements in set A, except those in set B.
Set C = difference(A, B)
'A ? B -- subset; true if every element in set A is also in set B.
Debug.Print subset(A, B)
'A = B -- equality; true if every element of set A is in set B and vice versa.
Debug.Print equality(A, B)
'Proper subset
Debug.Print properSubset(A, B)
'Modify -remove an element by key
A.Remove "Apple"
'Modify -remove the first element in the collection/set
A.Remove 1
End Sub```

## Wren

Translation of: Kotlin
Library: Wren-set

Note that the Set class in the above module uses a Map internally for storage. Consequently, iteration order is undefined.

```import "./set" for Set

var fruits = Set.new(["apple", "pear", "orange", "banana"])
System.print("fruits  : %(fruits)")
var fruits2 = Set.new(["melon", "orange", "lemon", "gooseberry"])
System.print("fruits2 : %(fruits2)\n")

System.print("fruits  contains 'banana'     : %(fruits.contains("banana"))")
System.print("fruits2 contains 'elderberry' : %(fruits2.contains("elderberry"))\n")

System.print("Union        : %(fruits.union(fruits2))")
System.print("Intersection : %(fruits.intersect(fruits2))")
System.print("Difference   : %(fruits.except(fruits2))\n")

System.print("fruits2 is a subset of fruits : %(fruits2.subsetOf(fruits))\n")
var fruits3 = fruits.copy()
System.print("fruits3 : %(fruits3)\n")
System.print("fruits2 and fruits are equal  : %(fruits2 == fruits)")
System.print("fruits3 and fruits are equal  : %(fruits3 == fruits)\n")

var fruits4 = Set.new(["apple", "orange"])
System.print("fruits4 : %(fruits4)\n")
System.print("fruits3 is a proper subset of fruits : %(fruits3.properSubsetOf(fruits))")
System.print("fruits4 is a proper subset of fruits : %(fruits4.properSubsetOf(fruits))\n")

var fruits5 = Set.new(["cherry", "blueberry", "raspberry"])
System.print("fruits5 : %(fruits5)\n")
System.print("fruits5 + 'guava'  : %(fruits5)")
fruits5.remove("cherry")
System.print("fruits5 - 'cherry' : %(fruits5)")
```
Output:
```fruits  : <banana, orange, pear, apple>
fruits2 : <lemon, gooseberry, orange, melon>

fruits  contains 'banana'     : true
fruits2 contains 'elderberry' : false

Union        : <banana, lemon, gooseberry, orange, melon, pear, apple>
Intersection : <orange>
Difference   : <banana, pear, apple>

fruits2 is a subset of fruits : false

fruits3 : <banana, orange, pear, apple>

fruits2 and fruits are equal  : false
fruits3 and fruits are equal  : true

fruits4 : <orange, apple>

fruits3 is a proper subset of fruits : false
fruits4 is a proper subset of fruits : true

fruits5 : <raspberry, blueberry, cherry>

fruits5 + 'guava'  : <raspberry, guava, blueberry, cherry>
fruits5 - 'cherry' : <raspberry, guava, blueberry>
```

