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A set is a collection (container) of certain values, without any particular order, and no repeated values.

It corresponds with a finite set in mathematics.

A set can be implemented as an associative array (partial mapping) in which the value of each key-value pair is ignored.

Given a set S, the power set (or powerset) of S, written P(S), or 2^S, is the set of all subsets of S.

Task

By using a library or built-in set type, or by defining a set type with necessary operations, write a function with a set S as input that yields the power set 2^S of S.

For example, the power set of {1,2,3,4} is

{{}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {1,4}, {2,4}, {1,2,4}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}.

For a set which contains n elements, the corresponding power set has 2ⁿ elements, including the edge cases of empty set.

The power set of the empty set is the set which contains itself (2⁰ = 1):

{\mathcal {P}}

(

\varnothing

) = {

\varnothing

}

And the power set of the set which contains only the empty set, has two subsets, the empty set and the set which contains the empty set (2¹ = 2):

{\mathcal {P}}

({

\varnothing

}) = {

\varnothing

, {

\varnothing

} }

Extra credit: Demonstrate that your language supports these last two powersets.

ABAP

This works for ABAP Version 7.40 and above

<lang ABAP> report z_powerset.

class set definition.

 public section.
   methods:
     constructor
       importing
         i_elements type stringtab optional,

     add_element
       importing
         i_element    type string
       returning
         value(r_set) type stringtab,

     remove_element
       importing
         i_element type string,

     contains_element
       importing
         i_element         type string
       returning
         value(r_contains) type abap_bool,

     get_size
       returning
         value(r_size) type int4,

     is_equal
       importing
         i_set             type ref to set
       returning
         value(r_is_equal) type abap_bool,

     get_elements
       returning
         value(r_elements) type stringtab,

     stringify
       returning
         value(r_string) type string.

 private section.
   data elements type stringtab.

endclass.

class set implementation.

 method constructor.
   loop at i_elements into data(element).
     me->add_element( element ).
   endloop.
 endmethod.

 method add_element.
   if not line_exists( me->elements[ table_line = i_element ] ).
     append i_element to me->elements.
   endif.
 endmethod.

 method remove_element.
   if line_exists( me->elements[ table_line = i_element ] ).
     delete me->elements where table_line = i_element.
   endif.
 endmethod.

 method contains_element.
   r_contains = cond abap_bool(
     when line_exists( me->elements[ table_line = i_element ] )
     then abap_true
     else abap_false ).
 endmethod.

 method get_size.
   r_size = lines( me->elements ).
 endmethod.

 method is_equal.
   if i_set->get_size( ) ne me->get_size( ).
     r_is_equal = abap_false.
     return.
   endif.

   loop at me->elements into data(element).
     if not i_set->contains_element( element ).
       r_is_equal = abap_false.
       return.
     endif.
   endloop.

   r_is_equal = abap_true.
 endmethod.

 method get_elements.
   r_elements = me->elements.
 endmethod.

 method stringify.
   r_string = cond string(
     when me->elements is initial
     then `∅`
     when me->elements eq value stringtab( ( `∅` ) )
     then `{ ∅ }`
     else reduce string(
       init result = `{ `
       for element in me->elements
       next result = cond string(
         when element eq ``
         then |{ result }∅, |
         when strlen( element ) eq 1 and element ne `∅`
         then |{ result }{ element }, |
         else |{ result }\{{ element }\}, | ) ) ).

   r_string = replace(
     val = r_string
     regex = `, $`
     with = ` }`).
 endmethod.

endclass.

start-of-selection.

 data(result_set) = new set( ).
 data(set1) = new set( ).
 data(set2) = new set( value stringtab( ( `∅` ) ) ).
 data(set3) = new set( value stringtab( ( `1` ) ( `2` ) ( `3` ) ( `3` ) ( `4` ) ( `4` ) ( `4` ) ) ).

 write |𝑷( { set1->stringify( ) } ) ->|.
 perform build_powerset using set1
                        changing result_set.

 write result_set->stringify( ).
 new-line.

 write |𝑷( { set2->stringify( ) } ) ->|.
 perform build_powerset using set2
                        changing result_set.

 write result_set->stringify( ).
 new-line.

 write |𝑷( { set3->stringify( ) } ) ->|.
 perform build_powerset using set3
                        changing result_set.

 write result_set->stringify( ).
 new-line.

form build_powerset using set type ref to set

                   changing powerset type ref to set.

 data(powerset_elements) = value stringtab( ( `` ) ).

 loop at set->get_elements( ) into data(element).
   do lines( powerset_elements ) times.
     if powerset_elements[ sy-index ] ne `∅`.
       append |{ powerset_elements[ sy-index ] }{ element }| to powerset_elements.
     else.
       append element to powerset_elements.
     endif.
   enddo.
 endloop.

 powerset = new set( powerset_elements ).

endform. </lang>

Output:

𝑷( ∅ ) -> { ∅ }
𝑷( { ∅ } ) -> { ∅, {∅} }
𝑷( { 1, 2, 3, 4 } ) -> { ∅, 1, 2, {12}, 3, {13}, {23}, {123}, 4, {14}, {24}, {124}, {34}, {134}, {234}, {1234} }

A solution (without recursion) that prints the power set of the n arguments passed by the command line. The idea is that the i'th bit of a natural between 0 and $2^{n}-1$ indicates whether or not we should put the i'th element of the command line inside the set.

<lang Ada> with Ada.Text_IO, Ada.Command_Line; use Ada.Text_IO, Ada.Command_Line;

procedure powerset is

 procedure print_subset (set : natural) is

-- each i'th binary digit of "set" indicates if the i'th integer belongs to "set" or not.

   k : natural := set;
   first : boolean := true;
 begin
   Put ("{");
   for i in 1..Argument_Count loop
     if k mod 2 = 1 then 
       if first then 
         first := false;
       else 
         Put (",");
       end if;        
       Put (Argument (i));
     end if;
     k := k / 2; -- we go to the next bit of "set"
   end loop;
   Put_Line("}");
 end print_subset;

begin

 for i in 0..2**Argument_Count-1 loop
     print_subset (i);
 end loop;

end powerset; </lang>

Output:

>./powerset a b c d
{}
{a}
{b}
{a,b}
{c}
{a,c}
{b,c}
{a,b,c}
{d}
{a,d}
{b,d}
{a,b,d}
{c,d}
{a,c,d}
{b,c,d}
{a,b,c,d}

ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

Requires: ALGOL 68g mk14.1+ <lang algol68>MODE MEMBER = INT;

PROC power set = ([]MEMBER s)[][]MEMBER:(

 [2**UPB s]FLEX[1:0]MEMBER r;
 INT upb r := 0;
 r[upb r +:= 1] := []MEMBER(());
 FOR i TO UPB s DO
   MEMBER e = s[i];
   FOR j TO upb r DO
     [UPB r[j] + 1]MEMBER x;
     x[:UPB x-1] := r[j];
     x[UPB x] := e; # append to the end of x #
     r[upb r +:= 1] := x # append to end of r #
   OD
 OD;
 r[upb r] := s;
 r

);

Example: #

test:(

 [][]MEMBER set = power set((1, 2, 4));
 FOR member TO UPB set DO
   INT upb = UPB set[member];
   FORMAT repr set = $"("f( upb=0 | $$ | $n(upb-1)(d", ")d$ )");"$;
   printf(($"set["d"] = "$,member, repr set, set[member],$l$))
 OD

)</lang>

Output:

set[1] = ();
set[2] = (1);
set[3] = (2);
set[4] = (1, 2);
set[5] = (4);
set[6] = (1, 4);
set[7] = (2, 4);
set[8] = (1, 2, 4);

AppleScript

Translation of: JavaScript

(functional composition examples)

Translation of: Haskell

<lang AppleScript>-- POWER SET -----------------------------------------------------------------

-- powerset :: [a] -> a on powerset(xs)

   script subSet
       on |λ|(acc, x)
           script cons
               on |λ|(y)
                   {x} & y
               end |λ|
           end script
           
           acc & map(cons, acc)
       end |λ|
   end script
   
   foldr(subSet, {{}}, xs)

end powerset

-- TEST ---------------------------------------------------------------------- on run

   script test
       on |λ|(x)
           set {setName, setMembers} to x
           {setName, powerset(setMembers)}
       end |λ|
   end script
   
   map(test, [¬
       ["Set [1,2,3]", {1, 2, 3}], ¬
       ["Empty set", {}], ¬
       ["Set containing only empty set", {{}}]])
   
   --> {{"Set [1,2,3]", {{}, {3}, {2}, {2, 3}, {1}, {1, 3}, {1, 2}, {1, 2, 3}}}, 
   -->  {"Empty set", {{}}}, 
   -->  {"Set containing only empty set", {{}, {{}}}}}

end run

-- GENERIC FUNCTIONS ---------------------------------------------------------

-- foldr :: (a -> b -> a) -> a -> [b] -> a on foldr(f, startValue, xs)

   tell mReturn(f)
       set v to startValue
       set lng to length of xs
       repeat with i from lng to 1 by -1
           set v to |λ|(v, item i of xs, i, xs)
       end repeat
       return v
   end tell

end foldr

-- map :: (a -> b) -> [a] -> [b] on map(f, xs)

   tell mReturn(f)
       set lng to length of xs
       set lst to {}
       repeat with i from 1 to lng
           set end of lst to |λ|(item i of xs, i, xs)
       end repeat
       return lst
   end tell

end map

-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)

   if class of f is script then
       f
   else
       script
           property |λ| : f
       end script
   end if

end mReturn</lang>

Output:

<lang AppleScript>{{"Set [1,2,3]", {{}, {3}, {2}, {2, 3}, {1}, {1, 3}, {1, 2}, {1, 2, 3}}},

{"Empty set", {{}}}, 
{"Set containing only empty set", {{}, {{}}}}}</lang>

ATS

<lang> (* ****** ****** *) //

include

"share/atspre_define.hats" // defines some names

include

"share/atspre_staload.hats" // for targeting C

include

"share/HATS/atspre_staload_libats_ML.hats" // for ... // (* ****** ****** *) // extern fun Power_set(xs: list0(int)): void // (* ****** ****** *)

// Helper: fast power function. fun power(n: int, p: int): int = if p = 1 then n else if p = 0 then 1 else if p % 2 = 0 then power(n*n, p/2) else n * power(n, p-1)

fun print_list(list: list0(int)): void =

 case+ list of
 | nil0() => println!(" ")
 | cons0(car, crd) =>
   let
     val () = begin print car; print ','; end
     val () = print_list(crd)
   in
   end

fun get_list_length(list: list0(int), length: int): int =

 case+ list of
 | nil0() => length
 | cons0(car, crd) => get_list_length(crd, length+1)

fun get_list_from_bit_mask(mask: int, list: list0(int), result: list0(int)): list0(int) =

 if mask = 0 then result
 else
   case+ list of
   | nil0() => result
   | cons0(car, crd) =>
     let
       val current: int = mask % 2
     in
       if current = 0 then
         get_list_from_bit_mask(mask >> 1, crd, result)
       else
         get_list_from_bit_mask(mask >> 1, crd, list0_cons(car, result))
     end

implement Power_set(xs) = let

 val len: int = get_list_length(xs, 0)
 val pow: int = power(2, len)
 fun loop(mask: int, list: list0(int)): void =
   if mask > 0 && mask >= pow then ()
   else
     let
       val () = print_list(get_list_from_bit_mask(mask, list, list0_nil()))
     in
       loop(mask+1, list)
     end
 in
   loop(0, xs)
 end

(* ****** ****** *)

implement main0() = let

 val xs: list0(int) = cons0(1, list0_pair(2, 3))

in

 Power_set(xs)

end (* end of [main0] *)

(* ****** ****** *) </lang>

AutoHotkey

ahk discussion <lang autohotkey>a = 1,a,-- ; elements separated by commas StringSplit a, a, `, ; a0 = #elements, a1,a2,... = elements of the set

t = { Loop % (1<<a0) { ; generate all 0-1 sequences

  x := A_Index-1
  Loop % a0
     t .= (x>>A_Index-1) & 1 ? a%A_Index% "," : ""
  t .= "}`n{"         ; new subsets in new lines

} MsgBox % RegExReplace(SubStr(t,1,StrLen(t)-1),",}","}")</lang>

AWK

<lang AWK>cat power_set.awk

!/usr/local/bin/gawk -f

User defined function

function tochar(l,n, r) {

while (l) { n--; if (l%2 != 0) r = r sprintf(" %c ",49+n); l = int(l/2) }; return r

}

For each input

{ for (i=0;i<=2^NF-1;i++) if (i == 0) printf("empty\n"); else printf("(%s)\n",tochar(i,NF)) } </lang>

Output:

$ gawk -f power_set.awk 
1 2 3 4
empty
( 4 )
( 3 )
( 4  3 )
( 2 )
( 4  2 )
( 3  2 )
( 4  3  2 )
( 1 )
( 4  1 )
( 3  1 )
( 4  3  1 )
( 2  1 )
( 4  2  1 )
( 3  2  1 )
( 4  3  2  1 )

BBC BASIC

The elements of a set are represented as the bits in an integer (hence the maximum size of set is 32). <lang bbcbasic> DIM list$(3) : list$() = "1", "2", "3", "4"

     PRINT FNpowerset(list$())
     END
     
     DEF FNpowerset(list$())
     IF DIM(list$(),1) > 31 ERROR 100, "Set too large to represent as integer"
     LOCAL i%, j%, s$
     s$ = "{"
     FOR i% = 0 TO (2 << DIM(list$(),1)) - 1
       s$ += "{"
       FOR j% = 0 TO DIM(list$(),1)
         IF i% AND (1 << j%) s$ += list$(j%) + ","
       NEXT
       IF RIGHT$(s$) = "," s$ = LEFT$(s$)
       s$ += "},"
     NEXT i%
     = LEFT$(s$) + "}"</lang>

