Rosetta Code

Set of real numbers

Revision as of 06:48, 30 June 2018 by rosettacode>Gerard Schildberger (→‎{{header|REXX}}: added/changed comments and whitespace, used a template for the output section.)

All real numbers form the uncountable set ℝ. Among its subsets, relatively simple are the convex sets, each expressed as a range between two real numbers a and b where a ≤ b. There are actually four cases for the meaning of "between", depending on open or closed boundary:

  • [a, b]: {x | a ≤ x and x ≤ b }
  • (a, b): {x | a < x and x < b }
  • [a, b): {x | a ≤ x and x < b }
  • (a, b]: {x | a < x and x ≤ b }
Task
Set of real numbers
You are encouraged to solve this task according to the task description, using any language you may know.

Note that if a = b, of the four only [a, a] would be non-empty.

Task

  • Devise a way to represent any set of real numbers, for the definition of 'any' in the implementation notes below.
  • Provide methods for these common set operations (x is a real number; A and B are sets):
  • x ∈ A: determine if x is an element of A
example: 1 is in [1, 2), while 2, 3, ... are not.
  • A ∪ B: union of A and B, i.e. {x | x ∈ A or x ∈ B}
example: [0, 2) ∪ (1, 3) = [0, 3); [0, 1) ∪ (2, 3] = well, [0, 1) ∪ (2, 3]
  • A ∩ B: intersection of A and B, i.e. {x | x ∈ A and x ∈ B}
example: [0, 2) ∩ (1, 3) = (1, 2); [0, 1) ∩ (2, 3] = empty set
  • A - B: difference between A and B, also written as A \ B, i.e. {x | x ∈ A and x ∉ B}
example: [0, 2) − (1, 3) = [0, 1]
  • Test your implementation by checking if numbers 0, 1, and 2 are in any of the following sets:
  • (0, 1] ∪ [0, 2)
  • [0, 2) ∩ (1, 2]
  • [0, 3) − (0, 1)
  • [0, 3) − [0, 1]

Implementation notes

  • 'Any' real set means 'sets that can be expressed as the union of a finite number of convex real sets'. Cantor's set needs not apply.
  • Infinities should be handled gracefully; indeterminate numbers (NaN) can be ignored.
  • You can use your machine's native real number representation, which is probably IEEE floating point, and assume it's good enough (it usually is).

Optional work

  • Create a function to determine if a given set is empty (contains no element).
  • Define A = {x | 0 < x < 10 and |sin(π x²)| > 1/2 }, B = {x | 0 < x < 10 and |sin(π x)| > 1/2}, calculate the length of the real axis covered by the set A − B. Note that

|sin(π x)| > 1/2 is the same as n + 1/6 < x < n + 5/6 for all integers n; your program does not need to derive this by itself.

C#

<lang csharp>using System;

namespace RosettaCode.SetOfRealNumbers { 

   public class Set<TValue>
   {
       public Set(Predicate<TValue> contains)
       {
           Contains = contains;
       }

       public Predicate<TValue> Contains
       {
           get;
           private set;
       }

       public Set<TValue> Union(Set<TValue> set)
       {
           return new Set<TValue>(value => Contains(value) || set.Contains(value));
       }

       public Set<TValue> Intersection(Set<TValue> set)
       {
           return new Set<TValue>(value => Contains(value) && set.Contains(value));
       }

       public Set<TValue> Difference(Set<TValue> set)
       {
           return new Set<TValue>(value => Contains(value) && !set.Contains(value));
       }
   }

}</lang> Test: <lang csharp>using Microsoft.VisualStudio.TestTools.UnitTesting; using RosettaCode.SetOfRealNumbers;

namespace RosettaCode.SetOfRealNumbersTest { 

   [TestClass]
   public class SetTest
   {
       [TestMethod]
       public void TestUnion()
       {
           var set =
               new Set<double>(value => 0d < value && value <= 1d).Union(
                   new Set<double>(value => 0d <= value && value < 2d));
           Assert.IsTrue(set.Contains(0d));
           Assert.IsTrue(set.Contains(1d));
           Assert.IsFalse(set.Contains(2d));
       }

       [TestMethod]
       public void TestIntersection()
       {
           var set =
               new Set<double>(value => 0d <= value && value < 2d).Intersection(
                   new Set<double>(value => 1d < value && value <= 2d));
           Assert.IsFalse(set.Contains(0d));
           Assert.IsFalse(set.Contains(1d));
           Assert.IsFalse(set.Contains(2d));
       }

       [TestMethod]
       public void TestDifference()
       {
           var set =
               new Set<double>(value => 0d <= value && value < 3d).Difference(
                   new Set<double>(value => 0d < value && value < 1d));
           Assert.IsTrue(set.Contains(0d));
           Assert.IsTrue(set.Contains(1d));
           Assert.IsTrue(set.Contains(2d));

           set =
               new Set<double>(value => 0d <= value && value < 3d).Difference(
                   new Set<double>(value => 0d <= value && value <= 1d));
           Assert.IsFalse(set.Contains(0d));
           Assert.IsFalse(set.Contains(1d));
           Assert.IsTrue(set.Contains(2d));
       }
   }

}</lang>

Clojure

Translation of: Racket

<lang Clojure>(ns rosettacode.real-set)

(defn >=|<= [lo hi] #(<= lo % hi))

(defn >|< [lo hi] #(< lo % hi))

(defn >=|< [lo hi] #(and (<= lo %) (< % hi)))

(defn >|<= [lo hi] #(and (< lo %) (<= % hi)))

(def ⋃ some-fn) (def ⋂ every-pred) (defn ∖ 

 ([s1] s1)  
 ([s1 s2]
    #(and (s1 %) (not (s2 %))))
 ([s1 s2 s3]
    #(and (s1 %) (not (s2 %)) (not (s3 %))))
 ([s1 s2 s3 & ss]
    (fn [x] (every? #(not (% x)) (list* s1 s2 s3 ss)))))

(clojure.pprint/pprint 

 (map #(map % [0 1 2])
         [(⋃ (>|<= 0 1) (>=|< 0 2))
          (⋂ (>=|< 0 2) (>|<= 1 2))
          (∖ (>=|< 0 3) (>|< 0 1))
          (∖ (>=|< 0 3) (>=|<= 0 1))])

(def ∅ (constantly false)) (def R (constantly true)) (def Z integer?) (def Q ratio?) (def I #(∖ R Z Q)) (def N #(∖ Z neg?))</lang>

Common Lisp

Common Lisp has a standard way to represent intervals. <lang lisp>(deftype set== (a b) `(real ,a ,b)) (deftype set<> (a b) `(real (,a) (,b))) (deftype set=> (a b) `(real ,a (,b))) (deftype set<= (a b) `(real (,a) ,b))

(deftype set-union (s1 s2) `(or ,s1 ,s2)) (deftype set-intersection (s1 s2) `(and ,s1 ,s2)) (deftype set-diff (s1 s2) `(and ,s1 (not ,s2)))

(defun in-set-p (x set) 

 (typep x set))

(defun test () 

 (let ((set '(set-union (set<= 0 1) (set=> 0 2))))
   (assert (in-set-p 0 set))
   (assert (in-set-p 1 set))
   (assert (not (in-set-p 2 set))))
 (let ((set '(set-intersection (set=> 0 2) (set<= 1 2))))
   (assert (not (in-set-p 0 set)))
   (assert (not (in-set-p 1 set)))
   (assert (not (in-set-p 2 set))))
 (let ((set '(set-diff (set=> 0 3) (set<> 0 1))))
   (assert (in-set-p 0 set))
   (assert (in-set-p 1 set))
   (assert (in-set-p 2 set)))
 (let ((set '(set-diff (set<= 0 3) (set== 0 1))))
   (assert (not (in-set-p 0 set)))
   (assert (not (in-set-p 1 set)))
   (assert (in-set-p 2 set))))</lang>

D

Translation of: C sharp

<lang d>struct Set(T) { 

   const pure nothrow bool delegate(in T) contains;

   bool opIn_r(in T x) const pure nothrow {
       return contains(x);
   }

   Set opBinary(string op)(in Set set)
   const pure nothrow if (op == "+" || op == "-") {
       static if (op == "+")
           return Set(x => contains(x) || set.contains(x));
       else
           return Set(x => contains(x) && !set.contains(x));
   }

   Set intersection(in Set set) const pure nothrow {
       return Set(x => contains(x) && set.contains(x));
   }

}

unittest { // Test union. 

   alias DSet = Set!double;
   const s = DSet(x => 0.0 < x && x <= 1.0) +
             DSet(x => 0.0 <= x && x < 2.0);
   assert(0.0 in s);
   assert(1.0 in s);
   assert(2.0 !in s);

}

unittest { // Test difference. 

   alias DSet = Set!double;
   const s1 = DSet(x => 0.0 <= x && x < 3.0) -
              DSet(x => 0.0 < x && x < 1.0);
   assert(0.0 in s1);
   assert(0.5 !in s1);
   assert(1.0 in s1);
   assert(2.0 in s1);

   const s2 = DSet(x => 0.0 <= x && x < 3.0) -
              DSet(x => 0.0 <= x && x <= 1.0);
   assert(0.0 !in s2);
   assert(1.0 !in s2);
   assert(2.0 in s2);

   const s3 = DSet(x => 0 <= x && x <= double.infinity) -
              DSet(x => 1.0 <= x && x <= 2.0);
   assert(0.0 in s3);
   assert(1.5 !in s3);
   assert(3.0 in s3);

}

unittest { // Test intersection. 

   alias DSet = Set!double;
   const s = DSet(x => 0.0 <= x && x < 2.0).intersection(
             DSet(x => 1.0 < x && x <= 2.0));
   assert(0.0 !in s);
   assert(1.0 !in s);
   assert(2.0 !in s);

}

void main() {}</lang>

EchoLisp

Implementation of sets operations, which apply to any subsets of ℜ defined by a predicate.

