Set of real numbers
You are encouraged to solve this task according to the task description, using any language you may know.
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 }
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 need 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
Providing an implementation of lambdas would be better, but this should do for now.
#include <math.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
struct RealSet {
bool(*contains)(struct RealSet*, struct RealSet*, double);
struct RealSet *left;
struct RealSet *right;
double low, high;
};
typedef enum {
CLOSED,
LEFT_OPEN,
RIGHT_OPEN,
BOTH_OPEN,
} RangeType;
double length(struct RealSet *self) {
const double interval = 0.00001;
double p = self->low;
int count = 0;
if (isinf(self->low) || isinf(self->high)) return -1.0;
if (self->high <= self->low) return 0.0;
do {
if (self->contains(self, NULL, p)) count++;
p += interval;
} while (p < self->high);
return count * interval;
}
bool empty(struct RealSet *self) {
if (self->low == self->high) {
return !self->contains(self, NULL, self->low);
}
return length(self) == 0.0;
}
static bool contains_closed(struct RealSet *self, struct RealSet *_, double d) {
return self->low <= d && d <= self->high;
}
static bool contains_left_open(struct RealSet *self, struct RealSet *_, double d) {
return self->low < d && d <= self->high;
}
static bool contains_right_open(struct RealSet *self, struct RealSet *_, double d) {
return self->low <= d && d < self->high;
}
static bool contains_both_open(struct RealSet *self, struct RealSet *_, double d) {
return self->low < d && d < self->high;
}
static bool contains_intersect(struct RealSet *self, struct RealSet *_, double d) {
return self->left->contains(self->left, NULL, d) && self->right->contains(self->right, NULL, d);
}
static bool contains_union(struct RealSet *self, struct RealSet *_, double d) {
return self->left->contains(self->left, NULL, d) || self->right->contains(self->right, NULL, d);
}
static bool contains_subtract(struct RealSet *self, struct RealSet *_, double d) {
return self->left->contains(self->left, NULL, d) && !self->right->contains(self->right, NULL, d);
}
struct RealSet* makeSet(double low, double high, RangeType type) {
bool(*contains)(struct RealSet*, struct RealSet*, double);
struct RealSet *rs;
switch (type) {
case CLOSED:
contains = contains_closed;
break;
case LEFT_OPEN:
contains = contains_left_open;
break;
case RIGHT_OPEN:
contains = contains_right_open;
break;
case BOTH_OPEN:
contains = contains_both_open;
break;
default:
return NULL;
}
rs = malloc(sizeof(struct RealSet));
rs->contains = contains;
rs->left = NULL;
rs->right = NULL;
rs->low = low;
rs->high = high;
return rs;
}
struct RealSet* makeIntersect(struct RealSet *left, struct RealSet *right) {
struct RealSet *rs = malloc(sizeof(struct RealSet));
rs->contains = contains_intersect;
rs->left = left;
rs->right = right;
rs->low = fmin(left->low, right->low);
rs->high = fmin(left->high, right->high);
return rs;
}
struct RealSet* makeUnion(struct RealSet *left, struct RealSet *right) {
struct RealSet *rs = malloc(sizeof(struct RealSet));
rs->contains = contains_union;
rs->left = left;
rs->right = right;
rs->low = fmin(left->low, right->low);
rs->high = fmin(left->high, right->high);
return rs;
}
struct RealSet* makeSubtract(struct RealSet *left, struct RealSet *right) {
struct RealSet *rs = malloc(sizeof(struct RealSet));
rs->contains = contains_subtract;
rs->left = left;
rs->right = right;
rs->low = left->low;
rs->high = left->high;
return rs;
}
int main() {
struct RealSet *a = makeSet(0.0, 1.0, LEFT_OPEN);
struct RealSet *b = makeSet(0.0, 2.0, RIGHT_OPEN);
struct RealSet *c = makeSet(1.0, 2.0, LEFT_OPEN);
struct RealSet *d = makeSet(0.0, 3.0, RIGHT_OPEN);
struct RealSet *e = makeSet(0.0, 1.0, BOTH_OPEN);
struct RealSet *f = makeSet(0.0, 1.0, CLOSED);
struct RealSet *g = makeSet(0.0, 0.0, CLOSED);
int i;
for (i = 0; i < 3; ++i) {
struct RealSet *t;
t = makeUnion(a, b);
printf("(0, 1] union [0, 2) contains %d is %d\n", i, t->contains(t, NULL, i));
free(t);
t = makeIntersect(b, c);
printf("[0, 2) intersect (1, 2] contains %d is %d\n", i, t->contains(t, NULL, i));