Output:

{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}

Bracmat

<lang bracmat>( ( powerset

 =   done todo first
   .   !arg:(?done.?todo)
     & (   !todo:%?first ?todo
         & (powerset$(!done !first.!todo),powerset$(!done.!todo))
       | !done
       )
 )

& out$(powerset$(.1 2 3 4)) );</lang>

Output:

Burlesque

C

<lang c>#include <stdio.h>

struct node { char *s; struct node* prev; };

void powerset(char **v, int n, struct node *up) { struct node me;

if (!n) { putchar('['); while (up) { printf(" %s", up->s); up = up->prev; } puts(" ]"); } else { me.s = *v; me.prev = up; powerset(v + 1, n - 1, up); powerset(v + 1, n - 1, &me); } }

int main(int argc, char **argv) { powerset(argv + 1, argc - 1, 0); return 0; }</lang>

Output:

% ./a.out 1 2 3
[ ]
[ 3 ]
[ 2 ]
[ 3 2 ]
[ 1 ]
[ 3 1 ]
[ 2 1 ]
[ 3 2 1 ]

C++

Non-recursive version

<lang cpp>#include <iostream>

include <set>
include <vector>
include <iterator>
include <algorithm>

typedef std::set<int> set_type; typedef std::set<set_type> powerset_type;

powerset_type powerset(set_type const& set) {

 typedef set_type::const_iterator set_iter;
 typedef std::vector<set_iter> vec;
 typedef vec::iterator vec_iter;

 struct local
 {
   static int dereference(set_iter v) { return *v; }
 };

 powerset_type result;

 vec elements;
 do
 {
   set_type tmp;
   std::transform(elements.begin(), elements.end(),
                  std::inserter(tmp, tmp.end()),
                  local::dereference);
   result.insert(tmp);
   if (!elements.empty() && ++elements.back() == set.end())
   {
     elements.pop_back();
   }
   else
   {
     set_iter iter;
     if (elements.empty())
     {
       iter = set.begin();
     }
     else
     {
       iter = elements.back();
       ++iter;
     }
     for (; iter != set.end(); ++iter)
     {
       elements.push_back(iter);
     }
   }
 } while (!elements.empty());

 return result;

}

int main() {

 int values[4] = { 2, 3, 5, 7 };
 set_type test_set(values, values+4);

 powerset_type test_powerset = powerset(test_set);

 for (powerset_type::iterator iter = test_powerset.begin();
      iter != test_powerset.end();
      ++iter)
 {
   std::cout << "{ ";
   char const* prefix = "";
   for (set_type::iterator iter2 = iter->begin();
        iter2 != iter->end();
        ++iter2)
   {
     std::cout << prefix << *iter2;
     prefix = ", ";
   }
   std::cout << " }\n";
 }

}</lang>

Output:

{  }
{ 2 }
{ 2, 3 }
{ 2, 3, 5 }
{ 2, 3, 5, 7 }
{ 2, 3, 7 }
{ 2, 5 }
{ 2, 5, 7 }
{ 2, 7 }
{ 3 }
{ 3, 5 }
{ 3, 5, 7 }
{ 3, 7 }
{ 5 }
{ 5, 7 }
{ 7 }

C++14 version

This simplified version has identical output to the previous code.

include <set>
include <iostream>

template <class S> auto powerset(const S& s) {

   std::set ret;
   ret.emplace();
   for (auto&& e: s) {
       std::set rs;
       for (auto x: ret) {
           x.insert(e);
           rs.insert(x);
       }
       ret.insert(begin(rs), end(rs));
   }
   return ret;

}
int main() {
~~std::set<int> s = {2, 3, 5, 7}; auto pset = powerset(s);~~
for (auto&& subset: pset) { std::cout << "{ "; char const* prefix = ""; for (auto&& e: subset) { std::cout << prefix << e; prefix = ", "; } std::cout << " }\n"; }
~~} </lang>~~
~~Recursive version~~
~~<lang cpp>#include <iostream>~~
~~include <set>~~
~~template<typename Set> std::set<Set> powerset(const Set& s, size_t n) {~~
typedef typename Set::const_iterator SetCIt; typedef typename std::set<Set>::const_iterator PowerSetCIt; std::set<Set> res; if(n > 0) { std::set<Set> ps = powerset(s, n-1); for(PowerSetCIt ss = ps.begin(); ss != ps.end(); ss++) for(SetCIt el = s.begin(); el != s.end(); el++) { Set subset(*ss); subset.insert(*el); res.insert(subset); } res.insert(ps.begin(), ps.end()); } else res.insert(Set()); return res;
~~} template<typename Set> std::set<Set> powerset(const Set& s) {~~
~~return powerset(s, s.size());~~
~~} </lang>~~
C#
~~<lang csharp> public IEnumerable<IEnumerable<T>> GetPowerSet<T>(List<T> list) {~~
return from m in Enumerable.Range(0, 1 << list.Count) select from i in Enumerable.Range(0, list.Count) where (m & (1 << i)) != 0 select list[i];
}
public void PowerSetofColors() {
var colors = new List<KnownColor> { KnownColor.Red, KnownColor.Green, KnownColor.Blue, KnownColor.Yellow }; var result = GetPowerSet(colors); Console.Write( string.Join( Environment.NewLine, result.Select(subset => string.Join(",", subset.Select(clr => clr.ToString()).ToArray())).ToArray()));
}
</lang>

~~Output:~~
Red Green Red,Green Blue Red,Blue Green,Blue Red,Green,Blue Yellow Red,Yellow Green,Yellow Red,Green,Yellow Blue,Yellow Red,Blue,Yellow Green,Blue,Yellow Red,Green,Blue,Yellow
An alternative implementation for an arbitrary number of elements:
<lang csharp>
~~public IEnumerable<IEnumerable<T>> GetPowerSet<T>(IEnumerable<T> input) { var seed = new List<IEnumerable<T>>() { Enumerable.Empty<T>() } as IEnumerable<IEnumerable<T>>;~~
~~return input.Aggregate(seed, (a, b) => a.Concat(a.Select(x => x.Concat(new List<T>() { b })))); }~~
</lang>

Non-recursive version
<lang csharp>
using System; class Powerset { static int count = 0, n = 4; static int [] buf = new int [n]; static void Main() { int ind = 0; int n_1 = n - 1; for (;;) { for (int i = 0; i <= ind; ++i) Console.Write("{0, 2}", buf [i]); Console.WriteLine(); count++; if (buf [ind] < n_1) { ind++; buf [ind] = buf [ind - 1] + 1; } else if (ind > 0) { ind--; buf [ind]++; } else break; } Console.WriteLine("n=" + n + " count=" + count); } }
~~</lang>~~

~~Recursive version <lang csharp> using System; class Powerset {~~
~~static int n = 4; static int [] buf = new int [n];~~
~~static void Main() { rec(0, 0); }~~
static void rec(int ind, int begin) { for (int i = begin; i < n; i++) { buf [ind] = i; for (int j = 0; j <= ind; j++) Console.Write("{0, 2}", buf [j]); Console.WriteLine(); rec(ind + 1, buf [ind] + 1); } }
~~}</lang>~~
~~Clojure~~
<lang Clojure>(use '[clojure.math.combinatorics :only [subsets] ])
(def S #{1 2 3 4})
user> (subsets S) (() (1) (2) (3) (4) (1 2) (1 3) (1 4) (2 3) (2 4) (3 4) (1 2 3) (1 2 4) (1 3 4) (2 3 4) (1 2 3 4))</lang>
Alternate solution, with no dependency on third-party library: <lang Clojure>(defn powerset [coll]
~~(reduce (fn [a x] (into a (map #(conj % x)) a)) #{#{}} coll))~~
~~(powerset #{1 2 3})</lang> <lang Clojure>#{#{} #{1} #{2} #{1 2} #{3} #{1 3} #{2 3} #{1 2 3}}</lang>~~
~~CoffeeScript~~
~~<lang coffeescript> print_power_set = (arr) ->~~
~~console.log "POWER SET of #{arr}" for subset in power_set(arr) console.log subset~~
~~power_set = (arr) ->~~
result = [] binary = (false for elem in arr) n = arr.length while binary.length <= n result.push bin_to_arr binary, arr i = 0 while true if binary[i] binary[i] = false i += 1 else binary[i] = true break binary[i] = true result
~~bin_to_arr = (binary, arr) ->~~
~~(arr[i] for i of binary when binary[arr.length - i - 1])~~
~~print_power_set [] print_power_set [4, 2, 1] print_power_set ['dog', 'c', 'b', 'a'] </lang>~~

~~Output:~~
<lang> > coffee power_set.coffee POWER SET of [] POWER SET of 4,2,1 [] [ 1 ] [ 2 ] [ 2, 1 ] [ 4 ] [ 4, 1 ] [ 4, 2 ] [ 4, 2, 1 ] POWER SET of dog,c,b,a [] [ 'a' ] [ 'b' ] [ 'b', 'a' ] [ 'c' ] [ 'c', 'a' ] [ 'c', 'b' ] [ 'c', 'b', 'a' ] [ 'dog' ] [ 'dog', 'a' ] [ 'dog', 'b' ] [ 'dog', 'b', 'a' ] [ 'dog', 'c' ] [ 'dog', 'c', 'a' ] [ 'dog', 'c', 'b' ] [ 'dog', 'c', 'b', 'a' ] </lang>
~~ColdFusion~~
~~Port from the JavaScript version, compatible with ColdFusion 8+ or Railo 3+ <lang javascript>public array function powerset(required array data) {~~
var ps = [""]; var d = arguments.data; var lenData = arrayLen(d); var lenPS = 0; for (var i=1; i LTE lenData; i++) { lenPS = arrayLen(ps); for (var j = 1; j LTE lenPS; j++) { arrayAppend(ps, listAppend(ps[j], d[i])); } } return ps;
}
var res = powerset([1,2,3,4]);</lang>

~~Output:~~
~~["","1","2","1,2","3","1,3","2,3","1,2,3","4","1,4","2,4","1,2,4","3,4","1,3,4","2,3,4","1,2,3,4"]~~
~~Common Lisp~~
~~<lang lisp>(defun powerset (s)~~
~~(if s (mapcan (lambda (x) (list (cons (car s) x) x)) (powerset (cdr s))) '(())))</lang>~~

~~Output:~~
~~> (powerset '(l i s p)) ((L I S P) (I S P) (L S P) (S P) (L I P) (I P) (L P) (P) (L I S) (I S) (L S) (S) (L I) (I) (L) NIL)~~
~~<lang lisp>(defun power-set (s)~~
(reduce #'(lambda (item ps) (append (mapcar #'(lambda (e) (cons item e)) ps) ps)) s :from-end t :initial-value '(())))</lang>

~~Output:~~
~~>(power-set '(1 2 3)) ((1 2 3) (1 2) (1 3) (1) (2 3) (2) (3) NIL)~~

~~Alternate, more recursive (same output): <lang lisp>(defun powerset (l)~~
~~(if (null l) (list nil) (let ((prev (powerset (cdr l))))~~
(append (mapcar #'(lambda (elt) (cons (car l) elt)) prev) prev))))</lang>

Imperative-style using LOOP: <lang lisp>(defun powerset (xs)
~~(loop for i below (expt 2 (length xs)) collect (loop for j below i for x in xs if (logbitp j i) collect x)))</lang>~~

~~Output:~~
~~>(powerset '(1 2 3) (NIL (1) (2) (1 2) (3) (1 3) (2 3) (1 2 3))~~
~~Yet another imperative solution, this time with dolist. <lang lisp>(defun power-set (list)~~
~~(let ((pow-set (list nil))) (dolist (element (reverse list) pow-set) (dolist (set pow-set) (push (cons element set) pow-set)))))</lang>~~

~~Output:~~
~~>(power-set '(1 2 3)) ((1) (1 3) (1 2 3) (1 2) (2) (2 3) (3) NIL)~~
D
This implementation defines a range which *lazily* enumerates the power set.
<lang d>import std.algorithm; import std.range;
auto powerSet(R)(R r) { return (1L<<r.length) .iota .map!(i => r.enumerate .filter!(t => (1<<t[0]) & i) .map!(t => t[1]) ); }
unittest { int[] emptyArr; assert(emptyArr.powerSet.equal!equal([emptyArr])); assert(emptyArr.powerSet.powerSet.equal!(equal!equal)([[], [emptyArr]])); }
void main(string[] args) { import std.stdio; args[1..$].powerSet.each!writeln; }</lang>
An alternative version, which implements the range construct from scratch:
<lang d>import std.range;
struct PowerSet(R) if (isRandomAccessRange!R) { R r; size_t position;
struct PowerSetItem { R r; size_t position;
private void advance() { while (!(position & 1)) { r.popFront(); position >>= 1; } }
@property bool empty() { return position == 0; } @property auto front() { advance(); return r.front; } void popFront() { advance(); r.popFront(); position >>= 1; } }
@property bool empty() { return position == (1 << r.length); } @property PowerSetItem front() { return PowerSetItem(r.save, position); } void popFront() { position++; } }
auto powerSet(R)(R r) { return PowerSet!R(r); }</lang>

~~Output:~~
~~$ rdmd powerset a b c [] ["a"] ["b"] ["a", "b"] ["c"] ["a", "c"] ["b", "c"] ["a", "b", "c"]~~
~~Déjà Vu~~
In Déjà Vu, sets are dictionaries with all values true and the default set to false.
<lang dejavu>powerset s: local :out [ set{ } ] for value in keys s: for subset in copy out: local :subset+1 copy subset set-to subset+1 value true push-to out subset+1 out
!. powerset set{ 1 2 3 4 }</lang>