Sets operations

<lang scheme> (lib 'match) ;; reader-infix macros

(reader-infix '∈ ) (reader-infix '∩ ) (reader-infix '∪ ) (reader-infix '⊖ ) ;; set difference

(define-syntax-rule (∈ x a) (a x)) (define-syntax-rule (∩ a b) (lambda(x) (and ( a x) (b x)))) (define-syntax-rule (∪ a b) (lambda(x) (or ( a x) (b x)))) (define-syntax-rule (⊖ a b) (lambda(x) (and ( a x) (not (b x)))))

predicates to define common sets

(define (∅ x) #f) ;; the empty set predicate (define (Z x) (integer? x)) (define (N x) (and (Z x) (>= x 0))) (define (Q x) (rational? x)) (define (ℜ x) #t)

predicates to define convex sets

(define (⟦...⟧ a b)(lambda(x) (and (>= x a) (<= x b)))) (define (⟦...⟦ a b)(lambda(x) (and (>= x a) (< x b)))) (define (⟧...⟧ a b)(lambda(x) (and (> x a) (<= x b)))) (define (⟧...⟦ a b)(lambda(x) (and (> x a) (< x b)))) </lang>

Output:
(3/7 ∈ ∅) → #f
(3/7 ∈ Q) → #t
(6.7 ∈ ℜ) → #t

(define A (⟦...⟧ 2 10)) ; closed interval
(define B (⟧...⟦ 5 15)) ; open interval

(8 ∈ A) → #t
(11 ∈ A)→ #f
(define AB (A ∩ B))
(8 ∈ AB) → #t
(3 ∈ AB) → #f
(5 ∈ AB) → #f ;; because B is ]5 .. 15]
(define A-B (A ⊖ B))
(5 ∈ A-B) → #t
(-666 ∈ (⟧...⟧ -Infinity 0 ))   → #t

;; task
 (0 ∈ ((⟧...⟧ 0 1)  ∪  (⟦...⟦ 0 2))) → #t
 (0 ∈ ((⟦...⟦ 0 2)  ∩  (⟧...⟧ 1 2))) → #f
 (0 ∈ ((⟦...⟦ 0 3)  ⊖  (⟧...⟦ 0 1))) → #t
 (0 ∈ ((⟦...⟦ 0 3)  ⊖  (⟦...⟧ 0 1))) → #f
 (1 ∈ ((⟧...⟧ 0 1)  ∪  (⟦...⟦ 0 2))) → #t
 (1 ∈ ((⟦...⟦ 0 2)  ∩  (⟧...⟧ 1 2))) → #f
 (1 ∈ ((⟦...⟦ 0 3)  ⊖  (⟧...⟦ 0 1))) → #t
 (1 ∈ ((⟦...⟦ 0 3)  ⊖  (⟦...⟧ 0 1))) → #f
 (2 ∈ ((⟧...⟧ 0 1)  ∪  (⟦...⟦ 0 2))) → #f
 (2 ∈ ((⟦...⟦ 0 2)  ∩  (⟧...⟧ 1 2))) → #f
 (2 ∈ ((⟦...⟦ 0 3)  ⊖  (⟧...⟦ 0 1))) → #t
 (2 ∈ ((⟦...⟦ 0 3)  ⊖  (⟦...⟧ 0 1))) → #t

Optional : measuring sets

<lang scheme>

The following applies to convex sets ⟧...⟦ Cx,
and families F of disjoint convex sets.
Cx are implemented as vectors [lo, hi]
(define-syntax-id _.lo [_ 0])
(define-syntax-id _.hi [_ 1])

;; Cx-ops
(define (Cx-new lo hi) (if (< lo hi) (vector lo hi) Cx-empty))
(define (Cx-empty? A) (>= A.lo A.hi))  
(define  Cx-empty #(+Infinity -Infinity)) 
(define (Cx-inter A B) (Cx-new (max A.lo B.lo) (min A.hi B.hi))) 
(define (Cx-measure A)  (if (< A.lo A.hi) (- A.hi A.lo) 0))

;; Families ops
(define (CF-measure FA) (for/sum ((A FA)) (Cx-measure A))) ;; because disjoint
;; intersection of two families
(define (CF-inter FA FB) (for*/list ((A FA)(B FB)) (Cx-inter A B)))
;; measure of FA/FB = m(FA) - m (FA ∩ FB)
(define (CF-measure-FA/FB FA FB)
	(- (CF-measure FA) (CF-measure (CF-inter FA FB))))
Application
FA = {x | 0 < x < 10 and |sin(π x²)| > 1/2 }
(define FA
	(for/list ((n 100)) 
         (Cx-new (sqrt (+ n (// 6))) (sqrt (+ n (// 5 6))))))
FB = {x | 0 < x < 10 and |sin(π x)| > 1/2 }
(define FB 
	(for/list ((n 10)) 
         (Cx-new (+ n (// 6)) (+ n (// 5 6)))))

→ (#(0.1667 0.8333) #(1.1667 1.8333) #(2.1667 2.8333) 

#(3.1667 3.8333) #(4.1667 4.8333) #(5.1667 5.8333) 
#(6.1667 6.8333) #(7.1667 7.8333) #(8.1667 8.8333) #(9.1667 9.8333))
	
(CF-measure-FA/FB FA FB)
     → 2.075864841184666

</lang>

Elena

ELENA 3.3 : <lang elena>import extensions.

extension setOp { 

   union : func
       = (:val)( self eval:val || func eval:val ).

   intersection : func
       = (:val)( self eval:val && func eval:val ).

   difference : func
       = (:val)( self eval:val && func eval:val; not ).

}

program = [ 

   // union
   var set := (:x)( (x >= 0.0r) && (x <= 1.0r) ) union(:x)( (x >= 0.0r) && (x < 2.0r) ).

   set(0.0r) assertTrue.
   set(1.0r) assertTrue.
   set(2.0r) assertFalse.

   // intersection
   var set2 := (:x)( (x >= 0.0r) && (x < 2.0r) ) intersection(:x)( (x >= 1.0r) && (x <= 2.0r) ).

   set2(0.0r) assertFalse.
   set2(1.0r) assertTrue.
   set2(2.0r) assertFalse.

   // difference
   var set3 := (:x)( (x >= 0.0r) && (x < 3.0r) ) difference(:x) ( (x >= 0.0r) && (x <= 1.0r) ).

   set3 (0.0r) assertFalse.
   set3 (1.0r) assertFalse.
   set3 (2.0r) assertTrue.

].</lang>

Go

Just the non-optional part: <lang go>package main

import "fmt"

type Set func(float64) bool

func Union(a, b Set) Set { return func(x float64) bool { return a(x) || b(x) } } func Inter(a, b Set) Set { return func(x float64) bool { return a(x) && b(x) } } func Diff(a, b Set) Set { return func(x float64) bool { return a(x) && !b(x) } } func open(a, b float64) Set { return func(x float64) bool { return a < x && x < b } } func closed(a, b float64) Set { return func(x float64) bool { return a <= x && x <= b } } func opCl(a, b float64) Set { return func(x float64) bool { return a < x && x <= b } } func clOp(a, b float64) Set { return func(x float64) bool { return a <= x && x < b } }

func main() { s := make([]Set, 4) s[0] = Union(opCl(0, 1), clOp(0, 2)) // (0,1] ∪ [0,2) s[1] = Inter(clOp(0, 2), opCl(1, 2)) // [0,2) ∩ (1,2] s[2] = Diff(clOp(0, 3), open(0, 1)) // [0,3) − (0,1) s[3] = Diff(clOp(0, 3), closed(0, 1)) // [0,3) − [0,1]

for i := range s { for x := float64(0); x < 3; x++ { fmt.Printf("%v ∈ s%d: %t\n", x, i, s[i](x)) } fmt.Println() } }</lang> Run in Go Playground.

Output:
0 ∈ s0: true
1 ∈ s0: true
2 ∈ s0: false

0 ∈ s1: false
1 ∈ s1: false
2 ∈ s1: false

0 ∈ s2: true
1 ∈ s2: true
2 ∈ s2: true

0 ∈ s3: false
1 ∈ s3: false
2 ∈ s3: true

This simple implementation doesn't support lengths so the although the A, B, and A−B sets can be defined and tested (see below), they can't be used to implement the optional part. <lang Go> A := Inter(open(0, 10), func(x float64) bool { return math.Abs(math.Sin(math.Pi*x*x)) > .5 }) B := Inter(open(0, 10), func(x float64) bool { return math.Abs(math.Sin(math.Pi*x)) > .5 }) C := Diff(A, B) // Can't get lengths, can only test for ∈ for x := float64(5.98); x < 6.025; x += 0.01 { fmt.Printf("%.2f ∈ A−B: %t\n", x, C(x)) }</lang>

Icon and Unicon

The following only works in Unicon. The code does a few crude simplifications of some representations, but more could be done.

<lang unicon>procedure main(A) 

   s1 := RealSet("(0,1]").union(RealSet("[0,2)"))
   s2 := RealSet("[0,2)").intersect(RealSet("(1,2)"))
   s3 := RealSet("[0,3)").difference(RealSet("(0,1)"))
   s4 := RealSet("[0,3)").difference(RealSet("[0,1]"))
   every s := s1|s2|s3|s4 do {
       every n := 0 to 2 do
           write(s.toString(),if s.contains(n) then " contains "
                                               else " doesn't contain ",n)
       write()
       }

end

class Range(a,b,lbnd,rbnd,ltest,rtest) 

   method contains(x); return ((ltest(a,x),rtest(x,b)),self); end
   method toString(); return lbnd||a||","||b||rbnd; end
   method notEmpty(); return (ltest(a,b),rtest(a,b),self); end
   method makeLTest(); return proc(if lbnd == "(" then "<" else "<=",2); end
   method makeRTest(); return proc(if rbnd == "(" then "<" else "<=",2); end

   method intersect(r)
       if a < r.a then (na := r.a, nlb := r.lbnd)
       else if a > r.a then (na := a, nlb := lbnd)
       else (na := a, nlb := if "(" == (lbnd|r.lbnd) then "(" else "[")
       if b < r.b then ( nb := b, nrb := rbnd)
       else if b > r.b then (nb := r.b, nrb := r.rbnd)
       else (nb := b, nrb := if ")" == (rbnd|r.rbnd) then ")" else "]")
       range := Range(nlb||na||","||nb||nrb)
       return range
   end

   method difference(r)
       if /r then return RealSet(toString())
       r1 := lbnd||a||","||min(b,r.a)||map(r.lbnd,"([","])")
       r2 := map(r.rbnd,")]","[(")||max(a,r.b)||","||b||rbnd
       return RealSet(r1).union(RealSet(r2))
   end

initially(s) 

   static lbnds, rbnds
   initial (lbnds := '([', rbnds := '])')
   if \s then {
       s ? {
           lbnd := (tab(upto(lbnds)),move(1))
           a := 1(tab(upto(',')),move(1))
           b := tab(upto(rbnds))
           rbnd := move(1)
           }
       ltest := proc(if lbnd == "(" then "<" else "<=",2)
       rtest := proc(if rbnd == ")" then "<" else "<=",2)
       }

end

class RealSet(ranges) 

   method contains(x); return ((!ranges).contains(x), self); end
   method notEmpty(); return ((!ranges).notEmpty(), self); end

   method toString()
       sep := s := ""
       every r := (!ranges).toString() do s ||:= .sep || 1(r, sep := " + ")
       return s
   end

   method clone()
       newR := RealSet()
       newR.ranges := (copy(\ranges) | [])
       return newR
   end

   method union(B)
       newR := clone()
       every put(newR.ranges, (!B.ranges).notEmpty())
       return newR
   end

   method intersect(B)
       newR := clone()
       newR.ranges := []
       every (r1 := !ranges, r2 := !B.ranges) do {
           range := r1.intersect(r2)
           put(newR.ranges, range.notEmpty())
           }
       return newR
   end

   method difference(B)
       newR := clone()
       newR.ranges := []
       every (r1 := !ranges, r2 := !B.ranges) do {
           rs := r1.difference(r2)
           if rs.notEmpty() then every put(newR.ranges, !rs.ranges)
           }
       return newR
   end

initially(s) 

   put(ranges := [],Range(\s).notEmpty())

end</lang>

Sample run: 

->srn
(0,1] + [0,2) contains 0
(0,1] + [0,2) contains 1
(0,1] + [0,2) doesn't contain 2

(1,2) doesn't contain 0
(1,2) doesn't contain 1
(1,2) doesn't contain 2

[0,0] + [1,3) contains 0
[0,0] + [1,3) contains 1
[0,0] + [1,3) contains 2

(1,3) doesn't contain 0
(1,3) doesn't contain 1
(1,3) contains 2

->

J

In essence, this looks like building a restricted set of statements. So we build a specialized parser and expression builder:

<lang j>has=: 1 :'(interval m)`:6' ing=: `

interval=: 3 :0 

 if.0<L.y do.y return.end.
 assert. 5=#words=. ;:y
 assert. (0 { words) e. ;:'[('
 assert. (2 { words) e. ;:','
 assert. (4 { words) e. ;:'])'
 'lo hi'=.(1 3{0".L:0 words)
 'cL cH'=.0 4{words e.;:'[]'
 (lo&(<`<:@.cL) *. hi&(>`>:@.cH))ing

)

union=: 4 :'(x has +. y has)ing' intersect=: 4 :'(x has *. y has)ing' without=: 4 :'(x has *. [: -. y has)ing'</lang>

With this in place, the required examples look like this:

<lang j> ('(0,1]' union '[0,2)')has 0 1 2 1 1 0 

  ('[0,2)' intersect '(1,2]')has 0 1 2

0 0 0 

  ('[0,3)' without '(0,1]')has 0 1 2

1 0 1 

  ('[0,3)' without '(0,1)')has 0 1 2

1 1 1 

  ('[0,3)' without '[0,1]')has 0 1 2

0 0 1</lang>

Note that without the arguments these wind up being expressions. For example:

<lang j> ('(0,1]' union '[0,2)')has (0&< *. 1&>:) +. 0&<: *. 2&></lang>

In other words, this is a statement built up from inequality terminals (where each inequality is bound to a constant) and the terminals are combined with logical operations.