free(t);
t = makeSubtract(d, e);
printf("[0, 3) - (0, 1) contains %d is %d\n", i, t->contains(t, NULL, i));
free(t);
t = makeSubtract(d, f);
printf("[0, 3) - [0, 1] contains %d is %d\n", i, t->contains(t, NULL, i));
free(t);
printf("\n");
}
printf("[0, 0] is empty %d\n", empty(g));
free(a);
free(b);
free(c);
free(d);
free(e);
free(f);
free(g);
return 0;
}
- Output:
(0, 1] union [0, 2) contains 0 is 1 [0, 2) intersect (1, 2] contains 0 is 0 [0, 3) - (0, 1) contains 0 is 1 [0, 3) - [0, 1] contains 0 is 0 (0, 1] union [0, 2) contains 1 is 1 [0, 2) intersect (1, 2] contains 1 is 0 [0, 3) - (0, 1) contains 1 is 1 [0, 3) - [0, 1] contains 1 is 0 (0, 1] union [0, 2) contains 2 is 0 [0, 2) intersect (1, 2] contains 2 is 0 [0, 3) - (0, 1) contains 2 is 1 [0, 3) - [0, 1] contains 2 is 1 [0, 0] is empty 0
C#
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));
}
}
}
Test:
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));
}
}
}
C++
#include <cassert>
#include <functional>
#include <iostream>
#define _USE_MATH_DEFINES
#include <math.h>
enum RangeType {
CLOSED,
BOTH_OPEN,
LEFT_OPEN,
RIGHT_OPEN
};
class RealSet {
private:
double low, high;
double interval = 0.00001;
std::function<bool(double)> predicate;
public:
RealSet(double low, double high, const std::function<bool(double)>& predicate) {
this->low = low;
this->high = high;
this->predicate = predicate;
}
RealSet(double start, double end, RangeType rangeType) {
low = start;
high = end;
switch (rangeType) {
case CLOSED:
predicate = [start, end](double d) { return start <= d && d <= end; };
break;
case BOTH_OPEN:
predicate = [start, end](double d) { return start < d && d < end; };
break;
case LEFT_OPEN:
predicate = [start, end](double d) { return start < d && d <= end; };
break;
case RIGHT_OPEN:
predicate = [start, end](double d) { return start <= d && d < end; };
break;
default:
assert(!"Unexpected range type encountered.");
}
}
bool contains(double d) const {
return predicate(d);
}
RealSet unionSet(const RealSet& rhs) const {
double low2 = fmin(low, rhs.low);
double high2 = fmax(high, rhs.high);
return RealSet(
low2, high2,
[this, &rhs](double d) { return predicate(d) || rhs.predicate(d); }
);
}
RealSet intersect(const RealSet& rhs) const {
double low2 = fmin(low, rhs.low);
double high2 = fmax(high, rhs.high);
return RealSet(
low2, high2,
[this, &rhs](double d) { return predicate(d) && rhs.predicate(d); }
);
}
RealSet subtract(const RealSet& rhs) const {
return RealSet(
low, high,
[this, &rhs](double d) { return predicate(d) && !rhs.predicate(d); }
);
}
double length() const {
if (isinf(low) || isinf(high)) return -1.0; // error value
if (high <= low) return 0.0;
double p = low;
int count = 0;
do {
if (predicate(p)) count++;
p += interval;
} while (p < high);
return count * interval;
}
bool empty() const {
if (high == low) {
return !predicate(low);
}
return length() == 0.0;
}
};
int main() {
using namespace std;
RealSet a(0.0, 1.0, LEFT_OPEN);
RealSet b(0.0, 2.0, RIGHT_OPEN);
RealSet c(1.0, 2.0, LEFT_OPEN);
RealSet d(0.0, 3.0, RIGHT_OPEN);
RealSet e(0.0, 1.0, BOTH_OPEN);
RealSet f(0.0, 1.0, CLOSED);
RealSet g(0.0, 0.0, CLOSED);
for (int i = 0; i <= 2; ++i) {
cout << "(0, 1] ∪ [0, 2) contains " << i << " is " << boolalpha << a.unionSet(b).contains(i) << "\n";
cout << "[0, 2) ∩ (1, 2] contains " << i << " is " << boolalpha << b.intersect(c).contains(i) << "\n";
cout << "[0, 3) - (0, 1) contains " << i << " is " << boolalpha << d.subtract(e).contains(i) << "\n";
cout << "[0, 3) - [0, 1] contains " << i << " is " << boolalpha << d.subtract(f).contains(i) << "\n";
cout << endl;
}
cout << "[0, 0] is empty is " << boolalpha << g.empty() << "\n";
cout << endl;
RealSet aa(
0.0, 10.0,
[](double x) { return (0.0 < x && x < 10.0) && abs(sin(M_PI * x * x)) > 0.5; }
);
RealSet bb(
0.0, 10.0,
[](double x) { return (0.0 < x && x < 10.0) && abs(sin(M_PI * x)) > 0.5; }
);
auto cc = aa.subtract(bb);
cout << "Approximate length of A - B is " << cc.length() << endl;
return 0;
}
- 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 Approximate length of A - B is 2.07587
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?))