~~Output:~~
~~[ set{ } set{ 4 } set{ 3 4 } set{ 3 } set{ 2 3 } set{ 2 3 4 } set{ 2 4 } set{ 2 } set{ 1 2 } set{ 1 2 4 } set{ 1 2 3 4 } set{ 1 2 3 } set{ 1 3 } set{ 1 3 4 } set{ 1 4 } set{ 1 } ]~~
E
<lang e>pragma.enable("accumulator")
def powerset(s) {
~~return accum [].asSet() for k in 0..!2**s.size() { _.with(accum [].asSet() for i ? ((2**i & k) > 0) => elem in s { _.with(elem) }) }~~
}</lang>
It would also be possible to define an object which is the powerset of a provided set without actually instantiating all of its members immediately.
~~EchoLisp~~
~~<lang scheme> (define (set-cons a A)~~
~~(make-set (cons a A)))~~
~~(define (power-set e)~~
~~(cond ((null? e) (make-set (list ∅))) (else (let [(ps (power-set (cdr e)))] (make-set (append ps (map set-cons (circular-list (car e)) ps)))))))~~
~~(define B (make-set ' ( 🍎 🍇 🎂 🎄 ))) (power-set B)~~
→ { ∅ { 🍇 } { 🍇 🍎 } { 🍇 🍎 🎂 } { 🍇 🍎 🎂 🎄 } { 🍇 🍎 🎄 } { 🍇 🎂 } { 🍇 🎂 🎄 } { 🍇 🎄 } { 🍎 } { 🍎 🎂 } { 🍎 🎂 🎄 } { 🍎 🎄 } { 🎂 } { 🎂 🎄 } { 🎄 } }
~~The Von Neumann universe~~
~~(define V0 (power-set null)) ;; null and ∅ are the same~~
~~→ { ∅ }~~
~~(define V1 (power-set V0))~~
~~→ { ∅ { ∅ } }~~
~~(define V2 (power-set V1))~~
~~→ { ∅ { ∅ } { ∅ { ∅ } } { { ∅ } } }~~
~~(define V3 (power-set V2))~~
~~→ { ∅ { ∅ } { ∅ { ∅ } } …🔃 )~~
~~(length V3) → 16 (define V4 (power-set V3)) (length V4) → 65536~~
length V5 = 2^65536

out of bounds

~~</lang>~~
~~Elixir~~
Translation of: Erlang
~~<lang elixir>defmodule RC do~~
use Bitwise def powerset1(list) do n = length(list) max = round(:math.pow(2,n)) for i <- 0..max-1, do: (for pos <- 0..n-1, band(i, bsl(1, pos)) != 0, do: Enum.at(list, pos) ) end def powerset2([]), do: [[]] def powerset2([h|t]) do pt = powerset2(t) (for x <- pt, do: [h|x]) ++ pt end def powerset3([]), do: [[]] def powerset3([h|t]) do pt = powerset3(t) powerset3(h, pt, pt) end defp powerset3(_, [], acc), do: acc defp powerset3(x, [h|t], acc), do: powerset3(x, t, [[x|h] | acc])
end
IO.inspect RC.powerset1([1,2,3]) IO.inspect RC.powerset2([1,2,3]) IO.inspect RC.powerset3([1,2,3]) IO.inspect RC.powerset1([]) IO.inspect RC.powerset1(["one"])</lang>

~~Output:~~
~~[[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] [[1, 2, 3], [1, 2], [1, 3], [1], [2, 3], [2], [3], []] [[1], [1, 3], [1, 2, 3], [1, 2], [2], [2, 3], [3], []] [[]] [[], ["one"]]~~
~~Erlang~~
~~Generates all subsets of a list with the help of binary:~~
For [1 2 3]: [ ] | 0 0 0 | 0 [ 3] | 0 0 1 | 1 [ 2 ] | 0 1 0 | 2 [ 2 3] | 0 1 1 | 3 [1 ] | 1 0 0 | 4 [1 3] | 1 0 1 | 5 [1 2 ] | 1 1 0 | 6 [1 2 3] | 1 1 1 | 7 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
~~<lang erlang>powerset(Lst) ->~~
~~N = length(Lst), Max = trunc(math:pow(2,N)), [[lists:nth(Pos+1,Lst) || Pos <- lists:seq(0,N-1), I band (1 bsl Pos) =/= 0] || I <- lists:seq(0,Max-1)].</lang>~~

~~Output:~~
[[], [1], [2], [1,2], [3], [1,3], [2,3], [1,2,3], [4], [1,4], [2,4], [1,2,4], [3,4], [1,3,4], [2,3,4], [1,2,3,4]]
Alternate shorter and more efficient version: <lang erlang>powerset([]) -> [[]]; powerset([H|T]) -> PT = powerset(T),
~~[ [H|X] || X <- PT ] ++ PT.</lang>~~
~~or even more efficient version: <lang erlang>powerset([]) -> [[]]; powerset([H|T]) -> PT = powerset(T),~~
~~powerset(H, PT, PT).~~
~~powerset(_, [], Acc) -> Acc; powerset(X, [H|T], Acc) -> powerset(X, T, [[X|H]|Acc]).</lang>~~
F#
almost exact copy of OCaml version <lang fsharp> let subsets xs = List.foldBack (fun x rest -> rest @ List.map (fun ys -> x::ys) rest) xs [[]] </lang>
alternatively with list comprehension
<lang fsharp> let rec pow =
~~function | [] -> [[]] | x::xs -> [for i in pow xs do yield! [i;x::i]]~~
~~</lang>~~
~~Factor~~
We use hash sets, denoted by HS{ } brackets, for our sets. members converts from a set to a sequence, and <hash-set> converts back. <lang factor>USING: kernel prettyprint sequences arrays sets hash-sets ; IN: powerset
add ( set elt -- newset ) 1array <hash-set> union ;

powerset ( set -- newset ) members { HS{ } } [ dupd [ add ] curry map append ] reduce <hash-set> ;</lang>
~~Usage: <lang factor>( scratchpad ) HS{ 1 2 3 4 } powerset . HS{~~
HS{ 1 2 3 4 } HS{ 1 2 } HS{ 1 3 } HS{ 2 3 } HS{ 1 2 3 } HS{ 1 4 } HS{ 2 4 } HS{ } HS{ 1 } HS{ 2 } HS{ 3 } HS{ 4 } HS{ 1 2 4 } HS{ 3 4 } HS{ 1 3 4 } HS{ 2 3 4 }
~~}</lang>~~
~~Fōrmulæ~~
In this page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
~~Forth~~
Works with: 4tH version 3.61.0

.

Translation of: C
~~<lang forth>: ?print dup 1 and if over args type space then ;~~
.set begin dup while ?print >r 1+ r> 1 rshift repeat drop drop ;

.powerset 0 do ." ( " 1 i .set ." )" cr loop ;

check-none dup 2 < abort" Usage: powerset [val] .. [val]" ;

check-size dup /cell 8 [*] >= abort" Set too large" ;

powerset 1 argn check-none check-size 1- lshift .powerset ;
~~powerset</lang>~~

~~Output:~~
~~$ 4th cxq powerset.4th 1 2 3 4 ( ) ( 1 ) ( 2 ) ( 1 2 ) ( 3 ) ( 1 3 ) ( 2 3 ) ( 1 2 3 ) ( 4 ) ( 1 4 ) ( 2 4 ) ( 1 2 4 ) ( 3 4 ) ( 1 3 4 ) ( 2 3 4 ) ( 1 2 3 4 )~~
~~Frink~~
Frink's set and array classes have built-in subsets[] methods that return all subsets. If called with an array, the results are arrays. If called with a set, the results are sets. <lang frink> a = new set[1,2,3,4] a.subsets[] </lang>
~~FunL~~
FunL uses Scala type scala.collection.immutable.Set as it's set type, which has a built-in method subsets returning an (Scala) iterator over subsets.
<lang funl>def powerset( s ) = s.subsets().toSet()</lang>
The powerset function could be implemented in FunL directly as:
<lang funl>def
~~powerset( {} ) = {{}} powerset( s ) = acc = powerset( s.tail() ) acc + map( x -> {s.head()} + x, acc )</lang>~~
or, alternatively as: <lang funl>import lists.foldr
def powerset( s ) = foldr( \x, acc -> acc + map( a -> {x} + a, acc), {{}}, s )
println( powerset({1, 2, 3, 4}) )</lang>

~~Output:~~
~~{{}, {4}, {1, 2}, {1, 3}, {2, 3, 4}, {3}, {1, 2, 3, 4}, {1, 4}, {1, 2, 3}, {2}, {1, 2, 4}, {1}, {3, 4}, {2, 3}, {2, 4}, {1, 3, 4}}~~
~~GAP~~
~~<lang gap># Built-in Combinations([1, 2, 3]);~~
~~[ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 3 ], [ 2 ], [ 2, 3 ], [ 3 ] ]~~
~~Note that it handles duplicates~~
~~Combinations([1, 2, 3, 1]);~~
[ [ ], [ 1 ], [ 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 2, 3 ], [ 1, 1, 3 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 3 ],

[ 2 ], [ 2, 3 ], [ 3 ] ]</lang>
Go
No native set type in Go. While the associative array trick mentioned in the task description works well in Go in most situations, it does not work here because we need sets of sets, and converting a general set to a hashable value for a map key is non-trivial.
Instead, this solution uses a simple (non-associative) slice as a set representation. To ensure uniqueness, the element interface requires an equality method, which is used by the set add method. Adding elements with the add method ensures the uniqueness property.
While the "add" and "has" methods make a usable set type, the power set method implemented here computes a result directly without using the add method. The algorithm ensures that the result will be a valid set as long as the input is a valid set. This allows the more efficient append function to be used. <lang go>package main
import (
~~"fmt" "strconv" "strings"~~
)
// types needed to implement general purpose sets are element and set
// element is an interface, allowing different kinds of elements to be // implemented and stored in sets. type elem interface {
// an element must be distinguishable from other elements to satisfy // the mathematical definition of a set. a.eq(b) must give the same // result as b.eq(a). Eq(elem) bool // String result is used only for printable output. Given a, b where // a.eq(b), it is not required that a.String() == b.String(). fmt.Stringer
}
// integer type satisfying element interface type Int int
func (i Int) Eq(e elem) bool {
~~j, ok := e.(Int) return ok && i == j~~
}
func (i Int) String() string {
~~return strconv.Itoa(int(i))~~
}
// a set is a slice of elem's. methods are added to implement // the element interface, to allow nesting. type set []elem
// uniqueness of elements can be ensured by using add method func (s *set) add(e elem) {
~~if !s.has(e) { *s = append(*s, e) }~~
}
func (s *set) has(e elem) bool {
~~for _, ex := range *s { if e.Eq(ex) { return true } } return false~~
}
func (s set) ok() bool {
~~for i, e0 := range s { for _, e1 := range s[i+1:] { if e0.Eq(e1) { return false } } } return true~~
}
// elem.Eq func (s set) Eq(e elem) bool {
t, ok := e.(set) if !ok { return false } if len(s) != len(t) { return false } for _, se := range s { if !t.has(se) { return false } } return true
}
// elem.String func (s set) String() string {
if len(s) == 0 { return "∅" } var buf strings.Builder buf.WriteRune('{') for i, e := range s { if i > 0 { buf.WriteRune(',') } buf.WriteString(e.String()) } buf.WriteRune('}') return buf.String()
}
// method required for task func (s set) powerSet() set {
r := set{set{}} for _, es := range s { var u set for _, er := range r { er := er.(set) u = append(u, append(er[:len(er):len(er)], es)) } r = append(r, u...) } return r
}
func main() {
var s set for _, i := range []Int{1, 2, 2, 3, 4, 4, 4} { s.add(i) } fmt.Println(" s:", s, "length:", len(s)) ps := s.powerSet() fmt.Println(" 𝑷(s):", ps, "length:", len(ps))
fmt.Println("\n(extra credit)") var empty set fmt.Println(" empty:", empty, "len:", len(empty)) ps = empty.powerSet() fmt.Println(" 𝑷(∅):", ps, "len:", len(ps)) ps = ps.powerSet() fmt.Println("𝑷(𝑷(∅)):", ps, "len:", len(ps))
fmt.Println("\n(regression test for earlier bug)") s = set{Int(1), Int(2), Int(3), Int(4), Int(5)} fmt.Println(" s:", s, "length:", len(s), "ok:", s.ok()) ps = s.powerSet() fmt.Println(" 𝑷(s):", "length:", len(ps), "ok:", ps.ok()) for _, e := range ps { if !e.(set).ok() { panic("invalid set in ps") } }
~~}</lang>~~

~~Output:~~
s: {1,2,3,4} length: 4 𝑷(s): {∅,{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}} length: 16 (extra credit) empty: ∅ len: 0 𝑷(∅): {∅} len: 1 𝑷(𝑷(∅)): {∅,{∅}} len: 2 (regression test for earlier bug) s: {1,2,3,4,5} length: 5 ok: true 𝑷(s): length: 32 ok: true
~~Groovy~~
Builds on the Combinations solution. Sets are not a "natural" collection type in Groovy. Lists are much more richly supported. Thus, this solution is liberally sprinkled with coercion from Set to List and from List to Set. <lang groovy>def comb comb = { m, List list ->
def n = list.size() m == 0 ? [[]] : (0..(n-m)).inject([]) { newlist, k -> def sublist = (k+1 == n) ? [] : list[(k+1)..<n] newlist += comb(m-1, sublist).collect { [list[k]] + it } }
}
def powerSet = { set ->
~~(0..(set.size())).inject([]){ list, i -> list + comb(i,set as List)}.collect { it as LinkedHashSet } as LinkedHashSet~~
}</lang>
Test program: <lang groovy>def vocalists = [ "C", "S", "N", "Y" ] as LinkedHashSet println "${vocalists}" println powerSet(vocalists)</lang>