Optional Work

Empty Set Detection

Here is an alternate formulation which allows detection of empty sets:

<lang j>has=: 1 :'(0 {:: interval m)`:6' ing=: `

edge=: 1&{::&interval edges=: /:~@~.@,&edge contour=: (, 2 (+/%#)\ ])@edge

interval=: 3 :0 

 if.0<L.y do.y return.end.
 assert. 5=#words=. ;:y
 assert. (0 { words) e. ;:'[('
 assert. (2 { words) e. ;:','
 assert. (4 { words) e. ;:'])'
 'lo hi'=.(1 3{0".L:0 words)
 'cL cH'=.0 4{words e.;:'[]'
 (lo&(<`<:@.cL) *. hi&(>`>:@.cH))ing ; lo,hi

)

union=: 4 :'(x has +. y has)ing; x edges y' intersect=: 4 :'(x has *. y has)ing; x edges y' without=: 4 :'(x has *. [: -. y has)ing; x edges y' in=: 4 :'y has x' isEmpty=: 1 -.@e. contour in ]</lang>

The above examples work identically with this version, but also:

<lang j> isEmpty '(0,1]' union '[0,2)' 0 

  isEmpty '[0,2)' intersect '(1,2]'

0 

  isEmpty '[0,2)' intersect '(2,3]'

1 

  isEmpty '[0,2)' intersect '[2,3]'

1 

  isEmpty '[0,2]' intersect '[2,3]'

0</lang>

Note that the the set operations no longer return a simple verb -- instead, they return a pair, where the first element represents the verb and the second element is a list of interval boundaries. We can tell if two adjacent bounds, from this list, bound a valid interval by checking any point between them.

Length of Set Difference

The optional work centers around expressions where the absolute value of sin pi * n is 0.5. It would be nice if J had an arcsine which gave all values within a range, but it does not have that. So:

<lang j> 1p_1 * _1 o. 0.5 0.166667</lang>

(Note on notation: 1 o. is sine in J, and 2 o. is cosine -- the mnemonic is that sine is an odd function and cosine is an even function, the practical value is that sine, cosine and sine/cosine pairs can all be generated from the same "real" valued function. Similarly, _1 o. is arcsine and _2 o. is arcsine. Also 1p_1 is the reciprocal of pi. So the above tells us that the principal value for arc sine 0.5 is one sixth.)

<lang j> (#~ 0.5 = 1 |@o. 1r6p1&*) i. 30 1 5 7 11 13 17 19 23 25 29 

  2 -~/\ (#~ 0.5 = 1 |@o. 1r6p1&*) i. 30

4 2 4 2 4 2 4 2 4</lang>

Here we see the integers which when multiplied by pi/6 give 0.5 for the absolute value of the sine, and their first difference. Thus:

<lang j>zeros0toN=: ((>: # ])[:+/\1,$&4 2@<.)&.(6&*)</lang>

is a function to generate the values which correspond to the boundaries of the intervals we want:

<lang j>zB=: zeros0toN 10 zA=: zeros0toN&.*: 10 

  zA

0.408248 0.912871 1.08012 1.35401 1.47196 1.68325 1.77951 1.95789 2.04124 2.1984... 

  zB

0.166667 0.833333 1.16667 1.83333 2.16667 2.83333 3.16667 3.83333 4.16667 4.8333... 

  #zA

200 

  #zB

20</lang>

And, here are the edges of the sets of intervals we need to consider.

To find the length of the the set A-B we can find the length of set A and subtract the length of the set A-B:

<lang j> (+/_2 -~/\zA) - +/,0>.zA (<.&{: - >.&{.)"1/&(_2 ]\ ]) zB 2.07586</lang>

Here, we have paired adjacent elements from the zero bounding list (non-overlapping infixes of length 2). For set A's length we sum the results of subtracting the smaller number of the pair from the larger. For set A-B's length we consider each combination of pairs from A and B and subtract the larger of the beginning values from the smaller of the ending values (and ignore any negative results).

Alternatively, if we use the set implementation with empty set detection, and the following definitions:

<lang j>intervalSet=: interval@('[',[,',',],')'"_)&": A=: union/_2 intervalSet/\ zA B=: union/_2 intervalSet/\ zB diff=: A without B</lang>

We can replace the above sentence to compute the length of the difference with:

<lang j> +/ ((2 (+/%#)\ edge diff) in diff) * 2 -~/\ edge diff 2.07588</lang>

(Note that this result is not exactly the same as the previous result. Determining why would be an interesting exercise in numerical analysis.)

Java

<lang java>import java.util.Objects; import java.util.function.Predicate;

public class RealNumberSet { 

   public enum RangeType {
       CLOSED,
       BOTH_OPEN,
       LEFT_OPEN,
       RIGHT_OPEN,
   }

   public static class RealSet {
       private Double low;
       private Double high;
       private Predicate<Double> predicate;
       private double interval = 0.00001;

       public RealSet(Double low, Double high, Predicate<Double> predicate) {
           this.low = low;
           this.high = high;
           this.predicate = predicate;
       }

       public RealSet(Double start, Double end, RangeType rangeType) {
           this(start, end, d -> {
               switch (rangeType) {
                   case CLOSED:
                       return start <= d && d <= end;
                   case BOTH_OPEN:
                       return start < d && d < end;
                   case LEFT_OPEN:
                       return start < d && d <= end;
                   case RIGHT_OPEN:
                       return start <= d && d < end;
                   default:
                       throw new IllegalStateException("Unhandled range type encountered.");
               }
           });
       }

       public boolean contains(Double d) {
           return predicate.test(d);
       }

       public RealSet union(RealSet other) {
           double low2 = Math.min(low, other.low);
           double high2 = Math.max(high, other.high);
           return new RealSet(low2, high2, d -> predicate.or(other.predicate).test(d));
       }

       public RealSet intersect(RealSet other) {
           double low2 = Math.min(low, other.low);
           double high2 = Math.max(high, other.high);
           return new RealSet(low2, high2, d -> predicate.and(other.predicate).test(d));
       }

       public RealSet subtract(RealSet other) {
           return new RealSet(low, high, d -> predicate.and(other.predicate.negate()).test(d));
       }

       public double length() {
           if (low.isInfinite() || high.isInfinite()) return -1.0; // error value
           if (high <= low) return 0.0;
           Double p = low;
           int count = 0;
           do {
               if (predicate.test(p)) count++;
               p += interval;
           } while (p < high);
           return count * interval;
       }

       public boolean isEmpty() {
           if (Objects.equals(high, low)) {
               return predicate.negate().test(low);
           }
           return length() == 0.0;
       }
   }

   public static void main(String[] args) {
       RealSet a = new RealSet(0.0, 1.0, RangeType.LEFT_OPEN);
       RealSet b = new RealSet(0.0, 2.0, RangeType.RIGHT_OPEN);
       RealSet c = new RealSet(1.0, 2.0, RangeType.LEFT_OPEN);
       RealSet d = new RealSet(0.0, 3.0, RangeType.RIGHT_OPEN);
       RealSet e = new RealSet(0.0, 1.0, RangeType.BOTH_OPEN);
       RealSet f = new RealSet(0.0, 1.0, RangeType.CLOSED);
       RealSet g = new RealSet(0.0, 0.0, RangeType.CLOSED);

       for (int i = 0; i <= 2; i++) {
           Double dd = (double) i;
           System.out.printf("(0, 1] ∪ [0, 2) contains %d is %s\n", i, a.union(b).contains(dd));
           System.out.printf("[0, 2) ∩ (1, 2] contains %d is %s\n", i, b.intersect(c).contains(dd));
           System.out.printf("[0, 3) − (0, 1) contains %d is %s\n", i, d.subtract(e).contains(dd));
           System.out.printf("[0, 3) − [0, 1] contains %d is %s\n", i, d.subtract(f).contains(dd));
           System.out.println();
       }

       System.out.printf("[0, 0] is empty is %s\n", g.isEmpty());
       System.out.println();

       RealSet aa = new RealSet(
           0.0, 10.0,
           x -> (0.0 < x && x < 10.0) && Math.abs(Math.sin(Math.PI * x * x)) > 0.5
       );
       RealSet bb = new RealSet(
           0.0, 10.0,
           x -> (0.0 < x && x < 10.0) && Math.abs(Math.sin(Math.PI * x)) > 0.5
       );
       RealSet cc = aa.subtract(bb);
       System.out.printf("Approx length of A - B is %f\n", cc.length());
   }

}</lang>

Output:
(0, 1] ∪ [0, 2) contains 0 is true
[0, 2) ∩ (1, 2] contains 0 is false
[0, 3) − (0, 1) contains 0 is true
[0, 3) − [0, 1] contains 0 is false

(0, 1] ∪ [0, 2) contains 1 is true
[0, 2) ∩ (1, 2] contains 1 is false
[0, 3) − (0, 1) contains 1 is true
[0, 3) − [0, 1] contains 1 is false

(0, 1] ∪ [0, 2) contains 2 is false
[0, 2) ∩ (1, 2] contains 2 is false
[0, 3) − (0, 1) contains 2 is true
[0, 3) − [0, 1] contains 2 is true

[0, 0] is empty is false

Approx length of A - B is 2.075870

Julia

<lang Julia> """ 

   struct ConvexRealSet

Convex real set (similar to a line segment). Parameters: lower bound, upper bound: floating point numbers 

           includelower, includeupper: boolean true or false to indicate whether
           the set has a closed boundary (set to true) or open (set to false).