Common Lisp
Common Lisp has a standard way to represent intervals.
(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))))
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() {}
Delphi
program Set_of_real_numbers;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
type
TSet = TFunc<Double, boolean>;
function Union(a, b: TSet): TSet;
begin
Result :=
function(x: double): boolean
begin
Result := a(x) or b(x);
end;
end;
function Inter(a, b: TSet): TSet;
begin
Result :=
function(x: double): boolean
begin
Result := a(x) and b(x);
end;
end;
function Diff(a, b: TSet): TSet;
begin
Result :=
function(x: double): boolean
begin
Result := a(x) and not b(x);
end;
end;
function Open(a, b: double): TSet;
begin
Result :=
function(x: double): boolean
begin
Result := (a < x) and (x < b);
end;
end;
function closed(a, b: double): TSet;
begin
Result :=
function(x: double): boolean
begin
Result := (a <= x) and (x <= b);
end;
end;
function opCl(a, b: double): TSet;
begin
Result :=
function(x: double): boolean
begin
Result := (a < x) and (x <= b);
end;
end;
function clOp(a, b: double): TSet;
begin
Result :=
function(x: double): boolean
begin
Result := (a <= x) and (x < b);
end;
end;
const
BOOLSTR: array[Boolean] of string = ('False', 'True');
begin
var s: TArray<TSet>;
SetLength(s, 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) n (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 var i := 0 to High(s) do
begin
for var x := 0 to 2 do
writeln(format('%d e s%d: %s', [x, i, BOOLSTR[s[i](x)]]));
writeln;
end;
readln;
end.
EchoLisp
Implementation of sets operations, which apply to any subsets of ℜ defined by a predicate.
Sets operations
(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))))
- 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
;; 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
Elena
ELENA 6.x :
import extensions;
extension setOp
{
union(func)
= (val => self(val) || func(val) );
intersection(func)
= (val => self(val) && func(val) );
difference(func)
= (val => self(val) && (func(val)).Inverted );
}
public 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();
}FreeBASIC
Type Func
As Integer ID
As Double ARGS(2)
End Type
Declare Function cf(f As Func, x As Double) As Boolean
Declare Function Union_(a As Func, b As Func, x As Double) As Boolean
Declare Function Inters(a As Func, b As Func, x As Double) As Boolean
Declare Function Differ(a As Func, b As Func, x As Double) As Boolean
Declare Function OpOp(a As Double, b As Double, x As Double) As Boolean
Declare Function ClCl(a As Double, b As Double, x As Double) As Boolean
Declare Function OpCl(a As Double, b As Double, x As Double) As Boolean
Declare Function ClOp(a As Double, b As Double, x As Double) As Boolean
Declare Function aspxx(a As Double) As Boolean
Declare Function aspx(a As Double) As Boolean
Function cf(f As Func, x As Double) As Boolean
Select Case f.ID
Case 1: Return OpOp(f.ARGS(0), f.ARGS(1), x)
Case 2: Return ClCl(f.ARGS(0), f.ARGS(1), x)
Case 3: Return OpCl(f.ARGS(0), f.ARGS(1), x)
Case 4: Return ClOp(f.ARGS(0), f.ARGS(1), x)
'Extra credit
Case 5: Return OpOp(f.ARGS(0), f.ARGS(1), x) And aspxx(x)
Case 6: Return OpOp(f.ARGS(0), f.ARGS(1), x) And aspx(x)
End Select
End Function
Function Union_(a As Func, b As Func, x As Double) As Boolean
Return cf(a, x) Or cf(b, x)
End Function
Function Inters(a As Func, b As Func, x As Double) As Boolean