~~Output:~~
~~[C, S, N, Y] [[], [C], [S], [N], [Y], [C, S], [C, N], [C, Y], [S, N], [S, Y], [N, Y], [C, S, N], [C, S, Y], [C, N, Y], [S, N, Y], [C, S, N, Y]]~~
Note: In this example, LinkedHashSet was used throughout for Set coercion. This is because LinkedHashSet preserves the order of input, like a List. However, if order does not matter you could replace all references to LinkedHashSet with Set.
~~Haskell~~
<lang Haskell>import Data.Set import Control.Monad
powerset :: Ord a => Set a -> Set (Set a) powerset = fromList . fmap fromList . listPowerset . toList
listPowerset :: [a] -> a listPowerset = filterM (const [True, False])</lang> listPowerset describes the result as all possible (using the list monad) filterings (using filterM) of the input list, regardless (using const) of each item's value. powerset simply converts the input and output from lists to sets.
Alternate Solution <lang Haskell>powerset [] = [[]] powerset (head:tail) = acc ++ map (head:) acc where acc = powerset tail</lang> or <lang Haskell>powerSet :: [a] -> a powerSet = foldr (\x acc -> acc ++ map (x:) acc) [[]]</lang>
which could also be understood, in point-free terms, as: <lang haskell>powerSet :: [a] -> a powerSet = foldr ((mappend <*>) . fmap . (:)) (pure [])</lang>
Examples:
*Main> listPowerset [1,2,3] [[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]] *Main> powerset (Data.Set.fromList [1,2,3]) {{},{1},{1,2},{1,2,3},{1,3},{2},{2,3},{3}}
Works with: GHC version 6.10
~~Prelude> import Data.List Prelude Data.List> subsequences [1,2,3] [[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]~~
Alternate solution
A method using only set operations and set mapping is also possible. Ideally, Set would be defined as a Monad, but that's impossible given the constraint that the type of inputs to Set.map (and a few other functions) be ordered. <lang Haskell>import qualified Data.Set as Set type Set=Set.Set unionAll :: (Ord a) => Set (Set a) -> Set a unionAll = Set.fold Set.union Set.empty
--slift is the analogue of liftA2 for sets. slift :: (Ord a, Ord b, Ord c) => (a->b->c) -> Set a -> Set b -> Set c slift f s0 s1 = unionAll (Set.map (\e->Set.map (f e) s1) s0)
--a -> {{},{a}} makeSet :: (Ord a) => a -> Set (Set a) makeSet = (Set.insert Set.empty) . Set.singleton.Set.singleton
powerSet :: (Ord a) => Set a -> Set (Set a) powerSet = (Set.fold (slift Set.union) (Set.singleton Set.empty)) . Set.map makeSet</lang> Usage: <lang Haskell> Prelude Data.Set> powerSet fromList [1,2,3] fromList [fromList [], fromList [1], fromList [1,2], fromList [1,2,3], fromList [1,3], fromList [2], fromList [2,3], fromList [3]]</lang>
~~Icon and Unicon~~
The two examples below show the similarities and differences between constructing an explicit representation of the solution, i.e. a set containing the powerset, and one using generators. The basic recursive algorithm is the same in each case, but wherever the first stores part of the result away, the second uses 'suspend' to immediately pass the result back to the caller. The caller may then decide to store the results in a set, a list, or dispose of each one as it appears.
~~Set building~~
The following version returns a set containing the powerset:
<lang Icon> procedure power_set (s)
result := set () if *s = 0 then insert (result, set ()) # empty set else { head := set(?s) # take a random element # and find powerset of remaining part of set tail_pset := power_set (x -- head) result ++:= tail_pset # add powerset of remainder to results every ps := !tail_pset do # and add head to each powerset from the remainder insert (result, ps ++ head) } return result
end </lang>
To test the above procedure:
<lang Icon> procedure main ()
~~every s := !power_set (set(1,2,3,4)) do { # requires '!' to generate items in the result set writes ("[ ") every writes (!s || " ") write ("]") }~~
~~end </lang>~~

~~Output:~~
~~[ 3 ] [ 4 3 ] [ 2 4 ] [ 2 3 ] [ 1 3 ] [ 4 ] [ 2 ] [ 2 1 3 ] [ 2 4 1 ] [ 4 1 3 ] [ 2 4 1 3 ] [ ] [ 2 4 3 ] [ 1 ] [ 4 1 ] [ 2 1 ]~~
~~Generator~~
An alternative version, which generates each item in the power set in turn:
<lang Icon> procedure power_set (s)
~~if *s = 0 then suspend set () else { head := set(?s) every ps := power_set (s -- head) do { suspend ps suspend ps ++ head } }~~
end
procedure main ()
~~every s := power_set (set(1,2,3,4)) do { # power_set's values are generated by 'every' writes ("[ ") every writes (!s || " ") write ("]") }~~
~~end </lang>~~
J
There are a number of ways to generate a power set in J. Here's one: <lang j>ps =: #~ 2 #:@i.@^ #</lang> For example: <lang j> ps 'ACE'
E C CE A AE AC ACE</lang>
In the typical use, this operation makes sense on collections of unique elements.
<lang J> ~.1 2 3 2 1 1 2 3
~~#ps 1 2 3 2 1~~
32
~~#ps ~.1 2 3 2 1~~
8</lang>
In other words, the power set of a 5 element set has 32 sets where the power set of a 3 element set has 8 sets. Thus if elements of the original "set" were not unique then sets of the power "set" will also not be unique sets.
~~Java~~
Works with: Java version 1.5+
~~Recursion~~
This implementation sorts each subset, but not the whole list of subsets (which would require a custom comparator). It also destroys the original set. <lang java5>public static ArrayList<String> getpowerset(int a[],int n,ArrayList<String> ps)
{ if(n<0) { return null; } if(n==0) { if(ps==null) ps=new ArrayList<String>(); ps.add(" "); return ps; } ps=getpowerset(a, n-1, ps); ArrayList<String> tmp=new ArrayList<String>(); for(String s:ps) { if(s.equals(" ")) tmp.add(""+a[n-1]); else tmp.add(s+a[n-1]); } ps.addAll(tmp); return ps; }</lang>
~~Iterative~~
The iterative implementation of the above idea. Each subset is in the order that the element appears in the input list. This implementation preserves the input. <lang java5> public static <T> List<List<T>> powerset(Collection<T> list) {
~~List<List<T>> ps = new ArrayList<List<T>>(); ps.add(new ArrayList<T>()); // add the empty set~~
~~// for every item in the original list for (T item : list) { List<List<T>> newPs = new ArrayList<List<T>>();~~
~~for (List<T> subset : ps) { // copy all of the current powerset's subsets newPs.add(subset);~~
~~// plus the subsets appended with the current item List<T> newSubset = new ArrayList<T>(subset); newSubset.add(item); newPs.add(newSubset); }~~
~~// powerset is now powerset of list.subList(0, list.indexOf(item)+1) ps = newPs; } return ps;~~
~~} </lang>~~
~~Binary String~~
This implementation works on idea that each element in the original set can either be in the power set or not in it. With n elements in the original set, each combination can be represented by a binary string of length n. To get all possible combinations, all you need is a counter from 0 to 2ⁿ - 1. If the k^th bit in the binary string is 1, the k^th element of the original set is in this combination. <lang java5>public static <T extends Comparable<? super T>> LinkedList<LinkedList<T>> BinPowSet( LinkedList<T> A){ LinkedList<LinkedList<T>> ans= new LinkedList<LinkedList<T>>(); int ansSize = (int)Math.pow(2, A.size()); for(int i= 0;i< ansSize;++i){ String bin= Integer.toBinaryString(i); //convert to binary while(bin.length() < A.size()) bin = "0" + bin; //pad with 0's LinkedList<T> thisComb = new LinkedList<T>(); //place to put one combination for(int j= 0;j< A.size();++j){ if(bin.charAt(j) == '1')thisComb.add(A.get(j)); } Collections.sort(thisComb); //sort it for easy checking ans.add(thisComb); //put this set in the answer list } return ans; }</lang>
~~JavaScript~~
~~ES5~~
~~Iteration~~
~~Uses a JSON stringifier from http://www.json.org/js.html~~
Works with: SpiderMonkey
~~<lang javascript>function powerset(ary) {~~
~~var ps = [[]]; for (var i=0; i < ary.length; i++) { for (var j = 0, len = ps.length; j < len; j++) { ps.push(ps[j].concat(ary[i])); } } return ps;~~
}
var res = powerset([1,2,3,4]);
load('json2.js'); print(JSON.stringify(res));</lang>

~~Output:~~
~~[[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3],[4],[1,4],[2,4],[1,2,4],[3,4],[1,3,4],[2,3,4],[1,2,3,4]]~~

~~Functional composition~~
Translation of: Haskell
~~<lang JavaScript>(function () {~~
~~// translating: powerset = foldr (\x acc -> acc ++ map (x:) acc) [[]]~~
~~function powerset(xs) { return xs.reduceRight(function (a, x) { return a.concat(a.map(function (y) { return [x].concat(y); })); }, [[]]); }~~

~~// TEST return { '[1,2,3] ->': powerset([1, 2, 3]), 'empty set ->': powerset([]), 'set which contains only the empty set ->': powerset([[]]) }~~
~~})();</lang>~~

~~Output:~~
~~<lang JavaScript>{~~
~~"[1,2,3] ->":[[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]], "empty set ->":[[]], "set which contains only the empty set ->":[[], [[]]]~~
~~}</lang>~~
~~ES6~~
~~<lang JavaScript>(() => {~~
~~'use strict';~~
~~// powerset :: [a] -> a const powerset = xs => xs.reduceRight((a, x) => a.concat(a.map(y => [x].concat(y))), [ [] ]);~~

~~// TEST return { '[1,2,3] ->': powerset([1, 2, 3]), 'empty set ->': powerset([]), 'set which contains only the empty set ->': powerset([ [] ]) };~~
~~})()</lang>~~

~~Output:~~
~~<lang JavaScript>{"[1,2,3] ->":[[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]], "empty set ->":[[]], "set which contains only the empty set ->":[[], [[]]]}</lang>~~
jq
~~<lang jq>def powerset:~~
~~reduce .[] as $i ([[]]; reduce .[] as $r (.; . + [$r + [$i]]));</lang>~~
~~Example:~~
~~[range(0;10)]|powerset|length # => 1024~~
~~Extra credit: <lang jq>~~
~~The power set of the empty set:~~
~~[] | powerset # => [[]]~~
~~The power set of the set which contains only the empty set:~~
~~[ [] ] | powerset # => [[],[[]]]</lang>~~
~~Recursive version~~
~~<lang jq>def powerset:~~
~~if length == 0 then [[]] else .[0] as $first | (.[1:] | powerset) | map([$first] + . ) + . end;</lang>~~
~~Example:~~
~~[1,2,3]|powerset # => [[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]]~~
~~Julia~~
~~<lang julia> function powerset{T}(x::Vector{T})~~
~~result = Vector{T}[[]] for elem in x, j in eachindex(result) push!(result, [result[j] ; elem]) end result~~
~~end </lang>~~

~~Output:~~
~~julia> show(powerset([1,2,3])) [Int64[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]~~
K
~~<lang K>~~
~~ps:{x@&:'+2_vs!_2^#x}~~
~~</lang> Usage: <lang K>~~
~~ps "ABC"~~
~~(""~~
~~,"C" ,"B" "BC" ,"A" "AC" "AB" "ABC")~~
~~</lang>~~
~~Kotlin~~
<lang scala>// version 1.1.3
class PowerSet<T>(val items: List<T>) {
private lateinit var combination: IntArray init { println("Power set of $items comprises:") for (m in 0..items.size) { combination = IntArray(m) generate(0, m) } } private fun generate(k: Int, m: Int) { if (k >= m) { println(combination.map { items[it] }) } else { for (j in 0 until items.size) if (k == 0 || j > combination[k - 1]) { combination[k] = j generate(k + 1, m) } } }
}
fun main(args: Array<String>) {
~~val itemsList = listOf( listOf(1, 2, 3, 4), emptyList<Int>(), listOf(emptyList<Int>()) ) for (items in itemsList) { PowerSet(items) println() }~~
~~}</lang>~~