""" mutable struct ConvexRealSet 

   lower::Float64
   includelower::Bool
   upper::Float64
   includeupper::Bool
   function ConvexRealSet(lo, up, incllo, inclup)
      this = new()
      this.upper = Float64(up)
      this.lower = Float64(lo)
      this.includelower = incllo
      this.includeupper = inclup
      this
   end

end


function ∈(s, xelem) 

   x = Float64(xelem)
   if(x == s.lower)
       if(s.includelower)
           return true
       else
           return false
       end
   elseif(x == s.upper)
       if(s.includeupper)
           return true
       else
           return false
       end
   end
   s.lower < x && x < s.upper

end


⋃(aset, bset, x) = (∈(aset, x) || ∈(bset, x))

⋂(aset, bset, x) = (∈(aset, x) && ∈(bset, x))

-(aset, bset, x) = (∈(aset, x) && !∈(bset, x))

isempty(s::ConvexRealSet) = (s.lower > s.upper) || 

                          ((s.lower == s.upper) && !s.includeupper && !s.includelower)


const s1 = ConvexRealSet(0.0, 1.0, false, true) const s2 = ConvexRealSet(0.0, 2.0, true, false) const s3 = ConvexRealSet(1.0, 2.0, false, true) const s4 = ConvexRealSet(0.0, 3.0, true, false) const s5 = ConvexRealSet(0.0, 1.0, false, false) const s6 = ConvexRealSet(0.0, 1.0, true, true) const sempty = ConvexRealSet(0.0, -1.0, true, true) const testlist = [0, 1, 2]


function testconvexrealset() 

   for i in testlist
       println("Testing with x = $i.\nResults:")
       println("    (0, 1] ∪ [0, 2): $(⋃(s1, s2, i))")
       println("    [0, 2) ∩ (1, 2]: $(⋂(s2, s3, i))")
       println("    [0, 3) − (0, 1): $(-(s4, s5, i))")
       println("    [0, 3) − [0, 1]: $(-(s4, s6, i))\n")
   end
   print("The set sempty is ")
   println(isempty(sempty) ? "empty." : "not empty.")

end


testconvexrealset() </lang>

Output:


Testing with x = 0.
Results:



   (0, 1] ∪ [0, 2): true
   [0, 2) ∩ (1, 2]: false
   [0, 3) − (0, 1): true
   [0, 3) − [0, 1]: false



Testing with x = 1.
Results:



   (0, 1] ∪ [0, 2): true
   [0, 2) ∩ (1, 2]: false
   [0, 3) − (0, 1): true
   [0, 3) − [0, 1]: false



Testing with x = 2.
Results:



   (0, 1] ∪ [0, 2): false
   [0, 2) ∩ (1, 2]: false
   [0, 3) − (0, 1): true
   [0, 3) − [0, 1]: true



The set sempty is empty.

Kotlin

The RealSet class has two constructors - a primary one which creates an object for an arbitrary predicate and a secondary one which creates an object for a simple range by generating the appropriate predicate and then invoking the primary one.

As far as the optional work is concerned, I decided to add a length property which gives only an approximate result. Basically, it works by keeping track of the low and high values of the set and then counting points at successive small intervals between these limits which satisfy the predicate. An isEmpty() function has also been added but as this depends, to some extent, on the length property it is not 100% reliable.

Clearly, the above approach is only suitable for sets with narrow ranges (as we have here) but does have the merit of not over-complicating the basic class. <lang scala>// version 1.1.4-3

typealias RealPredicate = (Double) -> Boolean

enum class RangeType { CLOSED, BOTH_OPEN, LEFT_OPEN, RIGHT_OPEN }

class RealSet(val low: Double, val high: Double, val predicate: RealPredicate) { 

   constructor (start: Double, end: Double, rangeType: RangeType): this(start, end,
       when (rangeType) {
           RangeType.CLOSED     -> fun(d: Double) = d in start..end
           RangeType.BOTH_OPEN  -> fun(d: Double) = start < d && d < end
           RangeType.LEFT_OPEN  -> fun(d: Double) = start < d && d <= end       
           RangeType.RIGHT_OPEN -> fun(d: Double) = start <= d && d < end
       }
   )

   fun contains(d: Double) = predicate(d)

   infix fun union(other: RealSet): RealSet {
       val low2 = minOf(low, other.low)
       val high2 = maxOf(high, other.high)
       return RealSet(low2, high2) { predicate(it) || other.predicate(it) }
   }
       
   infix fun intersect(other: RealSet): RealSet {
       val low2 = maxOf(low, other.low)
       val high2 = minOf(high, other.high)
       return RealSet(low2, high2) { predicate(it) && other.predicate(it) } 
   }

   infix fun subtract(other: RealSet) = RealSet(low, high) { predicate(it) && !other.predicate(it) }

   var interval = 0.00001

   val length: Double get() {
       if (!low.isFinite() || !high.isFinite()) return -1.0  // error value
       if (high <= low) return 0.0
       var p = low
       var count = 0
       do {
           if (predicate(p)) count++
           p += interval
       }
       while (p < high)
       return count * interval
   }

   fun isEmpty() = if (high == low) !predicate(low) else length == 0.0

}

fun main(args: Array<String>) { 

   val a = RealSet(0.0, 1.0, RangeType.LEFT_OPEN)
   val b = RealSet(0.0, 2.0, RangeType.RIGHT_OPEN)
   val c = RealSet(1.0, 2.0, RangeType.LEFT_OPEN)
   val d = RealSet(0.0, 3.0, RangeType.RIGHT_OPEN)
   val e = RealSet(0.0, 1.0, RangeType.BOTH_OPEN)
   val f = RealSet(0.0, 1.0, RangeType.CLOSED)
   val g = RealSet(0.0, 0.0, RangeType.CLOSED)

   for (i in 0..2) {
       val dd = i.toDouble()
       println("(0, 1] ∪ [0, 2) contains $i is ${(a union b).contains(dd)}")
       println("[0, 2) ∩ (1, 2] contains $i is ${(b intersect c).contains(dd)}")
       println("[0, 3) − (0, 1) contains $i is ${(d subtract e).contains(dd)}")
       println("[0, 3) − [0, 1] contains $i is ${(d subtract f).contains(dd)}\n")
   }

   println("[0, 0] is empty is ${g.isEmpty()}\n")  

   val aa = RealSet(0.0, 10.0) { x -> (0.0 < x && x < 10.0) && 
                                       Math.abs(Math.sin(Math.PI * x * x)) > 0.5  }
   val bb = RealSet(0.0, 10.0) { x -> (0.0 < x && x < 10.0) &&
                                       Math.abs(Math.sin(Math.PI * x)) > 0.5  }
   val cc = aa subtract bb
   println("Approx length of A - B is ${cc.length}")

}</lang>

Output:
(0, 1] ∪ [0, 2) contains 0 is true
[0, 2) ∩ (1, 2] contains 0 is false
[0, 3) − (0, 1) contains 0 is true
[0, 3) − [0, 1] contains 0 is false

(0, 1] ∪ [0, 2) contains 1 is true
[0, 2) ∩ (1, 2] contains 1 is false
[0, 3) − (0, 1) contains 1 is true
[0, 3) − [0, 1] contains 1 is false

(0, 1] ∪ [0, 2) contains 2 is false
[0, 2) ∩ (1, 2] contains 2 is false
[0, 3) − (0, 1) contains 2 is true
[0, 3) − [0, 1] contains 2 is true

[0, 0] is empty is false

Approx length of A - B is 2.07587

Mathematica

<lang Mathematica>(* defining functions *) setcc[a_, b_] := a <= x <= b setoo[a_, b_] := a < x < b setco[a_, b_] := a <= x < b setoc[a_, b_] := a < x <= b setSubtract[s1_, s2_] := s1 && Not[s2]; (* new function; subtraction not built in *) inSetQ[y_, set_] := set /. x -> y (* testing sets *) set1 = setoc[0, 1] || setco[0, 2] (* union built in as || shortcut (OR) *); Print[set1] Print["First trial set, (0, 1] ∪ [0, 2) , testing for {0,1,2}:"] Print[inSetQ[#, set1] & /@ {0, 1, 2}] set2 = setco[0, 2] && setoc[1, 2]; (* intersection built in as && shortcut (AND) *) Print[] Print[set2] Print["Second trial set, [0, 2) ∩ (1, 2], testing for {0,1,2}:"] Print[inSetQ[#, set2] & /@ {0, 1, 2}] Print[] set3 = setSubtract[setco[0, 3], setoo[0, 1]]; Print[set3] Print["Third trial set, [0, 3) \[Minus] (0, 1), testing for {0,1,2}"] Print[inSetQ[#, set3] & /@ {0, 1, 2}] Print[] set4 = setSubtract[setco[0, 3], setcc[0, 1]]; Print[set4] Print["Fourth trial set, [0,3)\[Minus][0,1], testing for {0,1,2}:"] Print[inSetQ[#, set4] & /@ {0, 1, 2}]</lang>

Output:
0<x<=1||0<=x<2
First trial set, (0, 1] ∪ [0, 2) , testing for {0,1,2}:
{True,True,False}

0<=x<2&&1<x<=2
Second trial set, [0, 2) ∩ (1, 2], testing for {0,1,2}:
{False,False,False}

0<=x<3&&!0<x<1
Third trial set, [0, 3) \[Minus] (0, 1), testing for {0,1,2}
{True,True,True}

0<=x<3&&!0<=x<=1
Fourth trial set, [0,3)\[Minus][0,1], testing for {0,1,2}:
{False,False,True}


PARI/GP

Define some sets and use built-in functions: <lang parigp>set11(x,a,b)=select(x -> a <= x && x <= b, x); set01(x,a,b)=select(x -> a < x && x <= b, x); set10(x,a,b)=select(x -> a <= x && x < b, x); set00(x,a,b)=select(x -> a < x && x < b, x);

V = [0, 1, 2]; 

   setunion(set01(V, 0, 1), set10(V, 0, 2))

setintersect(set10(V, 0, 2), set01(V, 1, 2)) 

   setminus(set10(V, 0, 3), set00(V, 0, 1))
   setminus(set10(V, 0, 3), set11(V, 0, 1))</lang>

Output:

[0, 1]
[]
[0, 1, 2]
[2]

Perl

<lang perl>use utf8;

  1. numbers used as boundaries to real sets. Each has 3 components:
  2. the real value x;
  3. a +/-1 indicating if it's x + ϵ or x - ϵ
  4. a 0/1 indicating if it's the left border or right border
  5. e.g. "[1.5, ..." is written "1.5, -1, 0", while "..., 2)" is "2, -1, 1"

package BNum;

use overload ( '""' => \&_str, '<=>' => \&_cmp, );

sub new { my $self = shift; bless [@_], ref $self || $self }

sub flip { my @a = @{+shift}; $a[2] = !$a[2]; bless \@a }

my $brackets = qw/ [ ( ) ] /; sub _str { my $v = sprintf "%.2f", $_[0][0]; $_[0][2] ? $v . ($_[0][1] == 1 ? "]" : ")") : ($_[0][1] == 1 ? "(" : "[" ) . $v; }

sub _cmp { my ($a, $b, $swap) = @_;

# if one of the argument is a normal number if ($swap) { return -_ncmp($a, $b) } if (!ref $b || !$b->isa(__PACKAGE__)) { return _ncmp($a, $b) }