## XPL0

Translation of: Wren
```proc PrintBool(Str, Test);
int  Str, Test;
[Text(0, Str);
Text(0, if Test then "true" else "false");
CrLf(0);
];

proc PrintSet(Str, Set, Names);
int  Str, Set, Names, I;
[Text(0, Str);
for I:= 0 to 31 do
if 1<<I & Set then
[Text(0, Names(I));  ChOut(0, ^ )];
CrLf(0);
];

int Names, Fruits, Fruits2, Fruits3, Fruits4, Fruits5;
def Apple=1<<0, Pear=1<<1, Orange=1<<2, Banana=1<<3, Melon=1<<4, Lemon=1<<5,
Gooseberry=1<<6, Elderberry=1<<7, Raspberry=1<<8, Blueberry=1<<9,
Cherry=1<<10, Guava=1<<11;
[Names:= ["Apple", "Pear", "Orange", "Banana", "Melon", "Lemon", "Gooseberry",
"Elderberry", "Raspberry", "Blueberry", "Cherry", "Guava"];
Fruits:= Apple ! Pear ! Orange ! Banana;
PrintSet("Fruits  : ", Fruits, Names);
Fruits2:= Melon ! Orange ! Lemon ! Gooseberry;
PrintSet("Fruits2 : ", Fruits2, Names);
CrLf(0);
PrintBool("Fruits  contains 'Banana'     : ", Fruits & Banana);
PrintBool("Fruits2 contains 'Elderberry' : ", Fruits2 & Elderberry);
CrLf(0);
PrintSet("Union        : ", Fruits ! Fruits2, Names);
PrintSet("Intersection : ", Fruits & Fruits2, Names);
PrintSet("Difference   : ", Fruits & ~Fruits2, Names);
CrLf(0);
PrintBool("Fruits2 is a subset of Fruits : ",
(Fruits2 & Fruits) # 0 & (Fruits2 & ~Fruits) = 0);
CrLf(0);
Fruits3:= Fruits;
PrintSet("Fruits3 : ", Fruits3, Names);
CrLf(0);
PrintBool("Fruits2 and Fruits are equal  : ", Fruits2 = Fruits);
PrintBool("Fruits3 and Fruits are equal  : ", Fruits3 = Fruits);
CrLf(0);
Fruits4:= Apple ! Orange;
PrintSet("Fruits4 : ", Fruits4, Names);
CrLf(0);
PrintBool("Fruits3 is a proper subset of Fruits : ",
(Fruits3 & Fruits) # 0 & (Fruits3 & ~Fruits) = 0 & Fruits3 # Fruits);
PrintBool("Fruits4 is a proper subset of Fruits : ",
(Fruits4 & Fruits) # 0 & (Fruits4 & ~Fruits) = 0 & Fruits4 # Fruits);
CrLf(0);
Fruits5:= Cherry ! Blueberry ! Raspberry;
PrintSet("Fruits5 : ", Fruits5, Names);
CrLf(0);
Fruits5:= Fruits5 + Guava;
PrintSet("Fruits5 + 'Guava'  : ", Fruits5, Names);
Fruits5:= Fruits5 - Cherry;     \Cherry better be present!
PrintSet("Fruits5 - 'Cherry' : ", Fruits5, Names);
]```
Output:
```Fruits  : Apple Pear Orange Banana
Fruits2 : Orange Melon Lemon Gooseberry

Fruits  contains 'Banana'     : true
Fruits2 contains 'Elderberry' : false

Union        : Apple Pear Orange Banana Melon Lemon Gooseberry
Intersection : Orange
Difference   : Apple Pear Banana

Fruits2 is a subset of Fruits : false

Fruits3 : Apple Pear Orange Banana

Fruits2 and Fruits are equal  : false
Fruits3 and Fruits are equal  : true

Fruits4 : Apple Orange

Fruits3 is a proper subset of Fruits : false
Fruits4 is a proper subset of Fruits : true

Fruits5 : Raspberry Blueberry Cherry

Fruits5 + 'Guava'  : Raspberry Blueberry Cherry Guava
Fruits5 - 'Cherry' : Raspberry Blueberry Guava
```

## zkl

A simplistic implementation that is fine for smallish sets

```var [const] unique = Utils.Helpers.listUnique;
class Set {
fcn init { var [const] set = (vm.arglist.copy() : unique(_)) }
fcn holds(x) { set.holds(x) }
fcn union(setB) { self(set.xplode(),setB.set.xplode()) }
fcn intersection(setB){ sb:=setB.set;
C:=self(); sc:=C.set;
foreach x in (set){ if (sb.holds(x)) sc.append(x) }
C
}
fcn diff(setB){ C:=self(); C.set.extend(set);
setB.set.pump(Void,C.set.remove);
C
}
fcn isSubset(setB){ sb:=setB.set;
set.pump(Void,'wrap(x){
if (not sb.holds(x)) return(Void.Stop,False); True
})
}
fcn __opEQ(setB) { ((set.len() == setB.set.len()) and self.isSubset(setB)) }
}```
```A := Set(1,2,3,3,3,4);
A.set.println();    //--> L(1,2,3,4)
A.holds(3).println();  //--> True
A.holds(9).println();  //--> False

B:=Set("one",2,"three");
A.union(B).set.println(); // -->L(1,2,3,4,"one","three")
B.union(A).set.println(); // -->L("one",2,"three",1,3,4)
A.union(B).diff(B.union(A)).set.println(); // -->L()

A.intersection(B).set.println(); //-->L(2)
B.intersection(A).set.println(); //-->L(2)

A.diff(B).set.println();  //-->L(1,3,4)
B.diff(A).set.println();  //-->L("one","three")

A.isSubset(B).println();  //-->False
A.isSubset(A).println();  //-->True
Set("three",2,2,2,2,2).isSubset(B).println();  //-->True

(A==B).println();  //-->False
(A==A).println();  //-->True
(A==Set(2,3,1,4)).println(); //-->True
```