~~Output:~~
Power set of [1, 2, 3, 4] comprises: [] [1] [2] [3] [4] [1, 2] [1, 3] [1, 4] [2, 3] [2, 4] [3, 4] [1, 2, 3] [1, 2, 4] [1, 3, 4] [2, 3, 4] [1, 2, 3, 4] Power set of [] comprises: [] Power set of [[]] comprises: [] [[]]
~~Logo~~
~~<lang logo>to powerset :set~~
~~if empty? :set [output [[]]] localmake "rest powerset butfirst :set output sentence map [sentence first :set ?] :rest :rest~~
end
show powerset [1 2 3] [[1 2 3] [1 2] [1 3] [1] [2 3] [2] [3] []]</lang>
~~Logtalk~~
~~<lang logtalk>:- object(set).~~
~~:- public(powerset/2).~~
~~powerset(Set, PowerSet) :- reverse(Set, RSet), powerset_1(RSet, [[]], PowerSet).~~
~~powerset_1([], PowerSet, PowerSet). powerset_1([X| Xs], Yss0, Yss) :- powerset_2(Yss0, X, Yss1), powerset_1(Xs, Yss1, Yss).~~
~~powerset_2([], _, []). powerset_2([Zs| Zss], X, [Zs, [X| Zs]| Yss]) :- powerset_2(Zss, X, Yss).~~
~~reverse(List, Reversed) :- reverse(List, [], Reversed).~~
~~reverse([], Reversed, Reversed). reverse([Head| Tail], List, Reversed) :- reverse(Tail, [Head| List], Reversed).~~
~~- end_object.</lang>~~
Usage example: <lang logtalk>| ?- set::powerset([1, 2, 3, 4], PowerSet).
PowerSet = [[],[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3],[4],[1,4],[2,4],[1,2,4],[3,4],[1,3,4],[2,3,4],[1,2,3,4]] yes</lang>
~~Lua~~
~~<lang lua> --returns the powerset of s, out of order. function powerset(s, start)~~
~~start = start or 1 if(start > #s) then return {{}} end local ret = powerset(s, start + 1) for i = 1, #ret do ret[#ret + 1] = {s[start], unpack(ret[i])} end return ret~~
end
--non-recurse implementation function powerset(s)
~~local t = {{}} for i = 1, #s do for j = 1, #t do t[#t+1] = {s[i],unpack(t[j])} end end return t~~
end
--alternative, copied from the Python implementation function powerset2(s)
~~local ret = {{}} for i = 1, #s do local k = #ret for j = 1, k do ret[k + j] = {s[i], unpack(ret[j])} end end return ret~~
~~end </lang>~~
M4
~~<lang M4>define(`for',~~
~~`ifelse($#, 0, ``$0, eval($2 <= $3), 1, `pushdef(`$1', `$2')$4`'popdef( `$1')$0(`$1', incr($2), $3, `$4')')')dnl~~
~~define(`nth',~~
~~`ifelse($1, 1, $2, `nth(decr($1), shift(shift($@)))')')dnl~~
~~define(`range',~~
~~`for(`x', eval($1 + 2), eval($2 + 2), `nth(x, $@)`'ifelse(x, eval($2+2), `', `,')')')dnl~~
~~define(`powerpart',~~
`{range(2, incr($1), $@)}`'ifelse(incr($1), $#, `', `for(`x', eval($1+2), $#, `,powerpart(incr($1), ifelse( eval(2 <= ($1 + 1)), 1, `range(2,incr($1), $@), ')`'nth(x, $@)`'ifelse( eval((x + 1) <= $#),1,`,range(incr(x), $#, $@)'))')')')dnl
~~define(`powerset',~~
~~`{powerpart(0, substr(`$1', 1, eval(len(`$1') - 2)))}')dnl~~
~~dnl powerset(`{a,b,c}')</lang>~~

~~Output:~~
~~{{},{a},{a,b},{a,b,c},{a,c},{b},{b,c},{c}}~~

~~Maple~~
~~<lang Maple> combinat:-powerset({1,2,3,4}); </lang>~~

~~Output:~~
~~{{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}~~
~~Mathematica~~
Built-in function that either gives all possible subsets, subsets with at most n elements, subsets with exactly n elements or subsets containing between n and m elements. Example of all subsets: <lang Mathematica>Subsets[{a, b, c}]</lang> gives: <lang Mathematica>{{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}</lang> Subsets[list, {n, Infinity}] gives all the subsets that have n elements or more.
Subsets[list, n] gives all the subsets that have at most n elements.
Subsets[list, {n}] gives all the subsets that have exactly n elements.
Subsets[list, {m,n}] gives all the subsets that have between m and n elements.
~~MATLAB~~
Sets are not an explicit data type in MATLAB, but cell arrays can be used for the same purpose. In fact, cell arrays have the benefit of containing any kind of data structure. So, this powerset function will work on a set of any type of data structure, without the need to overload any operators.
<lang MATLAB>function pset = powerset(theSet)
~~pset = cell(size(theSet)); %Preallocate memory~~
%Generate all numbers from 0 to 2^(num elements of the set)-1 for i = ( 0:(2^numel(theSet))-1 ) %Convert i into binary, convert each digit in binary to a boolean %and store that array of booleans indicies = logical(bitget( i,(1:numel(theSet)) )); %Use the array of booleans to extract the members of the original %set, and store the set containing these members in the powerset pset(i+1) = {theSet(indicies)}; end
end</lang>
Sample Usage: Powerset of the set of the empty set. <lang MATLAB>powerset({{}})
ans =
~~{} {1x1 cell} %This is the same as { {},{{}} }</lang>~~
Powerset of { {1,2},3 }. <lang MATLAB>powerset({{1,2},3})
ans =
~~{1x0 cell} {1x1 cell} {1x1 cell} {1x2 cell} %This is the same as { {},Template:1,2,{3},{{1,2},3} }</lang>~~
~~Maxima~~
~~<lang maxima>powerset({1, 2, 3, 4}); /* {{}, {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}} */</lang>~~
~~Nim~~
<lang nim>import sets, hashes
proc hash(x: HashSet[int]): Hash =
~~var h = 0 for i in x: h = h !& hash(i) result = !$h~~
~~proc powerset[T](inset: HashSet[T]): auto =~~
~~result = toSet([initSet[T]()])~~
~~for i in inset: var tmp = result for j in result: var k = j k.incl(i) tmp.incl(k) result = tmp~~
~~echo powerset(toSet([1,2,3,4]))</lang>~~
~~Objective-C~~
<lang objc>#import <Foundation/Foundation.h>
+ (NSArray *)powerSetForArray:(NSArray *)array { UInt32 subsetCount = 1 << array.count; NSMutableArray *subsets = [NSMutableArray arrayWithCapacity:subsetCount]; for(int subsetIndex = 0; subsetIndex < subsetCount; subsetIndex++) { NSMutableArray *subset = [[NSMutableArray alloc] init]; for (int itemIndex = 0; itemIndex < array.count; itemIndex++) { if((subsetIndex >> itemIndex) & 0x1) { [subset addObject:array[itemIndex]]; } } [subsets addObject:subset]; } return subsets; }</lang>
~~OCaml~~
The standard library already implements a proper Set datatype. As the base type is unspecified, the powerset must be parameterized as a module. Also, the library is lacking a map operation, which we have to implement first.
<lang ocaml>module PowerSet(S: Set.S) = struct
~~include Set.Make (S)~~
~~let map f s = let work x r = add (f x) r in fold work s empty ;;~~
~~let powerset s = let base = singleton (S.empty) in let work x r = union r (map (S.add x) r) in S.fold work s base ;;~~
end;; (* PowerSet *)</lang>
version for lists: <lang ocaml>let subsets xs = List.fold_right (fun x rest -> rest @ List.map (fun ys -> x::ys) rest) xs [[]]</lang>

~~OPL~~
<lang OPL> {string} s={"A","B","C","D"}; range r=1.. ftoi(pow(2,card(s))); {string} s2 [k in r] = {i | i in s: ((k div (ftoi(pow(2,(ord(s,i))))) mod 2) == 1)};
execute {
~~writeln(s2);~~
} </lang>
which gives
<lang result>
[{} {"A"} {"B"} {"A" "B"} {"C"} {"A" "C"} {"B" "C"} {"A" "B" "C"} {"D"} {"A"
~~"D"} {"B" "D"} {"A" "B" "D"} {"C" "D"} {"A" "C" "D"} {"B" "C" "D"} {"A" "B" "C" "D"}]~~
~~</lang>~~

Oz
~~Oz has a library for finite set constraints. Creating a power set is a trivial application of that: <lang oz>declare~~
%% Given a set as a list, returns its powerset (again as a list) fun {Powerset Set} proc {Describe Root} %% Describe sets by lower bound (nil) and upper bound (Set) Root = {FS.var.bounds nil Set} %% enumerate all possible sets {FS.distribute naive [Root]} end AllSets = {SearchAll Describe} in %% convert to list representation {Map AllSets FS.reflect.lowerBoundList} end
in
~~{Inspect {Powerset [1 2 3 4]}}</lang>~~
~~A more convential implementation without finite set constaints: <lang oz>fun {Powerset2 Set}~~
~~case Set of nil then [nil] [] H|T thens Acc = {Powerset2 T} in {Append Acc {Map Acc fun {$ A} H|A end}} end~~
~~end</lang>~~
~~PARI/GP~~
~~<lang parigp>vector(1<<#S,i,vecextract(S,i-1))</lang>~~
Works with: PARI/GP version 2.10.0+
~~The forsubset iterator was added in version 2.10.0 to efficiently iterate over combinations and power sets. <lang parigp>S=["a","b","c"] forsubset(#S,s,print1(vecextract(S,s)" "))</lang>~~

~~Output:~~
~~[] ["a"] ["b"] ["c"] ["a", "b"] ["a", "c"] ["b", "c"] ["a", "b", "c"]~~
~~Perl~~
~~Perl does not have a built-in set data-type. However, you can...~~
~~Module: Algorithm::Combinatorics~~
This module has an iterator over the power set. Note that it does not enforce that the input array is a set (no duplication). If each subset is processed immediately, this has an advantage of very low memory use. <lang perl>use Algorithm::Combinatorics "subsets"; my @S = ("a","b","c"); my @PS; my $iter = subsets(\@S); while (my $p = $iter->next) {
~~push @PS, "[@$p]"~~
~~} say join(" ",@PS);</lang>~~

~~Output:~~
~~[a b c] [b c] [a c] [c] [a b] [b] [a] []~~
~~Module: ntheory~~
The simplest solution is to use the one argument version of the combination iterator, which iterates over the power set. <lang perl>use ntheory "forcomb"; my @S = qw/a b c/; forcomb { print "[@S[@_]] " } scalar(@S); print "\n";</lang>

~~Output:~~
~~[] [a] [b] [c] [a b] [a c] [b c] [a b c]~~
~~Using the two argument version of the iterator gives a solution similar to the Perl6 and Python array versions. <lang perl>use ntheory "forcomb"; my @S = qw/a b c/; for $k (0..@S) {~~
~~# Iterate over each $#S+1,$k combination. forcomb { print "[@S[@_]] " } @S,$k;~~
~~} print "\n";</lang>~~

~~Output:~~
~~[] [a] [b] [c] [a b] [a c] [b c] [a b c]~~
Similar to the Pari/GP solution, one can also use vecextract with an integer mask to select elements. Note that it does not enforce that the input array is a set (no duplication). This also has low memory if each subset is processed immediately and the range is applied with a loop rather than a map. A solution using vecreduce could be done identical to the array reduce solution shown later. <lang perl>use ntheory "vecextract"; my @S = qw/a b c/; my @PS = map { "[".join(" ",vecextract(\@S,$_))."]" } 0..2**scalar(@S)-1; say join(" ",@PS);</lang>

~~Output:~~
~~[] [a] [b] [a b] [c] [a c] [b c] [a b c]~~
~~Module: Set::Object~~
The CPAN module Set::Object provides a set implementation for sets of arbitrary objects, for which a powerset function could be defined and used like so:
<lang perl>use Set::Object qw(set);
sub powerset {
~~my $p = Set::Object->new( set() ); foreach my $i (shift->elements) { $p->insert( map { set($_->elements, $i) } $p->elements ); } return $p;~~
}
my $set = set(1, 2, 3); my $powerset = powerset($set);
print $powerset->as_string, "\n";</lang>

~~Output:~~
~~Set::Object(Set::Object() Set::Object(1 2 3) Set::Object(1 2) Set::Object(1 3) Set::Object(1) Set::Object(2 3) Set::Object(2) Set::Object(3))~~
~~Simple custom hash-based set type~~
It's also easy to define a custom type for sets of strings or numbers, using a hash as the underlying representation (like the task description suggests):
<lang perl>package Set {
sub new { bless { map {$_ => undef} @_[1..$#_] }, shift; } sub elements { sort keys %{shift()} } sub as_string { 'Set(' . join(' ', sort keys %{shift()}) . ')' } # ...more set methods could be defined here...
}</lang>
(Note: For a ready-to-use module that uses this approach, and comes with all the standard set methods that you would expect, see the CPAN module Set::Tiny)
The limitation of this approach is that only primitive strings/numbers are allowed as hash keys in Perl, so a Set of Set's cannot be represented, and the return value of our powerset function will thus have to be a list of sets rather than being a Set object itself.
We could implement the function as an imperative foreach loop similar to the Set::Object based solution above, but using list folding (with the help of Perl's List::Util core module) seems a little more elegant in this case:
<lang perl>use List::Util qw(reduce);
sub powerset {
~~@{( reduce { [@$a, map { Set->new($_->elements, $b) } @$a ] } [Set->new()], shift->elements )};~~
}
my $set = Set->new(1, 2, 3); my @subsets = powerset($set);
print $_->as_string, "\n" for @subsets;</lang>

~~Output:~~
~~Set() Set(1) Set(2) Set(1 2) Set(3) Set(1 3) Set(2 3) Set(1 2 3)~~
~~Arrays~~
If you don't actually need a proper set data-type that guarantees uniqueness of its elements, the simplest approach is to use arrays to store "sets" of items, in which case the implementation of the powerset function becomes quite short.
Recursive solution: <lang perl>sub powerset {
~~@_ ? map { $_, [$_[0], @$_] } powerset(@_[1..$#_]) : [];~~
}</lang>
List folding solution:
<lang perl>use List::Util qw(reduce);
sub powerset {
~~@{( reduce { [@$a, map([@$_, $b], @$a)] } [[]], @_ )}~~
}</lang>
Usage & output: <lang perl>my @set = (1, 2, 3); my @powerset = powerset(@set);
sub set_to_string {
~~"{" . join(", ", map { ref $_ ? set_to_string(@$_) : $_ } @_) . "}"~~
}
print set_to_string(@powerset), "\n";</lang>

~~Output:~~
~~{{}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}~~
~~Lazy evaluation~~
If the initial set is quite large, constructing it's powerset all at once can consume lots of memory.
If you want to iterate through all of the elements of the powerset of a set, and don't mind each element being generated immediately before you process it, and being thrown away immediately after you're done with it, you can use vastly less memory. This is similar to the earlier solutions using the Algorithm::Combinatorics and ntheory modules.
The following algorithm uses one bit of memory for every element of the original set (technically it uses several bytes per element with current versions of Perl). This is essentially doing a vecextract operation by hand.
<lang perl>use strict; use warnings; sub powerset(&@) {
my $callback = shift; my $bitmask = ; my $bytes = @_/8; { my @indices = grep vec($bitmask, $_, 1), 0..$#_; $callback->( @_[@indices] ); ++vec($bitmask, $_, 8) and last for 0 .. $bytes; redo if @indices != @_; }
}
print "powerset of empty set:\n"; powerset { print "[@_]\n" }; print "powerset of set {1,2,3,4}:\n"; powerset { print "[@_]\n" } 1..4; my $i = 0; powerset { ++$i } 1..9; print "The powerset of a nine element set contains $i elements.\n"; </lang>