$a->[0] <=> $b->[0] || $a->[1] <=> $b->[1] }

sub _ncmp { # $a is a BNum, $b is something comparable to a real my ($a, $b) = @_; $a->[0] <=> $b || $a->[1] <=> 0 }

package RealSet; use Carp; use overload ( '""' => \&_str, '|' => \&_or, '&' => \&_and, '~' => \&_neg, '-' => \&_diff, 'bool' => \&not_empty, # set is true if not empty );

my %pm = qw/ [ -1 ( 1 ) -1 ] 1 /; sub range { my ($cls, $a, $b, $spec) = @_; $spec =~ /^( \[ | \( )( \) | \] )$/x or croak "bad spec $spec";

$a = BNum->new($a, $pm{$1}, 0); $b = BNum->new($b, $pm{$2}, 1); normalize($a < $b ? [$a, $b] : []) }

sub normalize { my @a = @{+shift}; # remove invalid or duplicate borders, such as "[2, 1]" or "3) [3" # note that "(a" == "a]" and "a)" == "[a", but "a)" < "(a" and # "[a" < "a]" for (my $i = $#a; $i > 0; $i --) { splice @a, $i - 1, 2 if $a[$i] <= $a[$i - 1] } bless \@a }

sub not_empty { scalar @{ normalize shift } }

sub _str { my (@a, @s) = @{+shift} or return '()'; join " ∪ ", map { shift(@a).", ".shift(@a) } 0 .. $#a/2 }

sub _or { # we may have nested ranges now; let only outmost ones survive my $d = 0; normalize [ map { $_->[2] ? --$d ? () : ($_) : $d++ ? () : ($_) } sort{ $a <=> $b } @{+shift}, @{+shift} ]; }

sub _neg { normalize [ BNum->new('-inf', 1, 0), map($_->flip, @{+shift}), BNum->new('inf', -1, 1), ] }

sub _and { my $d = 0; normalize [ map { $_->[2] ? --$d ? ($_) : () : $d++ ? ($_) : () } sort{ $a <=> $b } @{+shift}, @{+shift} ]; }

sub _diff { shift() & ~shift() }

sub has { my ($a, $b) = @_; for (my $i = 0; $i < $#$a; $i += 2) { return 1 if $a->[$i] <= $b && $b <= $a->[$i + 1] } return 0 }

sub len { my ($a, $l) = shift; for (my $i = 0; $i < $#$a; $i += 2) { $l += $a->[$i+1][0] - $a->[$i][0] } return $l }

package main; use List::Util 'reduce';

sub rng { RealSet->range(@_) } my @sets = ( rng(0, 1, '(]') | rng(0, 2, '[)'), rng(0, 2, '[)') & rng(0, 2, '(]'), rng(0, 3, '[)') - rng(0, 1, '()'), rng(0, 3, '[)') - rng(0, 1, '[]'), );

for my $i (0 .. $#sets) { print "Set $i = ", $sets[$i], ": "; for (0 .. 2) { print "has $_; " if $sets[$i]->has($_); } print "\n"; }

  1. optional task

print "\n####\n"; sub brev { # show only head and tail if string too long my $x = shift; return $x if length $x < 60; substr($x, 0, 30)." ... ".substr($x, -30, 30) }

  1. "|sin(x)| > 1/2" means (n + 1/6) pi < x < (n + 5/6) pi

my $x = reduce { $a | $b } map(rng(sqrt($_ + 1./6), sqrt($_ + 5./6), '()'), 0 .. 101); $x &= rng(0, 10, '()');

print "A\t", '= {x | 0 < x < 10 and |sin(π x²)| > 1/2 }', "\n\t= ", brev($x), "\n";

my $y = reduce { $a | $b } map { rng($_ + 1./6, $_ + 5./6, '()') } 0 .. 11; $y &= rng(0, 10, '()');

print "B\t", '= {x | 0 < x < 10 and |sin(π x)| > 1/2 }', "\n\t= ", brev($y), "\n";

my $z = $x - $y; print "A - B\t= ", brev($z), "\n\tlength = ", $z->len, "\n"; print $z ? "not empty\n" : "empty\n";</lang>output<lang>Set 0 = [0.00, 2.00): has 0; has 1; Set 1 = (0.00, 2.00): has 1; Set 2 = [0.00, 0.00] ∪ [1.00, 3.00): has 0; has 1; has 2; Set 3 = (1.00, 3.00): has 2;

A = {x | 0 < x < 10 and |sin(π x²)| > 1/2 } 

       = (0.41, 0.91) ∪ (1.08, 1.35) ∪  ...  ∪ (9.91, 9.94) ∪ (9.96, 9.99)

B = {x | 0 < x < 10 and |sin(π x)| > 1/2 } 

       = (0.17, 0.83) ∪ (1.17, 1.83) ∪  ...  ∪ (8.17, 8.83) ∪ (9.17, 9.83)

A - B = [0.83, 0.91) ∪ (1.08, 1.17] ∪ ... ∪ (9.91, 9.94) ∪ (9.96, 9.99) 

       length = 2.07586484118467
       not empty</lang>

Perl 6

Works with: Rakudo version 2018.04

<lang perl6>class Iv { 

   has $.range handles <min max excludes-min excludes-max minmax ACCEPTS>;
   method empty {

$.min after $.max or $.min === $.max && ($.excludes-min || $.excludes-max) 

   }
   multi method Bool() { not self.empty };
   method length() { $.max - $.min }
   method gist() {

($.excludes-min ?? '(' !! '[') ~ $.min ~ ',' ~ $.max ~ ($.excludes-max ?? ')' !! ']'); 

   }

}

class IvSet { 

   has Iv @.intervals;

   sub canon (@i) {

my @new = consolidate(|@i).grep(*.so); @new.sort(*.range.min); 

   }

   method new(@ranges) {

my @iv = canon @ranges.map: { Iv.new(:range($_)) } self.bless(:intervals(@iv)); 

   }

   method complement {

my @new; my @old = @!intervals; if not @old { return iv -Inf..Inf; } my $pre; push @old, $(Inf^..Inf) unless @old[*-1].max === Inf; if @old[0].min === -Inf { $pre = @old.shift; } else { $pre = -Inf..^-Inf; } while @old { my $old = @old.shift; my $excludes-min = !$pre.excludes-max; my $excludes-max = !$old.excludes-min; push @new, $(Range.new($pre.max,$old.min,:$excludes-min,:$excludes-max)); $pre = $old; } IvSet.new(@new); 

   }

   method ACCEPTS(IvSet:D $me: $candidate) {

so $.intervals.any.ACCEPTS($candidate); 

   }
   method empty { so $.intervals.all.empty }
   multi method Bool() { not self.empty };

   method length() { [+] $.intervals».length }
   method gist() { join ' ', $.intervals».gist }

}

sub iv(**@ranges) { IvSet.new(@ranges) }

multi infix:<∩> (Iv $a, Iv $b) { 

   if $a.min ~~ $b or $a.max ~~ $b or $b.min ~~ $a or $b.max ~~ $a {

my $min = $a.range.min max $b.range.min; my $max = $a.range.max min $b.range.max; my $excludes-min = not $min ~~ $a & $b; my $excludes-max = not $max ~~ $a & $b; Iv.new(:range(Range.new($min,$max,:$excludes-min, :$excludes-max))); 

   }

} multi infix:<∪> (Iv $a, Iv $b) { 

   my $min = $a.range.min min $b.range.min;
   my $max = $a.range.max max $b.range.max;
   my $excludes-min = not $min ~~ $a | $b;
   my $excludes-max = not $max ~~ $a | $b;
   Iv.new(:range(Range.new($min,$max,:$excludes-min, :$excludes-max)));

}

multi infix:<∩> (IvSet $ars, IvSet $brs) { 

   my @overlap;
   for $ars.intervals -> $a {

for $brs.intervals -> $b { if $a.min ~~ $b or $a.max ~~ $b or $b.min ~~ $a or $b.max ~~ $a { my $min = $a.range.min max $b.range.min; my $max = $a.range.max min $b.range.max; my $excludes-min = not $min ~~ $a & $b; my $excludes-max = not $max ~~ $a & $b; push @overlap, $(Range.new($min,$max,:$excludes-min, :$excludes-max)); } } 

   }
   IvSet.new(@overlap)

}

multi infix:<∪> (IvSet $a, IvSet $b) { 

   iv |$a.intervals».range, |$b.intervals».range;

}

multi consolidate() { () } multi consolidate($this is copy, *@those) { 

   gather {
       for consolidate |@those -> $that {
           if $this ∩ $that { $this ∪= $that }
           else             { take $that }
       }
       take $this;
   }

}

multi infix:<−> (IvSet $a, IvSet $b) { $a ∩ $b.complement }

multi prefix:<−> (IvSet $a) { $a.complement; }

constant ℝ = iv -Inf..Inf;

my $s1 = iv(0^..1) ∪ iv(0..^2); my $s2 = iv(0..^2) ∩ iv(1^..2); my $s3 = iv(0..^3) − iv(0^..^1); my $s4 = iv(0..^3) − iv(0..1) ;

say "\t\t\t\t0\t1\t2"; say "(0, 1] ∪ [0, 2) -> $s1.gist()\t", 0 ~~ $s1,"\t", 1 ~~ $s1,"\t", 2 ~~ $s1; say "[0, 2) ∩ (1, 2] -> $s2.gist()\t", 0 ~~ $s2,"\t", 1 ~~ $s2,"\t", 2 ~~ $s2; say "[0, 3) − (0, 1) -> $s3.gist()\t", 0 ~~ $s3,"\t", 1 ~~ $s3,"\t", 2 ~~ $s3; say "[0, 3) − [0, 1] -> $s4.gist()\t", 0 ~~ $s4,"\t", 1 ~~ $s4,"\t", 2 ~~ $s4;

say ;

say "ℝ is not empty: ", !ℝ.empty; say "[0,3] − ℝ is empty: ", not iv(0..3) − ℝ;

my $A = iv(0..10) ∩ 

  iv |(0..10).map({ $_ - 1/6 .. $_ + 1/6 }).cache;

my $B = iv 0..sqrt(1/6), |(1..99).map({ $(sqrt($_-1/6) .. sqrt($_ + 1/6)) }), sqrt(100-1/6)..10;

say 'A − A is empty: ', not $A − $A;

say ;

my $C = $A − $B; say "A − B ="; say " ",.gist for $C.intervals; say "Length A − B = ", $C.length;</lang>

Output:
				0	1	2
(0, 1] ∪ [0, 2) -> [0,2)	True	True	False
[0, 2) ∩ (1, 2] -> (1,2)	False	False	False
[0, 3) − (0, 1) -> [0,0] [1,3)	True	True	True
[0, 3) − [0, 1] -> (1,3)	False	False	True