~~Output:~~
powerset of empty set: [] powerset of set {1,2,3,4}: [] [1] [2] [1 2] [3] [1 3] [2 3] [1 2 3] [4] [1 4] [2 4] [1 2 4] [3 4] [1 3 4] [2 3 4] [1 2 3 4] The powerset of a nine element set contains 512 elements.
~~The technique shown above will work with arbitrarily large sets, and uses a trivial amount of memory.~~
~~Perl 6~~
Works with: rakudo version 2014-02-25
~~<lang perl6>sub powerset(Set $s) { $s.combinations.map(*.Set).Set } say powerset set <a b c d>;</lang>~~

~~Output:~~
~~set(set(), set(a), set(b), set(c), set(d), set(a, b), set(a, c), set(a, d), set(b, c), set(b, d), set(c, d), set(a, b, c), set(a, b, d), set(a, c, d), set(b, c, d), set(a, b, c, d))~~
~~If you don't care about the actual Set type, the .combinations method by itself may be good enough for you: <lang perl6>.say for <a b c d>.combinations</lang>~~

~~Output:~~
~~a b c d a b a c a d b c b d c d a b c a b d a c d b c d a b c d~~
~~Phix~~
<lang Phix>sequence powerset integer step = 1
function pst(object key, object /*data*/, object /*user_data*/)
integer k = 1 while k<length(powerset) do k += step for j=1 to step do powerset[k] = append(powerset[k],key) k += 1 end for end while step *= 2 return 1
end function
function power_set(integer d)
~~powerset = repeat({},power(2,dict_size(d))) step = 1 traverse_dict(routine_id("pst"),0,d) return powerset~~
end function
integer d1234 = new_dict() setd(1,0,d1234) setd(2,0,d1234) setd(3,0,d1234) setd(4,0,d1234) ?power_set(d1234) integer d0 = new_dict() ?power_set(d0) setd({},0,d0) ?power_set(d0)</lang>

~~Output:~~
~~{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}} {{}} {{},{{}}}~~
~~PHP~~
~~<lang PHP> <?php function get_subset($binary, $arr) {~~
// based on true/false values in $binary array, include/exclude // values from $arr $subset = array(); foreach (range(0, count($arr)-1) as $i) { if ($binary[$i]) { $subset[] = $arr[count($arr) - $i - 1]; } } return $subset;
}
function print_array($arr) {
if (count($arr) > 0) { echo join(" ", $arr); } else { echo "(empty)"; } echo '
';
}
function print_power_sets($arr) {
echo "POWER SET of [" . join(", ", $arr) . "]
"; foreach (power_set($arr) as $subset) { print_array($subset); }
}
function power_set($arr) {
$binary = array(); foreach (range(1, count($arr)) as $i) { $binary[] = false; } $n = count($arr); $powerset = array(); while (count($binary) <= count($arr)) { $powerset[] = get_subset($binary, $arr); $i = 0; while (true) { if ($binary[$i]) { $binary[$i] = false; $i += 1; } else { $binary[$i] = true; break; } } $binary[$i] = true; } return $powerset;
}
print_power_sets(array()); print_power_sets(array('singleton')); print_power_sets(array('dog', 'c', 'b', 'a')); ?> </lang>

~~Output:~~
~~<lang> POWER SET of [] POWER SET of [singleton] (empty) singleton POWER SET of [dog, c, b, a] (empty) a b a b c a c b c a b c dog a dog b dog a b dog c dog a c dog b c dog a b c dog </lang>~~
~~PicoLisp~~
~~<lang PicoLisp>(de powerset (Lst)~~
~~(ifn Lst (cons) (let L (powerset (cdr Lst)) (conc (mapcar '((X) (cons (car Lst) X)) L) L ) ) ) )</lang>~~
~~PL/I~~
Translation of: REXX
~~<lang pli>*process source attributes xref or(!);~~
/*-------------------------------------------------------------------- * 06.01.2014 Walter Pachl translated from REXX *-------------------------------------------------------------------*/ powerset: Proc Options(main); Dcl (hbound,index,left,substr) Builtin; Dcl sysprint Print; Dcl s(4) Char(5) Var Init('one','two','three','four'); Dcl ps Char(1000) Var; Dcl (n,chunk,p) Bin Fixed(31); n=hbound(s); /* number of items in the list. */ ps='{} '; /* start with a null power set. */ Do chunk=1 To n; /* loop through the ... . */ ps=ps!!combn(chunk); /* a CHUNK at a time. */ End; Do While(ps>); p=index(ps,' '); Put Edit(left(ps,p-1))(Skip,a); ps=substr(ps,p+1); End;
combn: Proc(y) Returns(Char(1000) Var); /*-------------------------------------------------------------------- * returns the list of subsets with y elements of set s *-------------------------------------------------------------------*/ Dcl (y,base,bbase,ym,p,j,d,u) Bin Fixed(31); Dcl (z,l) Char(1000) Var Init(); Dcl a(20) Bin Fixed(31) Init((20)0); Dcl i Bin Fixed(31); base=hbound(s)+1; bbase=base-y; ym=y-1; Do p=1 To y; a(p)=p; End; Do j=1 By 1; l=; Do d=1 To y; u=a(d); l=l!!','!!s(u); End; z=z!!'{'!!substr(l,2)!!'} '; a(y)=a(y)+1; If a(y)=base Then If combu(ym) Then Leave; End; /* Put Edit('combn',y,z)(Skip,a,f(2),x(1),a); */ Return(z);
combu: Proc(d) Recursive Returns(Bin Fixed(31)); Dcl (d,u) Bin Fixed(31); If d=0 Then Return(1); p=a(d); Do u=d To y; a(u)=p+1; If a(u)=bbase+u Then Return(combu(u-1)); p=a(u); End; Return(0); End; End;
~~End;</lang>~~

~~Output:~~
~~{} {one} {two} {three} {four} {one,two} {one,three} {one,four} {two,three} {two,four} {three,four} {one,two,three} {one,two,four} {one,three,four} {two,three,four} {one,two,three,four}~~
~~PowerShell~~
~~<lang PowerShell> function power-set ($array) {~~
if($array) { $n = $array.Count function state($set, $i){ if($i -gt -1) { state $set ($i-1) state ($set+@($array[$i])) ($i-1) } else { "$($set | sort)" } } $set = state @() ($n-1) $power = 0..($set.Count-1) | foreach{@(0)} $i = 0 $set | sort | foreach{$power[$i++] = $_.Split()} $power | sort {$_.Count} } else {@()}
} $OFS = " " $setA = power-set @(1,2,3,4) "number of sets in setA: $($setA.Count)" "sets in setA:" $OFS = ", " $setA | foreach{"{"+"$_"+"}"} $setB = @() "number of sets in setB: $($setB.Count)" "sets in setB:" $setB | foreach{"{"+"$_"+"}"} $setC = @(@(), @(@())) "number of sets in setC: $($setC.Count)" "sets in setC:" $setC | foreach{"{"+"$_"+"}"} $OFS = " " </lang> Output:
number of sets in setA: 16 sets in setA: {} {1} {2} {3} {4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2, 3, 4} number of sets in setB: 0 sets in setB: number of sets in setC: 2 sets in setC: {} {}
~~Prolog~~
~~Logical (cut-free) Definition~~
The predicate powerset(X,Y) defined here can be read as "Y is the powerset of X", it being understood that lists are used to represent sets.
The predicate subseq(X,Y) is true if and only if the list X is a subsequence of the list Y.
The definitions here are elementary, logical (cut-free), and efficient (within the class of comparably generic implementations). <lang Prolog>powerset(X,Y) :- bagof( S, subseq(S,X), Y).
subseq( [], []). subseq( [], [_|_]). subseq( [X|Xs], [X|Ys] ) :- subseq(Xs, Ys). subseq( [X|Xs], [_|Ys] ) :- append(_, [X|Zs], Ys), subseq(Xs, Zs). </lang>

~~Output:~~
?- powerset([1,2,3], X). X = [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]. % Symbolic: ?- powerset( [X,Y], S). S = [[], [X], [X, Y], [Y]]. % In reverse: ?- powerset( [X,Y], [[], [1], [1, 2], [2]] ). X = 1, Y = 2.
~~Single-Functor Definition~~
~~<lang Prolog>power_set( [], [[]]). power_set( [X|Xs], PS) :-~~
~~power_set(Xs, PS1), maplist( append([X]), PS1, PS2 ), % i.e. prepend X to each PS1 append(PS1, PS2, PS).</lang>~~

~~Output:~~
~~?- power_set([1,2,3,4,5,6,7,8], X), length(X,N), writeln(N). 256~~
~~Constraint Handling Rules~~
CHR is a programming language created by Professor Thom Frühwirth.
Works with SWI-Prolog and module chr written by Tom Schrijvers and Jan Wielemaker. <lang Prolog>:- use_module(library(chr)).
~~- chr_constraint chr_power_set/2, chr_power_set/1, clean/0.~~
clean @ clean \ chr_power_set(_) <=> true. clean @ clean <=> true.
only_one @ chr_power_set(A) \ chr_power_set(A) <=> true.

creation @ chr_power_set([H | T], A) <=>
~~append(A, [H], B),~~
~~chr_power_set(T, A),~~
~~chr_power_set(T, B),~~
chr_power_set(B).

empty_element @ chr_power_set([], _) <=> chr_power_set([]). </lang>

~~Output:~~
~~?- chr_power_set([1,2,3,4], []), findall(L, find_chr_constraint(chr_power_set(L)), LL), clean. LL = [[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],[]] .~~
~~PureBasic~~
~~This code is for console mode. <lang PureBasic>If OpenConsole()~~
Define argc=CountProgramParameters() If argc>=(SizeOf(Integer)*8) Or argc<1 PrintN("Set out of range.") End 1 Else Define i, j, text$ Define.q bset=1<<argc Print("{") For i=0 To bset-1 ; check all binary combinations If Not i: text$= "{" Else : text$=", {" EndIf k=0 For j=0 To argc-1 ; step through each bit If i&(1<<j) If k: text$+", ": EndIf ; pad the output text$+ProgramParameter(j): k+1 ; append each matching bit EndIf Next j Print(text$+"}") Next i PrintN("}") EndIf
~~EndIf</lang>~~

~~Output:~~
~~C:\Users\PureBasic_User\Desktop>"Power Set.exe" 1 2 3 4 {{}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}, {4}, {1, 4}, {2, 4}, {1, 2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}~~
~~Python~~
~~<lang python>def list_powerset(lst):~~
# the power set of the empty set has one element, the empty set result = [[]] for x in lst: # for every additional element in our set # the power set consists of the subsets that don't # contain this element (just take the previous power set) # plus the subsets that do contain the element (use list # comprehension to add [x] onto everything in the # previous power set) result.extend([subset + [x] for subset in result]) return result
~~the above function in one statement~~
~~def list_powerset2(lst):~~
~~return reduce(lambda result, x: result + [subset + [x] for subset in result], lst, [[]])~~
~~def powerset(s):~~
~~return frozenset(map(frozenset, list_powerset(list(s))))</lang>~~
list_powerset computes the power set of a list of distinct elements. powerset simply converts the input and output from lists to sets. We use the frozenset type here for immutable sets, because unlike mutable sets, it can be put into other sets.

~~Example:~~
>>> list_powerset([1,2,3]) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] >>> powerset(frozenset([1,2,3])) frozenset([frozenset([3]), frozenset([1, 2]), frozenset([]), frozenset([2, 3]), frozenset([1]), frozenset([1, 3]), frozenset([1, 2, 3]), frozenset([2])])
~~Further Explanation~~
If you take out the requirement to produce sets and produce list versions of each powerset element, then add a print to trace the execution, you get this simplified version of the program above where it is easier to trace the inner workings <lang python>def powersetlist(s):
~~r = [[]] for e in s: print "r: %-55r e: %r" % (r,e) r += [x+[e] for x in r] return r~~
~~s= [0,1,2,3] print "\npowersetlist(%r) =\n %r" % (s, powersetlist(s))</lang>~~