ℝ is not empty: True
[0,3] − ℝ is empty: True
A − A is empty: True

A − B =
  [0.833333,0.912870929175277)
  (1.08012344973464,1.166667]
  [1.833333,1.95789002074512)
  (2.04124145231932,2.166667]
  (2.85773803324704,2.97209241668783)
  (3.02765035409749,3.13581462037113)
  [3.833333,3.85140666943045)
  (3.89444048184931,3.97911212877111)
  (4.02077936060494,4.10284454169706)
  (4.14326763155202,4.166667]
  [4.833333,4.88193950529227)
  (4.91596040125088,4.98330546257535)
  (5.01663898109747,5.08265022732563)
  (5.11533641774094,5.166667]
  (5.84522597225006,5.90197706987526)
  (5.93014895821906,5.98609499868932)
  (6.01387285088957,6.06904715201104)
  (6.09644705272396,6.15088069574865)
  [6.833333,6.84348838921594)
  (6.8677992593455,6.91616464041548)
  (6.94022093788567,6.98808509774554)
  (7.01189465598754,7.05927286151579)
  (7.08284312029193,7.12974987873581)
  (7.15308791129165,7.166667]
  [7.833333,7.86341740805697)
  (7.88458411500991,7.92674796706274)
  (7.94774601171091,7.98957654280459)
  (8.01040989379861,8.05191488612077)
  (8.07258735887489,8.11377429642539)
  (8.13428956127495,8.166667]
  (8.8411914732499,8.87881373457813)
  (8.89756521002609,8.93495010245347)
  (8.95358401237553,8.99073597284078)
  (9.00925450115972,9.04617783007461)
  (9.06458309392477,9.10128196098403)
  (9.1195760135363,9.15605446321358)
  [9.833333,9.84039294608367)
  (9.85731538841416,9.89107341663853)
  (9.9079092984679,9.94149552800449)
  (9.9582461641931,9.99166319154791)
Length A − B = 2.07586484118467

Python

<lang python>class Setr(): 

   def __init__(self, lo, hi, includelo=True, includehi=False):
       self.eqn = "(%i<%sX<%s%i)" % (lo,
                                     '=' if includelo else ,
                                     '=' if includehi else ,
                                     hi)

   def __contains__(self, X):
       return eval(self.eqn, locals())

   # union
   def __or__(self, b):
       ans = Setr(0,0)
       ans.eqn = "(%sor%s)" % (self.eqn, b.eqn)
       return ans

   # intersection
   def __and__(self, b):
       ans = Setr(0,0)
       ans.eqn = "(%sand%s)" % (self.eqn, b.eqn)
       return ans

   # difference
   def __sub__(self, b):
       ans = Setr(0,0)
       ans.eqn = "(%sand not%s)" % (self.eqn, b.eqn)
       return ans

   def __repr__(self):
       return "Setr%s" % self.eqn


sets = [ 

   Setr(0,1, 0,1) | Setr(0,2, 1,0),
   Setr(0,2, 1,0) & Setr(1,2, 0,1),
   Setr(0,3, 1,0) - Setr(0,1, 0,0),
   Setr(0,3, 1,0) - Setr(0,1, 1,1),

] settexts = '(0, 1] ∪ [0, 2);[0, 2) ∩ (1, 2];[0, 3) − (0, 1);[0, 3) − [0, 1]'.split(';')

for s,t in zip(sets, settexts): 

   print("Set %s %s. %s" % (t,
                            ', '.join("%scludes %i"
                                    % ('in' if v in s else 'ex', v)
                                    for v in range(3)),
                            s.eqn))</lang>
Output
Set (0, 1] ∪ [0, 2) includes 0, includes 1, excludes 2. ((0<X<=1)or(0<=X<2))
Set [0, 2) ∩ (1, 2] excludes 0, excludes 1, excludes 2. ((0<=X<2)and(1<X<=2))
Set [0, 3) − (0, 1) includes 0, includes 1, includes 2. ((0<=X<3)and not(0<X<1))
Set [0, 3) − [0, 1] excludes 0, excludes 1, includes 2. ((0<=X<3)and not(0<=X<=1))

Racket

This is a simple representation of sets as functions (so obviously no good way to the the extra set length). <lang Racket>

  1. lang racket
Use a macro to allow infix operators

(require (only-in racket [#%app #%%app])) (define-for-syntax infixes '()) (define-syntax (definfix stx) 

 (syntax-case stx ()
   [(_ (x . xs) body ...) #'(definfix x (λ xs body ...))]
   [(_ x body) (begin (set! infixes (cons #'x infixes)) #'(define x body))]))

(define-syntax (#%app stx) 

 (syntax-case stx ()
   [(_ X op Y)
    (and (identifier? #'op) (ormap (λ(o) (free-identifier=? #'op o)) infixes))
    #'(#%%app op X Y)]
   [(_ f x ...) #'(#%%app f x ...)]))


Ranges
(X +-+ Y) => [X,Y]; (X --- Y) => (X,Y); and same for `+--' and `--+'
Simple implementation as functions
Constructors

(definfix ((+-+ X Y) n) (<= X n Y))  ; [X,Y] (definfix ((--- X Y) n) (< X n Y))  ; (X,Y) (definfix ((+-- X Y) n) (and (<= X n) (< n Y))) ; [X,Y) (definfix ((--+ X Y) n) (and (< X n) (<= n Y))) ; (X,Y] (definfix ((== X) n) (= X n))  ; [X,X]

Set operations

(definfix ((∪ . Rs) n) (ormap (λ(p) (p n)) Rs)) (definfix ((∩ . Rs) n) (andmap (λ(p) (p n)) Rs)) (definfix ((∖ R1 R2) n) (and (R1 n) (not (R2 n)))) ; set-minus, not backslash (define ((¬ R) n) (not (R n)))

Special sets

(define (∅ n) #f) (define (ℜ n) #t)

(define-syntax-rule (try set) 

 (apply printf "~a => ~a ~a ~a\n" (~s #:width 23 'set)
        (let ([pred set]) (for/list ([i 3]) (if (pred i) 'Y 'N)))))

(try ((0 --+ 1) ∪ (0 +-- 2))) (try ((0 +-- 2) ∩ (1 --+ 2))) (try ((0 +-- 3) ∖ (0 --- 1))) (try ((0 +-- 3) ∖ (0 +-+ 1))) </lang>

Output: 

((0 --+ 1) ∪ (0 +-- 2)) => Y Y N
((0 +-- 2) ∩ (1 --+ 2)) => N N N
((0 +-- 3) ∖ (0 --- 1)) => Y Y Y
((0 +-- 3) ∖ (0 +-+ 1)) => N N Y

REXX

no error checking, no ∞

<lang rexx>/*REXX program demonstrates a way to represent any set of real numbers and usage. */ call quertySet 1, 3, '[1,2)' call quertySet , , '[0,2) union (1,3)' call quertySet , , '[0,1) union (2,3]' call quertySet , , '[0,2] inter (1,3)' call quertySet , , '(1,2) ∩ (2,3]' call quertySet , , '[0,2) \ (1,3)' say; say center(' start of required tasks ', 40, "═") call quertySet , , '(0,1] union [0,2)' call quertySet , , '[0,2) ∩ (1,3)' call quertySet , , '[0,3] - (0,1)' call quertySet , , '[0,3] - [0,1]' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ emptySet: parse arg _; nam= valSet(_, 00); return @.3>@.4 /*──────────────────────────────────────────────────────────────────────────────────────*/ isInSet: parse arg #,x; call valSet x 

          if \datatype(#, 'N')       then call set_bad "number isn't not numeric:" #
          if (@.1=='(' &  #<=@.2) |,
             (@.1=='[' &  #< @.2) |,
             (@.4==')' &  #>=@.3) |,
             (@.4==']' &  #> @.3)    then return 0
          return 1

/*──────────────────────────────────────────────────────────────────────────────────────*/ quertySet: parse arg lv,hv,s1 oop s2 .; op=oop; upper op; cop= 

          if lv==  then lv=0;      if hv==  then hv= 2;      if op==  then cop=  0
          if wordpos(op, '| or UNION')                 \==0                 then cop= "|"
          if wordpos(op, '& ∩ AND INTER INTERSECTION') \==0                 then cop= "&"
          if wordpos(op, '\ - DIF DIFF DIFFERENCE')    \==0                 then cop= "\"
          say
                  do i=lv  to hv;  b = isInSet(i, s1)
                  if cop\==0  then do
                                   b2= isInSet(i, s2)
                                   if cop=='&'  then b= b & b2
                                   if cop=='|'  then b= b | b2
                                   if cop=='\'  then b= b & \b2
                                   end
                  express = s1 center(oop, max(5, length(oop) ) )    s2
                  say right(i, 5)    ' is in set'     express": "   word('no yes', b+1)
                  end   /*i*/
          return

/*──────────────────────────────────────────────────────────────────────────────────────*/ valSet: parse arg q; q=space(q, 0); L=length(q); @.0= ','; @.4= right(q,1) 

          parse var q    @.1  2  @.2  ','  @.3  (@.4)
          if @.2>@.3  then parse var L   . @.0  @.2  @.3
          return space(@.1 @.2 @.0 @.3 @.4, 0)</lang>
output   is the same as the next REXX version (below).

has error checking, ∞ support

<lang rexx>/*REXX program demonstrates a way to represent any set of real numbers and usage. */ call quertySet 1, 3, '[1,2)' call quertySet , , '[0,2) union (1,3)' call quertySet , , '[0,1) union (2,3]' call quertySet , , '[0,2] inter (1,3)' call quertySet , , '(1,2) ∩ (2,3]' call quertySet , , '[0,2) \ (1,3)' say; say center(' start of required tasks ', 40, "═") call quertySet , , '(0,1] union [0,2)' call quertySet , , '[0,2) ∩ (1,3)' call quertySet , , '[0,3] - (0,1)' call quertySet , , '[0,3] - [0,1]' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ badSet: say; say '***error*** bad format of SET_def: ('arg(1)")"; exit /*──────────────────────────────────────────────────────────────────────────────────────*/ emptySet: parse arg _; nam= valSet(_, 00); return @.3>@.4 /*──────────────────────────────────────────────────────────────────────────────────────*/ isInSet: parse arg #,x; call valSet x 

          if \datatype(#, 'N')       then call set_bad "number isn't not numeric:" #
          if (@.1=='(' &  #<=@.2) |,
             (@.1=='[' &  #< @.2) |,
             (@.4==')' &  #>=@.3) |,
             (@.4==']' &  #> @.3)    then return 0
          return 1

/*──────────────────────────────────────────────────────────────────────────────────────*/ quertySet: parse arg lv,hv,s1 oop s2 .; op=oop; upper op; cop= 

          if lv==  then lv=0;      if hv==  then hv= 2;      if op==  then cop=  0
          if wordpos(op, '| or UNION')                 \==0                 then cop= "|"
          if wordpos(op, '& ∩ AND INTER INTERSECTION') \==0                 then cop= "&"
          if wordpos(op, '\ - DIF DIFF DIFFERENCE')    \==0                 then cop= "\"
          say
                  do i=lv  to hv;  b = isInSet(i, s1)
                  if cop\==0  then do
                                   b2= isInSet(i, s2)
                                   if cop=='&'  then b= b & b2
                                   if cop=='|'  then b= b | b2
                                   if cop=='\'  then b= b & \b2
                                   end
                  express = s1 center(oop, max(5, length(oop) ) )    s2
                  say right(i, 5)    ' is in set'     express": "   word('no yes', b+1)
                  end   /*i*/
          return

/*──────────────────────────────────────────────────────────────────────────────────────*/ valSet: parse arg q; q=space(q, 0); L= length(q); @.0= ',' 

          infinity = copies(9, digits() - 1)'e'copies(9, digits() - 1)0
          if L<2                    then call set_bad  'invalid expression'
          @.4= right(q, 1)
          parse var q  @.1  2  @.2  ','  @.3  (@.4)
          if @.1\=='(' & @.1\=="["  then call set_bad  'left boundry'
          if @.4\==')' & @.4\=="]"  then call set_bad  'right boundry'
                   do j=2  to 3;    u=@.j;               upper u
                   if right(@.j, 1)=='∞' | u="INFINITY"  then @.j= '-'infinity
                   if \datatype(@.j, 'N')  then call set_bad  "value not numeric:"    @.j
                   end  /*j*/
          if @.2>@.3  then parse var   L  .  @.0  @.2  @.3
          return space(@.1 @.2 @.0 @.3 @.4,  0)</lang>
output   when using the (internal) default inputs:
    1  is in set [1,2)       :  yes
    2  is in set [1,2)       :  no
    3  is in set [1,2)       :  no