~~Output:~~
r: [[]] e: 0 r: [[], [0]] e: 1 r: [[], [0], [1], [0, 1]] e: 2 r: [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]] e: 3 powersetlist([0, 1, 2, 3]) = [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]
~~Binary Count method~~
If you list the members of the set and include them according to if the corresponding bit position of a binary count is true then you generate the powerset. (Note that only frozensets can be members of a set in the second function) <lang python>def powersequence(val):
Generate a 'powerset' for sequence types that are indexable by integers. Uses a binary count to enumerate the members and returns a list
Examples: >>> powersequence('STR') # String [, 'S', 'T', 'ST', 'R', 'SR', 'TR', 'STR'] >>> powersequence([0,1,2]) # List [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]] >>> powersequence((3,4,5)) # Tuple [(), (3,), (4,), (3, 4), (5,), (3, 5), (4, 5), (3, 4, 5)] >>> vtype = type(val); vlen = len(val); vrange = range(vlen) return [ reduce( lambda x,y: x+y, (val[i:i+1] for i in vrange if 2**i & n), vtype()) for n in range(2**vlen) ]
~~def powerset(s):~~
~~Generate the powerset of s~~
Example: >>> powerset(set([6,7,8])) set([frozenset([7]), frozenset([8, 6, 7]), frozenset([6]), frozenset([6, 7]), frozenset([]), frozenset([8]), frozenset([8, 7]), frozenset([8, 6])]) return set( frozenset(x) for x in powersequence(list(s)) )</lang>
~~Recursive Alternative~~
This is an (inefficient) recursive version that almost reflects the recursive definition of a power set as explained in http://en.wikipedia.org/wiki/Power_set#Algorithms. It does not create a sorted output.
<lang python> def p(l):
~~if not l: return [[]] return p(l[1:]) + [[l[0]] + x for x in p(l[1:])]~~
~~</lang>~~
~~Python: Standard documentation~~
Pythons documentation has a method that produces the groupings, but not as sets:
<lang python>>>> from pprint import pprint as pp >>> from itertools import chain, combinations >>> >>> def powerset(iterable):
~~"powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)" s = list(iterable) return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))~~
~~>>> pp(set(powerset({1,2,3,4}))) {(),~~
~~(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,)}~~
~~>>> </lang>~~
Qi
Translation of: Scheme
~~<lang qi> (define powerset~~
~~[] -> [[]] [A|As] -> (append (map (cons A) (powerset As)) (powerset As)))~~
~~</lang>~~
R
~~Non-recursive version~~
The conceptual basis for this algorithm is the following: <lang>for each element in the set: for each subset constructed so far: new subset = (subset + element) </lang>
This method is much faster than a recursive method, though the speed is still O(2^n).
<lang R>powerset = function(set){ ps = list() ps1 = numeric() #Start with the empty set. for(element in set){ #For each element in the set, take all subsets temp = vector(mode="list",length=length(ps)) #currently in "ps" and create new subsets (in "temp") for(subset in 1:length(ps)){ #by adding "element" to each of them. tempsubset = c(pssubset,element) } ps=c(ps,temp) #Add the additional subsets ("temp") to "ps". } return(ps) }
powerset(1:4) </lang>
The list "temp" is a compromise between the speed costs of doing arithmetic and of creating new lists (since R lists are immutable, appending to a list means actually creating a new list object). Thus, "temp" collects new subsets that are later added to the power set. This improves the speed by 4x compared to extending the list "ps" at every step.
~~Recursive version~~
Library: sets
The sets package includes a recursive method to calculate the power set. However, this method takes ~100 times longer than the non-recursive method above. <lang R>library(sets)</lang> An example with a vector. <lang R>v <- (1:3)^2 sv <- as.set(v) 2^sv</lang>
~~{{}, {1}, {4}, {9}, {1, 4}, {1, 9}, {4, 9}, {1, 4, 9}}~~
~~An example with a list. <lang R>l <- list(a=1, b="qwerty", c=list(d=TRUE, e=1:3)) sl <- as.set(l) 2^sl</lang>~~
~~{{}, {1}, {"qwerty"}, {<<list(2)>>}, {1, <<list(2)>>}, {"qwerty", 1}, {"qwerty", <<list(2)>>}, {"qwerty", 1, <<list(2)>>}}~~
~~Racket~~
~~<lang racket>~~
~~Direct translation of 'functional' ruby method~~
~~(define (powerset s)~~
~~(for/fold ([outer-set (set(set))]) ([element s]) (set-union outer-set (list->set (set-map outer-set (λ(inner-set) (set-add inner-set element)))))))~~
~~</lang>~~
~~Rascal~~
<lang rascal> import Set;
public set[set[&T]] PowerSet(set[&T] s) = power(s); </lang>

~~Output:~~
~~<lang rascal> rascal>PowerSet({1,2,3,4}) set[set[int]]: {~~
~~{4,3}, {4,2,1}, {4,3,1}, {4,2}, {4,3,2}, {4,1}, {4,3,2,1}, {4}, {3}, {2,1}, {3,1}, {2}, {3,2}, {1}, {3,2,1}, {}~~
~~} </lang>~~
~~REXX~~
<lang rexx>/*REXX program displays a power set; items may be anything (but can't have blanks).*/ parse arg S /*allow the user specify optional set. */ if S= then S= 'one two three four' /*Not specified? Then use the default.*/ @= '{}' /*start process with a null power set. */ N= words(S); do chunk=1 for N /*traipse through the items in the set.*/
~~@=@ combN(N, chunk) /*take N items, a CHUNK at a time. */ end /*chunk*/~~
~~w= length(2**N) /*the number of items in the power set.*/~~
~~do k=1 for words(@) /* [↓] show combinations, one per line*/ say right(k, w) word(@, k) /*display a single combination to term.*/ end /*k*/~~
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ combN: procedure expose S; parse arg x,y; base= x + 1; bbase= base - y
!.= 0 do p=1 for y; !.p= p end /*p*/ $= do j=1; L= do d=1 for y; L= L','word(S, !.d) end /*d*/
$=$ '{'strip(L, "L", ',')"}"; !.y= !.y + 1 if !.y==base then if .combU(y - 1) then leave end /*j*/ return strip($) /*return with a partial powerset chunk.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ .combU: procedure expose !. y bbase; parse arg d; if d==0 then return 1
p= !.d do u=d to y; !.u= p + 1 if !.u==bbase+u then return .combU(u-1) p= !.u end /*u*/ return 0</lang>

~~output when using the default input:~~
1 {} 2 {one} 3 {two} 4 {three} 5 {four} 6 {one,two} 7 {one,three} 8 {one,four} 9 {two,three} 10 {two,four} 11 {three,four} 12 {one,two,three} 13 {one,two,four} 14 {one,three,four} 15 {two,three,four} 16 {one,two,three,four}
~~Ring~~
~~<lang ring>~~
~~Project : Power set~~
list = ["1", "2", "3", "4"] see powerset(list)
func powerset(list)
s = "{" for i = 1 to (2 << len(list)) - 1 step 2 s = s + "{" for j = 1 to len(list) if i & (1 << j) s = s + list[j] + "," ok next if right(s,1) = "," s = left(s,len(s)-1) ok s = s + "}," next return left(s,len(s)-1) + "}"
~~</lang> Output:~~
~~{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}~~
~~Ruby~~
~~<lang ruby># Based on http://johncarrino.net/blog/2006/08/11/powerset-in-ruby/~~
See the link if you want a shorter version.

This was intended to show the reader how the method works.
~~class Array~~
# Adds a power_set method to every array, i.e.: [1, 2].power_set def power_set # Injects into a blank array of arrays. # acc is what we're injecting into # you is each element of the array inject([[]]) do |acc, you| ret = [] # Set up a new array to add into acc.each do |i| # For each array in the injected array, ret << i # Add itself into the new array ret << i + [you] # Merge the array with a new array of the current element end ret # Return the array we're looking at to inject more. end end # A more functional and even clearer variant. def func_power_set inject([[]]) { |ps,item| # for each item in the Array ps + # take the powerset up to now and add ps.map { |e| e + [item] } # it again, with the item appended to each element } end
~~end~~
~~A direct translation of the "power array" version above~~
~~require 'set' class Set~~
~~def powerset inject(Set[Set[]]) do |ps, item| ps.union ps.map {|e| e.union (Set.new [item])} end end~~
end
p [1,2,3,4].power_set p %w(one two three).func_power_set
p Set[1,2,3].powerset</lang>

~~Output:~~
[[], [4], [3], [3, 4], [2], [2, 4], [2, 3], [2, 3, 4], [1], [1, 4], [1, 3], [1, 3, 4], [1, 2], [1, 2, 4], [1, 2, 3], [1, 2, 3, 4]] [[], ["one"], ["two"], ["one", "two"], ["three"], ["one", "three"], ["two", "three"], ["one", "two", "three"]] #<Set: {#<Set: {}>, #<Set: {1}>, #<Set: {2}>, #<Set: {1, 2}>, #<Set: {3}>, #<Set: {1, 3}>, #<Set: {2, 3}>, #<Set: {1, 2, 3}>}>
~~SAS~~
<lang SAS> options mprint mlogic symbolgen source source2;
%macro SubSets (FieldCount = ); data _NULL_; Fields = &FieldCount; SubSets = 2**Fields; call symput ("NumSubSets", SubSets); run;
%put &NumSubSets;
data inital; %do j = 1 %to &FieldCount; F&j. = 1; %end; run;
data SubSets; set inital; RowCount =_n_; call symput("SetCount",RowCount); run;
%put SetCount ;
%do %while (&SetCount < &NumSubSets);
data loop; %do j=1 %to &FieldCount; if rand('GAUSSIAN') > rand('GAUSSIAN') then F&j. = 1; %end;
data SubSets_ ; set SubSets loop; run;
proc sort data=SubSets_ nodupkey; by F1 - F&FieldCount.; run;
data Subsets; set SubSets_; RowCount =_n_; run;
proc sql noprint; select max(RowCount) into :SetCount from SubSets; quit; run;
%end; %Mend SubSets; </lang>
You can then call the macro as: <lang SAS> %SubSets(FieldCount = 5); </lang>
The output will be the dataset SUBSETS and will have a 5 columns F1, F2, F3, F4, F5 and 32 columns, one with each combination of 1 and missing values.

~~Output:~~
Obs F1 F2 F3 F4 F5 RowCount 1 . . . . . 1 2 . . . . 1 2 3 . . . 1 . 3 4 . . . 1 1 4 5 . . 1 . . 5 6 . . 1 . 1 6 7 . . 1 1 . 7 8 . . 1 1 1 8 9 . 1 . . . 9 10 . 1 . . 1 10 11 . 1 . 1 . 11 12 . 1 . 1 1 12 13 . 1 1 . . 13 14 . 1 1 . 1 14 15 . 1 1 1 . 15 16 . 1 1 1 1 16 17 1 . . . . 17 18 1 . . . 1 18 19 1 . . 1 . 19 20 1 . . 1 1 20 21 1 . 1 . . 21 22 1 . 1 . 1 22 23 1 . 1 1 . 23 24 1 . 1 1 1 24 25 1 1 . . . 25 26 1 1 . . 1 26 27 1 1 . 1 . 27 28 1 1 . 1 1 28 29 1 1 1 . . 29 30 1 1 1 . 1 30 31 1 1 1 1 . 31 32 1 1 1 1 1 32
~~Scala~~
<lang scala>import scala.compat.Platform.currentTime
object Powerset extends App {
~~def powerset[A](s: Set[A]) = s.foldLeft(Set(Set.empty[A])) { case (ss, el) => ss ++ ss.map(_ + el)}~~
assert(powerset(Set(1, 2, 3, 4)) == Set(Set.empty, Set(1), Set(2), Set(3), Set(4), Set(1, 2), Set(1, 3), Set(1, 4), Set(2, 3), Set(2, 4), Set(3, 4), Set(1, 2, 3), Set(1, 3, 4), Set(1, 2, 4), Set(2, 3, 4), Set(1, 2, 3, 4))) println(s"Successfully completed without errors. [total ${currentTime - executionStart} ms]")
}</lang>
Another option that produces lazy sequence of the sets: <lang scala>def powerset[A](s: Set[A]) = (0 to s.size).map(s.toSeq.combinations(_)).reduce(_ ++ _).map(_.toSet)</lang>
A tail-recursive version: <lang scala>def powerset[A](s: Set[A]) = {
def powerset_rec(acc: List[Set[A]], remaining: List[A]): List[Set[A]] = remaining match { case Nil => acc case head :: tail => powerset_rec(acc ++ acc.map(_ + head), tail) } powerset_rec(List(Set.empty[A]), s.toList)
~~}</lang>~~
~~Scheme~~
Translation of: Common Lisp
~~<lang scheme>(define (power-set set)~~
~~(if (null? set) '(()) (let ((rest (power-set (cdr set)))) (append (map (lambda (element) (cons (car set) element)) rest) rest))))~~
(display (power-set (list 1 2 3))) (newline)
(display (power-set (list "A" "C" "E"))) (newline)</lang>

~~Output:~~
~~((1 2 3) (1 2) (1 3) (1) (2 3) (2) (3) ()) ((A C E) (A C) (A E) (A) (C E) (C) (E) ())~~
~~Call/cc generation:<lang lisp>(define (power-set lst)~~
~~(define (iter yield) (let recur ((a '()) (b lst)) (if (null? b) (set! yield~~
~~(call-with-current-continuation (lambda (resume) (set! iter resume) (yield a)))) (begin (recur (append a (list (car b))) (cdr b)) (recur a (cdr b)))))~~
~~;; signal end of generation (yield 'end-of-seq))~~
~~(lambda () (call-with-current-continuation iter)))~~
~~(define x (power-set '(1 2 3))) (let loop ((a (x)))~~
~~(if (eq? a 'end-of-seq) #f (begin (display a) (newline) (loop (x)))))</lang>~~

~~Output:~~
~~(1 2) (1 3) (1) (2 3) (2) (3) ()~~
~~Iterative:<lang scheme> (define (power_set_iter set)~~
~~(let loop ((res '(())) (s set)) (if (empty? s) res (loop (append (map (lambda (i) (cons (car s) i)) res) res) (cdr s)))))~~
~~</lang>~~

~~Output:~~
'((e d c b a) (e d c b) (e d c a) (e d c) (e d b a) (e d b) (e d a) (e d) (e c b a) (e c b) (e c a) (e c) (e b a) (e b) (e a) (e) (d c b a) (d c b) (d c a) (d c) (d b a) (d b) (d a) (d) (c b a) (c b) (c a) (c) (b a) (b) (a) ())
~~Seed7~~
<lang seed7>$ include "seed7_05.s7i";
const func array bitset: powerSet (in bitset: baseSet) is func
result var array bitset: pwrSet is [] (bitset.value); local var integer: element is 0; var integer: index is 0; var bitset: aSet is bitset.value; begin for element range baseSet do for key index range pwrSet do aSet := pwrSet[index]; if element not in aSet then incl(aSet, element); pwrSet &:= aSet; end if; end for; end for; end func;
~~const proc: main is func~~
~~local var bitset: aSet is bitset.value; begin for aSet range powerSet({1, 2, 3, 4}) do writeln(aSet); end for; end func;</lang>~~