    0  is in set [0,2) union (1,3):  yes
    1  is in set [0,2) union (1,3):  yes
    2  is in set [0,2) union (1,3):  yes

    0  is in set [0,1) union (2,3]:  yes
    1  is in set [0,1) union (2,3]:  no
    2  is in set [0,1) union (2,3]:  no

    0  is in set [0,2] inter (1,3):  no
    1  is in set [0,2] inter (1,3):  no
    2  is in set [0,2] inter (1,3):  yes

    0  is in set (1,2)   ∩   (2,3]:  no
    1  is in set (1,2)   ∩   (2,3]:  no
    2  is in set (1,2)   ∩   (2,3]:  no

    0  is in set [0,2)   \   (1,3):  yes
    1  is in set [0,2)   \   (1,3):  yes
    2  is in set [0,2)   \   (1,3):  no

═══════ start of required tasks ════════

    0  is in set (0,1] union [0,2):  yes
    1  is in set (0,1] union [0,2):  yes
    2  is in set (0,1] union [0,2):  no

    0  is in set [0,2)   ∩   (1,3):  no
    1  is in set [0,2)   ∩   (1,3):  no
    2  is in set [0,2)   ∩   (1,3):  no

    0  is in set [0,3]   -   (0,1):  yes
    1  is in set [0,3]   -   (0,1):  yes
    2  is in set [0,3]   -   (0,1):  yes

    0  is in set [0,3]   -   [0,1]:  no
    1  is in set [0,3]   -   [0,1]:  no
    2  is in set [0,3]   -   [0,1]:  yes

Ruby

Works with: Ruby version 1.9.3

<lang ruby>class Rset 

 Set = Struct.new(:lo, :hi, :inc_lo, :inc_hi) do
   def include?(x)
     (inc_lo ? lo<=x : lo<x) and (inc_hi ? x<=hi : x<hi)
   end
   def length
     hi - lo
   end
   def to_s
     "#{inc_lo ? '[' : '('}#{lo},#{hi}#{inc_hi ? ']' : ')'}"
   end
 end
 
 def initialize(lo=nil, hi=nil, inc_lo=false, inc_hi=false)
   if lo.nil? and hi.nil?
     @sets = []            # empty set
   else
     raise TypeError      unless lo.is_a?(Numeric) and hi.is_a?(Numeric)
     raise ArgumentError  unless valid?(lo, hi, inc_lo, inc_hi)
     @sets = [Set[lo, hi, !!inc_lo, !!inc_hi]]         # !! -> Boolean values
   end
 end
 
 def self.[](lo, hi, inc_hi=true)
   self.new(lo, hi, true, inc_hi)
 end
 
 def self.parse(str)
   raise ArgumentError  unless str =~ /(\[|\()(.+),(.+)(\]|\))/
   b0, lo, hi, b1 = $~.captures        # $~ : Regexp.last_match
   lo = Rational(lo)
   lo = lo.numerator  if lo.denominator == 1
   hi = Rational(hi)
   hi = hi.numerator  if hi.denominator == 1
   self.new(lo, hi, b0=='[', b1==']')
 end
 
 def initialize_copy(obj)
   super
   @sets = @sets.map(&:dup)
 end
 
 def include?(x)
   @sets.any?{|set| set.include?(x)}
 end
 
 def empty?
   @sets.empty?
 end
 
 def union(other)
   sets = (@sets+other.sets).map(&:dup).sort_by{|set| [set.lo, set.hi]}
   work = []
   pre = sets.shift
   sets.each do |post|
     if valid?(pre.hi, post.lo, !pre.inc_hi, !post.inc_lo)
       work << pre
       pre = post
     else
       pre.inc_lo |= post.inc_lo  if pre.lo == post.lo
       if pre.hi < post.hi
         pre.hi = post.hi
         pre.inc_hi = post.inc_hi
       elsif pre.hi == post.hi
         pre.inc_hi |= post.inc_hi
       end
     end
   end
   work << pre  if pre
   new_Rset(work)
 end
 alias | union
 
 def intersection(other)
   sets = @sets.map(&:dup)
   work = []
   other.sets.each do |oset|
     sets.each do |set|
       if set.hi < oset.lo or oset.hi < set.lo
         # ignore
       elsif oset.lo < set.lo and set.hi < oset.hi
         work << set
       else
         lo = [set.lo, oset.lo].max
         if set.lo == oset.lo
           inc_lo = set.inc_lo && oset.inc_lo
         else
           inc_lo = (set.lo < oset.lo) ? oset.inc_lo : set.inc_lo
         end
         hi = [set.hi, oset.hi].min
         if set.hi == oset.hi
           inc_hi = set.inc_hi && oset.inc_hi
         else
           inc_hi = (set.hi < oset.hi) ? set.inc_hi : oset.inc_hi
         end
         work << Set[lo, hi, inc_lo, inc_hi]  if valid?(lo, hi, inc_lo, inc_hi)
       end
     end
   end
   new_Rset(work)
 end
 alias & intersection
 
 def difference(other)
   sets = @sets.map(&:dup)
   other.sets.each do |oset|
     work = []
     sets.each do |set|
       if set.hi < oset.lo or oset.hi < set.lo
         work << set
       elsif oset.lo < set.lo and set.hi < oset.hi
         # delete
       else
         if set.lo < oset.lo
           inc_hi = (set.hi==oset.lo and !set.inc_hi) ? false : !oset.inc_lo
           work << Set[set.lo, oset.lo, set.inc_lo, inc_hi]
         elsif valid?(set.lo, oset.lo, set.inc_lo, !oset.inc_lo)
           work << Set[set.lo, set.lo, true, true]
         end
         if oset.hi < set.hi
           inc_lo = (oset.hi==set.lo and !set.inc_lo) ? false : !oset.inc_hi
           work << Set[oset.hi, set.hi, inc_lo, set.inc_hi]
         elsif valid?(oset.hi, set.hi, !oset.inc_hi, set.inc_hi)
           work << Set[set.hi, set.hi, true, true]
         end
       end
     end
     sets = work
   end
   new_Rset(sets)
 end
 alias - difference
 
 # symmetric difference
 def ^(other)
   (self - other) | (other - self)
 end
 
 def ==(other)
   self.class == other.class and @sets == other.sets
 end
 
 def length
   @sets.inject(0){|len, set| len + set.length}
 end
 
 def to_s
   "#{self.class}#{@sets.join}"
 end
 alias inspect to_s
 
 protected
 
 attr_accessor :sets
 
 private
 
 def new_Rset(sets)
   rset = self.class.new          # empty set
   rset.sets = sets
   rset
 end
 
 def valid?(lo, hi, inc_lo, inc_hi)
   lo < hi or (lo==hi and inc_lo and inc_hi)
 end

end

def Rset(lo, hi, inc_hi=false) 

 Rset.new(lo, hi, false, inc_hi)

end</lang>

Test case: <lang ruby>p a = Rset[1,2,false] [1,2,3].each{|x|puts "#{x} => #{a.include?(x)}"} puts a = Rset[0,2,false] #=> Rset[0,2) b = Rset(1,3) #=> Rset(1,3) c = Rset[0,1,false] #=> Rset[0,1) d = Rset(2,3,true) #=> Rset(2,3] puts "#{a} | #{b} -> #{a | b}" puts "#{c} | #{d} -> #{c | d}" puts puts "#{a} & #{b} -> #{a & b}" puts "#{c} & #{d} -> #{c & d}" puts "(#{c} & #{d}).empty? -> #{(c&d).empty?}" puts puts "#{a} - #{b} -> #{a - b}" puts "#{a} - #{a} -> #{a - a}" e = Rset(0,3,true) f = Rset[1,2] puts "#{e} - #{f} -> #{e - f}"

puts "\nTest :" test_set = [["(0, 1]", "|", "[0, 2)"], 

           ["[0, 2)", "&", "(1, 2]"],
           ["[0, 3)", "-", "(0, 1)"],
           ["[0, 3)", "-", "[0, 1]"] ]

test_set.each do |sa,ope,sb| 

 str = "#{sa} #{ope} #{sb}"
 e = eval("Rset.parse(sa) #{ope} Rset.parse(sb)")
 puts "%s -> %s" % [str, e]
 (0..2).each{|i| puts "  #{i} : #{e.include?(i)}"}

end

puts test_set = ["x = Rset[0,2] | Rset(3,7) | Rset[8,10]", 

           "y = Rset(7,9) | Rset(5,6) | Rset[1,4]",
           "x | y", "x & y", "x - y", "y - x", "x ^ y",
           "y ^ x == (x | y) - (x & y)"]

x = y = nil test_set.each {|str| puts "#{str} -> #{eval(str)}"}

puts inf = 1.0 / 0.0 # infinity puts "a = #{a = Rset(-inf,inf)}" puts "b = #{b = Rset.parse('[1/3,11/7)')}" puts "a - b -> #{a - b}"</lang>

Output:
Rset[1,2)
1 => true
2 => false
3 => false

Rset[0,2) | Rset(1,3) -> Rset[0,3)
Rset[0,1) | Rset(2,3] -> Rset[0,1)(2,3]

Rset[0,2) & Rset(1,3) -> Rset(1,2)
Rset[0,1) & Rset(2,3] -> Rset
(Rset[0,1) & Rset(2,3]).empty? -> true

Rset[0,2) - Rset(1,3) -> Rset[0,1]
Rset[0,2) - Rset[0,2) -> Rset
Rset(0,3] - Rset[1,2] -> Rset(0,1)(2,3]

Test :
(0, 1] | [0, 2) -> Rset[0,2)
  0 : true
  1 : true
  2 : false
[0, 2) & (1, 2] -> Rset(1,2)
  0 : false
  1 : false
  2 : false
[0, 3) - (0, 1) -> Rset[0,0][1,3)
  0 : true
  1 : true
  2 : true
[0, 3) - [0, 1] -> Rset(1,3)
  0 : false
  1 : false
  2 : true

x = Rset[0,2] | Rset(3,7) | Rset[8,10] -> Rset[0,2](3,7)[8,10]
y = Rset(7,9) | Rset(5,6) | Rset[1,4] -> Rset[1,4](5,6)(7,9)
x | y -> Rset[0,7)(7,10]
x & y -> Rset[1,2](3,4](5,6)[8,9)
x - y -> Rset[0,1)(4,5][6,7)[9,10]
y - x -> Rset(2,3](7,8)
x ^ y -> Rset[0,1)(2,3](4,5][6,7)(7,8)[9,10]
y ^ x == (x | y) - (x & y) -> true

a = Rset(-Infinity,Infinity)
b = Rset[1/3,11/7)
a - b -> Rset(-Infinity,1/3)[11/7,Infinity)

Optional work:

Works with: Ruby version 2.1+

(with Rational suffix.) <lang ruby>str, e = "e = Rset.new", nil puts "#{str} -> #{eval(str)}\t\t# create empty set" str = "e.empty?" puts "#{str} -> #{eval(str)}" puts

include Math lohi = Enumerator.new do |y| 

 t = 1 / sqrt(6)
 0.step do |n|
   y << [sqrt(12*n+1) * t, sqrt(12*n+5) * t]
   y << [sqrt(12*n+7) * t, sqrt(12*n+11) * t]
 end

end

a = Rset.new loop do 

 lo, hi = lohi.next
 break  if 10 <= lo
 a |= Rset(lo, hi)

end a &= Rset(0,10)

b = (0...10).inject(Rset.new){|res,i| res |= Rset(i+1/6r,i+5/6r)}

puts "a  : #{a}" puts "a.length : #{a.length}" puts "b  : #{b}" puts "b.length : #{b.length}" puts "a - b  : #{a - b}" puts "(a-b).length : #{(a-b).length}"</lang>

Output:
e = Rset.new -> Rset		# create empty set
e.empty? -> true

a        : Rset(0.4082482904638631,0.912870929175277)(1.0801234497346435,1.3540064007726602)(1.4719601443879746,1.6832508230603467) ... (9.907909298467901,9.941495528004495)(9.958246164193106,9.991663191547909)
a.length : 6.50103079235655
b        : Rset(1/6,5/6)(7/6,11/6)(13/6,17/6)(19/6,23/6)(25/6,29/6)(31/6,35/6)(37/6,41/6)(43/6,47/6)(49/6,53/6)(55/6,59/6)
b.length : 20/3
a - b    : Rset[5/6,0.912870929175277)(1.0801234497346435,7/6][11/6,1.9578900207451218)(2.041241452319315,13/6] ... (9.907909298467901,9.941495528004495)(9.958246164193106,9.991663191547909)
(a-b).length : 2.0758648411846745

Tcl

This code represents each set of real numbers as a collection of ranges, where each range is quad of the two boundary values and whether each of those boundaries is a closed boundary. (Using expressions internally would make the code much shorter, at the cost of being much less tractable when it comes to deriving information like the length of the real line “covered” by the set.) A side-effect of the representation is that the length of the list that represents the set is, after normalization, the number of discrete ranges in the set. <lang tcl>package require Tcl 8.5

proc inRange {x range} { 

   lassign $range a aClosed b bClosed
   expr {($aClosed ? $a<=$x : $a<$x) && ($bClosed ? $x<=$b : $x<$b)}

} proc normalize {A} { 

   set A [lsort -index 0 -real [lsort -index 1 -integer -decreasing $A]]
   for {set i 0} {$i < [llength $A]} {incr i} {

lassign [lindex $A $i] a aClosed b bClosed if {$b < $a || ($a == $b && !($aClosed && $bClosed))} { set A [lreplace $A $i $i] incr i -1 } 

   }
   for {set i 0} {$i < [llength $A]} {incr i} {

for {set j [expr {$i+1}]} {$j < [llength $A]} {incr j} { set R [lindex $A $i] lassign [lindex $A $j] a aClosed b bClosed if {[inRange $a $R]} { if {![inRange $b $R]} { lset A $i 2 $b lset A $i 3 $bClosed } set A [lreplace $A $j $j] incr j -1 } } 

   }
   return $A

}

proc realset {args} { 

   set RE {^\s*([\[(])\s*([-\d.e]+|-inf)\s*,\s*([-\d.e]+|inf)\s*([\])])\s*$}
   set result {}
   foreach s $args {

if { [regexp $RE $s --> left a b right] && [string is double $a] && [string is double $b] } then { lappend result [list \ $a [expr {$left eq "\["}] $b [expr {$right eq "\]"}]] } else { error "bad range descriptor" } 

   }
   return $result

} proc elementOf {x A} { 

   foreach range $A {

if {[inRange $x $range]} {return 1} 

   }
   return 0

} proc union {A B} { 

   return [normalize [concat $A $B]]

} proc intersection {A B} { 

   set B [normalize $B]
   set C {}
   foreach RA [normalize $A] {

lassign $RA Aa AaClosed Ab AbClosed foreach RB $B { lassign $RB Ba BaClosed Bb BbClosed if {$Aa > $Bb || $Ba > $Ab} continue set RC {} lappend RC [expr {max($Aa,$Ba)}] if {$Aa==$Ba} { lappend RC [expr {min($AaClosed,$BaClosed)}] } else { lappend RC [expr {$Aa>$Ba ? $AaClosed : $BaClosed}] } lappend RC [expr {min($Ab,$Bb)}] if {$Ab==$Bb} { lappend RC [expr {min($AbClosed,$BbClosed)}] } else { lappend RC [expr {$Ab<$Bb ? $AbClosed : $BbClosed}] } lappend C $RC } 

   }
   return [normalize $C]

} proc difference {A B} { 

   set C {}
   set B [normalize $B]
   foreach arange [normalize $A] {

if {[isEmpty [intersection [list $arange] $B]]} { lappend C $arange continue } lassign $arange Aa AaClosed Ab AbClosed foreach brange $B { lassign $brange Ba BaClosed Bb BbClosed if {$Bb < $Aa || ($Bb==$Aa && !($AaClosed && $BbClosed))} { continue } if {$Ab < $Ba || ($Ab==$Ba && !($BaClosed && $AbClosed))} { lappend C [list $Aa $AaClosed $Ab $AbClosed] unset arange break } if {$Aa==$Bb} { set AaClosed 0 continue } elseif {$Ab==$Ba} { set AbClosed 0 lappend C [list $Aa $AaClosed $Ab $AbClosed] unset arange continue } if {$Aa<$Ba} { lappend C [list $Aa $AaClosed $Ba [expr {!$BaClosed}]] if {$Ab>$Bb} { set Aa $Bb set AaClosed [expr {!$BbClosed}] } else { unset arange break } } elseif {$Aa==$Ba} { lappend C [list $Aa $AaClosed $Ba [expr {!$BaClosed}]] set Aa $Bb set AaClosed [expr {!$BbClosed}] } else { set Aa $Bb set AaClosed [expr {!$BbClosed}] } } if {[info exist arange]} { lappend C [list $Aa $AaClosed $Ab $AbClosed] } 

   }
   return [normalize $C]

} proc isEmpty A { 

   expr {![llength [normalize $A]]}

} proc length A { 

   set len 0.0
   foreach range [normalize $A] {

lassign $range a _ b _ set len [expr {$len + ($b-$a)}] 

   }
   return $len

}</lang> Basic problems: <lang tcl>foreach {str Set} { 

   {(0, 1] ∪ [0, 2)} {

union [realset {(0,1]}] [realset {[0,2)}] 

   }
   {[0, 2) ∩ (1, 2]} {

intersection [realset {[0,2)}] [realset {(1,2]}] 

   }
   {[0, 3) − (0, 1)} {

difference [realset {[0,3)}] [realset {(0,1)}] 

   }
   {[0, 3) − [0, 1]} {

difference [realset {[0,3)}] [realset {[0,1]}] 

   }

} { 

   set Set [eval $Set]
   foreach x {0 1 2} {

puts "$x : $str :\t[elementOf $x $Set]" 

   }

}</lang> Extra credit: <lang tcl>proc spi2 {from to} { 

   for {set i $from} {$i<=$to} {incr i} {

lappend result [list [expr {$i+1./6}] 0 [expr {$i+5./6}] 0] 

   }
   return [intersection [list [list $from 0 $to 0]] $result]

} proc applyfunc {var func} { 

   upvar 1 $var A
   for {set i 0} {$i < [llength $A]} {incr i} {

lassign [lindex $A $i] a - b - lset A $i 0 [$func $a] lset A $i 2 [$func $b] 

   }

} set A [spi2 0 100] applyfunc A ::tcl::mathfunc::sqrt set B [spi2 0 10] set AB [difference $A $B] puts "[llength $AB] contiguous subsets, total length [length $AB]"</lang> Output: 

0 : (0, 1] ∪ [0, 2) :	1
1 : (0, 1] ∪ [0, 2) :	1
2 : (0, 1] ∪ [0, 2) :	0
0 : [0, 2) ∩ (1, 2] :	0
1 : [0, 2) ∩ (1, 2] :	0
2 : [0, 2) ∩ (1, 2] :	0
0 : [0, 3) − (0, 1) :	1
1 : [0, 3) − (0, 1) :	1
2 : [0, 3) − (0, 1) :	1
0 : [0, 3) − [0, 1] :	0
1 : [0, 3) − [0, 1] :	0
2 : [0, 3) − [0, 1] :	1
40 contiguous subsets, total length 2.075864841184667

zkl

Translation of: D

No ∞ <lang zkl>class RealSet{ 

  fcn init(fx){ var [const] contains=fx; }
  fcn holds(x){ contains(x) }
  fcn __opAdd(rs){ RealSet('wrap(x){ contains(x) or rs.contains(x) }) }
  fcn __opSub(rs){ RealSet('wrap(x){ contains(x) and not rs.contains(x) }) }
  fcn intersection(rs) { RealSet('wrap(x){ contains(x) and rs.contains(x) }) }

}</lang> The python method could used but the zkl compiler is slow when used in code to generate code.

The method used is a bit inefficient because it closes the contains function of the other set so you can build quite a long call chain as you create new sets. <lang zkl>tester := TheVault.Test.UnitTester.UnitTester(); 

   // test union

s:=RealSet(fcn(x){ 0.0 < x <= 1.0 }) + 

  RealSet(fcn(x){ 0.0 <= x <  1.0 });

tester.testRun(s.holds(0.0),Void,True,__LINE__); tester.testRun(s.holds(1.0),Void,True,__LINE__); tester.testRun(s.holds(2.0),Void,False,__LINE__); 

   // test difference

s1 := RealSet(fcn(x){ 0.0 <= x < 3.0 }) - 

     RealSet(fcn(x){ 0.0 <  x < 1.0 });

tester.testRun(s1.holds(0.0),Void,True,__LINE__); tester.testRun(s1.holds(0.5),Void,False,__LINE__); tester.testRun(s1.holds(1.0),Void,True,__LINE__); tester.testRun(s1.holds(2.0),Void,True,__LINE__);

s2 := RealSet(fcn(x){ 0.0 <= x < 3.0 }) - 

     RealSet(fcn(x){ 0.0 <= x <= 1.0 });

tester.testRun(s2.holds(0.0),Void,False,__LINE__); tester.testRun(s2.holds(1.0),Void,False,__LINE__); tester.testRun(s2.holds(2.0),Void,True,__LINE__); 

   // test intersection

s := RealSet(fcn(x){ 0.0 <= x < 2.0 }).intersection( 

    RealSet(fcn(x){ 1.0 <  x <= 2.0 }));

tester.testRun(s.holds(0.0),Void,False,__LINE__); tester.testRun(s.holds(1.0),Void,False,__LINE__); tester.testRun(s.holds(2.0),Void,False,__LINE__);</lang>

Output:
$ zkl bbb
===================== Unit Test 1 =====================
Test 1 passed!
===================== Unit Test 2 =====================
Test 2 passed!
...
===================== Unit Test 12 =====================
Test 12 passed!
===================== Unit Test 13 =====================
Test 13 passed!
Retrieved from "https://rosettacode.org/wiki/Set_of_real_numbers?oldid=201884"