~~Output:~~
~~{} {1} {2} {1, 2} {3} {1, 3} {2, 3} {1, 2, 3} {4} {1, 4} {2, 4} {1, 2, 4} {3, 4} {1, 3, 4} {2, 3, 4} {1, 2, 3, 4}~~
~~SETL~~
<lang haskell>Pfour := pow({1, 2, 3, 4}); Pempty := pow({}); PPempty := pow(Pempty);
print(Pfour); print(Pempty); print(PPempty);</lang>

~~Output:~~
~~{{} {1} {2} {3} {4} {1 2} {1 3} {1 4} {2 3} {2 4} {3 4} {1 2 3} {1 2 4} {1 3 4} {2 3 4} {1 2 3 4}} {{}} {{} {{}}}~~
~~Sidef~~
~~<lang ruby>var arr = %w(a b c) for i in (0 .. arr.len) {~~
~~say arr.combinations(i)~~
~~}</lang>~~

~~Output:~~
~~[[]] [["a"], ["b"], ["c"]] [["a", "b"], ["a", "c"], ["b", "c"]] [["a", "b", "c"]]~~
~~Simula~~
~~<lang simula>SIMSET BEGIN~~
~~LINK CLASS LOF_INT(N); INTEGER N;;~~
~~LINK CLASS LOF_LOF_INT(H); REF(HEAD) H;;~~
REF(HEAD) PROCEDURE MAP(P_LI, P_LLI); REF(HEAD) P_LI; REF(HEAD) P_LLI; BEGIN REF(HEAD) V_RESULT; V_RESULT :- NEW HEAD; IF NOT P_LLI.EMPTY THEN BEGIN REF(LOF_LOF_INT) V_LLI; V_LLI :- P_LLI.FIRST QUA LOF_LOF_INT; WHILE V_LLI =/= NONE DO BEGIN REF(HEAD) V_NEWLIST; V_NEWLIST :- NEW HEAD; ! ADD THE SAME 1ST ELEMENT TO EVERY NEWLIST ; NEW LOF_INT(P_LI.FIRST QUA LOF_INT.N).INTO(V_NEWLIST); IF NOT V_LLI.H.EMPTY THEN BEGIN REF(LOF_INT) V_LI; V_LI :- V_LLI.H.FIRST QUA LOF_INT; WHILE V_LI =/= NONE DO BEGIN NEW LOF_INT(V_LI.N).INTO(V_NEWLIST); V_LI :- V_LI.SUC; END; END; NEW LOF_LOF_INT(V_NEWLIST).INTO(V_RESULT); V_LLI :- V_LLI.SUC; END; END; MAP :- V_RESULT; END MAP;
REF(HEAD) PROCEDURE SUBSETS(P_LI); REF(HEAD) P_LI; BEGIN REF(HEAD) V_RESULT; IF P_LI.EMPTY THEN BEGIN V_RESULT :- NEW HEAD; NEW LOF_LOF_INT(NEW HEAD).INTO(V_RESULT); END ELSE BEGIN REF(HEAD) V_SUBSET, V_MAP; REF(LOF_INT) V_LI; V_SUBSET :- NEW HEAD; V_LI :- P_LI.FIRST QUA LOF_INT; ! SKIP OVER 1ST ELEMENT ; IF V_LI =/= NONE THEN V_LI :- V_LI.SUC; WHILE V_LI =/= NONE DO BEGIN NEW LOF_INT(V_LI.N).INTO(V_SUBSET); V_LI :- V_LI.SUC; END; V_RESULT :- SUBSETS(V_SUBSET); V_MAP :- MAP(P_LI, V_RESULT); IF NOT V_MAP.EMPTY THEN BEGIN REF(LOF_LOF_INT) V_LLI; V_LLI :- V_MAP.FIRST QUA LOF_LOF_INT; WHILE V_LLI =/= NONE DO BEGIN NEW LOF_LOF_INT(V_LLI.H).INTO(V_RESULT); V_LLI :- V_LLI.SUC; END; END; END; SUBSETS :- V_RESULT; END SUBSETS;
PROCEDURE PRINT_LIST(P_LI); REF(HEAD) P_LI; BEGIN OUTTEXT("["); IF NOT P_LI.EMPTY THEN BEGIN INTEGER I; REF(LOF_INT) V_LI; I := 0; V_LI :- P_LI.FIRST QUA LOF_INT; WHILE V_LI =/= NONE DO BEGIN IF I > 0 THEN OUTTEXT(","); OUTINT(V_LI.N, 0); V_LI :- V_LI.SUC; I := I+1; END; END; OUTTEXT("]"); END PRINT_LIST;
PROCEDURE PRINT_LIST_LIST(P_LLI); REF(HEAD) P_LLI; BEGIN OUTTEXT("["); IF NOT P_LLI.EMPTY THEN BEGIN INTEGER I; REF(LOF_LOF_INT) V_LLI; I := 0; V_LLI :- P_LLI.FIRST QUA LOF_LOF_INT; WHILE V_LLI =/= NONE DO BEGIN IF I > 0 THEN BEGIN OUTTEXT(","); ! OUTIMAGE; END; PRINT_LIST(V_LLI.H); V_LLI :- V_LLI.SUC; I := I+1; END; END; OUTTEXT("]"); OUTIMAGE; END PRINT_LIST_LIST;
~~INTEGER N; REF(HEAD) V_RANGE; REF(HEAD) V_LISTS;~~
V_RANGE :- NEW HEAD; V_LISTS :- SUBSETS(V_RANGE); PRINT_LIST_LIST(V_LISTS); OUTIMAGE; FOR N := 1 STEP 1 UNTIL 4 DO BEGIN NEW LOF_INT(N).INTO(V_RANGE); V_LISTS :- SUBSETS(V_RANGE); PRINT_LIST_LIST(V_LISTS); OUTIMAGE; END;
~~END. </lang>~~

~~Output:~~
~~[[]] [[],[1]] [[],[2],[1],[1,2]] [[],[3],[2],[2,3],[1],[1,3],[1,2],[1,2,3]] [[],[4],[3],[3,4],[2],[2,4],[2,3],[2,3,4],[1],[1,4],[1,3],[1,3,4],[1,2],[1,2,4], [1,2,3],[1,2,3,4]]~~
~~Smalltalk~~
Works with: GNU Smalltalk
Code from Bonzini's blog
<lang smalltalk>Collection extend [
~~power [ ^(0 to: (1 bitShift: self size) - 1) readStream collect: [ :each || i | i := 0. self select: [ :elem | (each bitAt: (i := i + 1)) = 1 ] ] ]~~
].</lang>
<lang smalltalk>#(1 2 4) power do: [ :each |
~~each asArray printNl ].~~
~~( 'A' 'C' 'E' ) power do: [ :each |~~
~~each asArray printNl ].</lang>~~
~~Standard ML~~
~~version for lists: <lang sml>fun subsets xs = foldr (fn (x, rest) => rest @ map (fn ys => x::ys) rest) [[]] xs</lang>~~
~~Swift~~
Works with: Swift version Revision 4 - tested with Xcode 9.2 playground
~~<lang Swift>func powersetFrom<T>(_ elements: Set<T>) -> Set<Set<T>> {~~
guard elements.count > 0 else { return [[]] } var powerset: Set<Set<T>> = [[]] for element in elements { for subset in powerset { powerset.insert(subset.union([element])) } } return powerset
}
// Example: powersetFrom([1, 2, 4])</lang>

~~Output:~~
~~{ {2, 4} {4, 1} {4}, {2, 4, 1} {2, 1} Set([]) {1} {2} }~~
~~<lang Swift>//Example: powersetFrom(["a", "b", "d"])</lang>~~

~~Output:~~
~~{ {"b", "d"} {"b"} {"d"}, {"a"} {"b", "d", "a"} Set([]) {"d", "a"} {"b", "a"} }~~
~~Tcl~~
~~<lang tcl>proc subsets {l} {~~
~~set res [list [list]] foreach e $l { foreach subset $res {lappend res [lappend subset $e]} } return $res~~
~~} puts [subsets {a b c d}]</lang>~~

~~Output:~~
~~{} a b {a b} c {a c} {b c} {a b c} d {a d} {b d} {a b d} {c d} {a c d} {b c d} {a b c d}~~
~~Binary Count Method~~
~~<lang tcl>proc powersetb set {~~
set res {} for {set i 0} {$i < 2**[llength $set]} {incr i} { set pos -1 set pset {} foreach el $set { if {$i & 1<<[incr pos]} {lappend pset $el} } lappend res $pset } return $res
~~}</lang>~~
~~TXR~~
The power set function can be written concisely like this:
<lang txr>(defun power-set (s)
~~(mappend* (op comb s) (range 0 (length s))))</lang>~~
This generates the lists of combinations of all possible lengths, from 0 to the length of s and catenates them. The comb function generates a lazy list, so it is appropriate to use mappend* (the lazy version of mappend) to keep the behavior lazy.
A complete program which takes command line arguments and prints the power set in comma-separated brace notation:
<lang txr>@(do (defun power-set (s)
~~(mappend* (op comb s) (range 0 (length s)))))~~
~~@(bind pset @(power-set *args*)) @(output) @ (repeat) {@(rep)@pset, @(last)@pset@(empty)@(end)} @ (end) @(end)</lang>~~

~~Output:~~
~~$ txr rosetta/power-set.txr 1 2 3 {1, 2, 3} {1, 2} {1, 3} {1} {2, 3} {2} {3} {}~~
The above power-set function generalizes to strings and vectors.
<lang txr>@(do (defun power-set (s)
~~(mappend* (op comb s) (range 0 (length s)))) (prinl (power-set "abc")) (prinl (power-set "b")) (prinl (power-set "")) (prinl (power-set #(1 2 3))))</lang>~~

~~Output:~~
~~$ txr power-set-generic.txr ("" "a" "b" "c" "ab" "ac" "bc" "abc") ("" "b") ("") (#() #(1) #(2) #(3) #(1 2) #(1 3) #(2 3) #(1 2 3))~~
~~UnixPipes~~
<lang ksh> | cat A a b c
| cat A |\
xargs -n 1 ksh -c 'echo \{`cat A`\}' |\ xargs |\ sed -e 's; ;,;g' \ -e 's;^;echo ;g' \ -e 's;\},;}\\ ;g' |\ ksh |unfold `wc -l A` |\ xargs -n1 -I{} ksh -c 'echo {} |\ unfold 1 |sort -u |xargs' |sort -u
~~a a b a b c a c b b c c </lang>~~
~~UNIX Shell~~
From here <lang bash>p() { [ $# -eq 0 ] && echo || (shift; p "$@") | while read r ; do echo -e "$1 $r\n$r"; done }</lang> Usage <lang bash>|p `cat` | sort | uniq A C E ^D</lang>
~~Ursala~~
Sets are a built in type constructor in Ursala, represented as lexically sorted lists with duplicates removed. The powerset function is a standard library function, but could be defined as shown below. <lang Ursala>powerset = ~&NiC+ ~&i&& ~&at^?\~&aNC ~&ahPfatPRXlNrCDrT</lang> test program: <lang Ursala>#cast %sSS
test = powerset {'a','b','c','d'}</lang>

~~Output:~~
{ {}, {'a'}, {'a','b'}, {'a','b','c'}, {'a','b','c','d'}, {'a','b','d'}, {'a','c'}, {'a','c','d'}, {'a','d'}, {'b'}, {'b','c'}, {'b','c','d'}, {'b','d'}, {'c'}, {'c','d'}, {'d'}}
V
V has a built in called powerlist <lang v>[A C E] powerlist =[[A C E] [A C] [A E] [A] [C E] [C] [E] []]</lang>
its implementation in std.v is (like joy) <lang v>[powerlist
~~[null?] [unitlist] [uncons] [dup swapd [cons] map popd swoncat] linrec].~~
~~</lang>~~
~~VBScript~~
<lang vb>Function Dec2Bin(n) q = n Dec2Bin = "" Do Until q = 0 Dec2Bin = CStr(q Mod 2) & Dec2Bin q = Int(q / 2) Loop Dec2Bin = Right("00000" & Dec2Bin,6) End Function
Function PowerSet(s) arrS = Split(s,",") PowerSet = "{" For i = 0 To 2^(UBound(arrS)+1)-1 If i = 0 Then PowerSet = PowerSet & "{}," Else binS = Dec2Bin(i) PowerSet = PowerSet & "{" c = 0 For j = Len(binS) To 1 Step -1 If CInt(Mid(binS,j,1)) = 1 Then PowerSet = PowerSet & arrS(c) & "," End If c = c + 1 Next PowerSet = Mid(PowerSet,1,Len(PowerSet)-1) & "}," End If Next PowerSet = Mid(PowerSet,1,Len(PowerSet)-1) & "}" End Function
WScript.StdOut.Write PowerSet("1,2,3,4")</lang>

~~Output:~~
~~{{},{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}}~~
~~zkl~~
~~Using a combinations function, build the power set from combinations of 1,2,... items. <lang zkl>fcn pwerSet(list){~~
~~(0).pump(list.len(),List, Utils.Helpers.pickNFrom.fp1(list), T(Void.Write,Void.Write) ) .append(list)~~
~~}</lang>~~

~~Output:~~
~~foreach n in (5){ ps:=pwerSet((1).pump(n,List)); ps.println(" Size = ",ps.len()); }~~

~~Output:~~
L(L()) Size = 1 L(L(),L(1)) Size = 2 L(L(),L(1),L(2),L(1,2)) Size = 4 L(L(),L(1),L(2),L(3),L(1,2),L(1,3),L(2,3),L(1,2,3)) Size = 8 L(L(),L(1),L(2),L(3),L(4),L(1,2),L(1,3),L(1,4),L(2,3),L(2,4), L(3,4),L(1,2,3),L(1,2,4),L(1,3,4),L(2,3,4),L(1,2,3,4)) Size = 16