Return cf(a, x) And cf(b, x)
End Function
Function Differ(a As Func, b As Func, x As Double) As Boolean
Return cf(a, x) And (Not cf(b, x))
End Function
Function OpOp(a As Double, b As Double, x As Double) As Boolean
Return a < x And x < b
End Function
Function ClCl(a As Double, b As Double, x As Double) As Boolean
Return a <= x And x <= b
End Function
Function OpCl(a As Double, b As Double, x As Double) As Boolean
Return a < x And x <= b
End Function
Function ClOp(a As Double, b As Double, x As Double) As Boolean
Return a <= x And x < b
End Function
'Extra credit
Function aspxx(a As Double) As Boolean
Return Abs(Sin(3.14159 * a * a)) > 0.5
End Function
Function aspx(a As Double) As Boolean
Return Abs(Sin(3.14159 * a)) > 0.5
End Function
' Set definitions and test methods
Dim As Func s(6, 2)
s(1, 0).ID = 3: s(1, 0).ARGS(0) = 0: s(1, 0).ARGS(1) = 1
s(1, 1).ID = 4: s(1, 1).ARGS(0) = 0: s(1, 1).ARGS(1) = 2
s(2, 0).ID = 4: s(2, 0).ARGS(0) = 0: s(2, 0).ARGS(1) = 2
s(2, 1).ID = 3: s(2, 1).ARGS(0) = 1: s(2, 1).ARGS(1) = 2
s(3, 0).ID = 4: s(3, 0).ARGS(0) = 0: s(3, 0).ARGS(1) = 3
s(3, 1).ID = 1: s(3, 1).ARGS(0) = 0: s(3, 1).ARGS(1) = 1
s(4, 0).ID = 4: s(4, 0).ARGS(0) = 0: s(4, 0).ARGS(1) = 3
s(4, 1).ID = 2: s(4, 1).ARGS(0) = 0: s(4, 1).ARGS(1) = 1
s(5, 0).ID = 2: s(5, 0).ARGS(0) = 0: s(5, 0).ARGS(1) = 0
'Extra credit
s(6, 1).ID = 5: s(6, 1).ARGS(0) = 0: s(6, 1).ARGS(1) = 10
s(6, 2).ID = 6: s(6, 2).ARGS(0) = 0: s(6, 2).ARGS(1) = 10
Dim As Integer i, x, r
For x = 0 To 2
i = 1
r = Union_(s(i, 1), s(i, 2), x)
Print Using "# in (#_,#] u [#_,#) : &"; x; s(i, 0).ARGS(0); s(i, 0).ARGS(1); s(i, 1).ARGS(0); s(1, 1).ARGS(1); Cbool(r)
Next x
Print
For x = 0 To 2
i = 2
r = Inters(s(i, 1), s(i, 2), x)
Print Using "# in (#_,#] u [#_,#) : &"; x; s(i, 0).ARGS(0); s(i, 0).ARGS(1); s(i, 1).ARGS(0); s(1, 1).ARGS(1); Cbool(r)
Next x
Print
For x = 0 To 2
i = 3
r = Differ(s(i, 1), s(i, 2), x)
Print Using "# in (#_,#] u [#_,#) : &"; x; s(i, 0).ARGS(0); s(i, 0).ARGS(1); s(i, 1).ARGS(0); s(1, 1).ARGS(1); Cbool(r)
Next x
Print
For x = 0 To 2
i = 4
r = Differ(s(i, 1), s(i, 2), x)
Print Using "# in (#_,#] u [#_,#) : &"; x; s(i, 0).ARGS(0); s(i, 0).ARGS(1); s(i, 1).ARGS(0); s(1, 1).ARGS(1); Cbool(r)
Next x
Print
x = 0
i = 5
r = Differ(s(i, 1), s(i, 2), x)
Print Using "[#_,#] is empty : &"; s(i, 0).ARGS(0); s(i, 0).ARGS(1); Cbool(r)
Print
'Extra credit
Dim As Double z = 0, paso = 0.00001
Dim As Integer count = 0
While z <= 10
If Differ(s(6, 1), s(6, 2), z) Then count += 1
z += paso
Wend
Print "Approximate length of A-B: "; count * paso
Sleep
- Output:
0 in (0,1] u [0,2) : true 1 in (0,1] u [0,2) : true 2 in (0,1] u [0,2) : false 0 in (0,2] u [1,2) : false 1 in (0,2] u [1,2) : false 2 in (0,2] u [1,2) : false 0 in (0,3] u [0,2) : false 1 in (0,3] u [0,2) : false 2 in (0,3] u [0,2) : false 0 in (0,3] u [0,2) : true 1 in (0,3] u [0,2) : true 2 in (0,3] u [0,2) : false [0,0] is empty : false Approximate length of A-B: 2.07586
F#
open System
let union s1 s2 =
fun x -> (s1 x) || (s2 x);
let difference s1 s2 =
fun x -> (s1 x) && not (s2 x)
let intersection s1 s2 =
fun x -> (s1 x) && (s2 x)
[<EntryPoint>]
let main _ =
//test set union
let u1 = union (fun x -> 0.0 < x && x <= 1.0) (fun x -> 0.0 <= x && x < 2.0)
assert (u1 0.0)
assert (u1 1.0)
assert (not (u1 2.0))
//test set difference
let d1 = difference (fun x -> 0.0 <= x && x < 3.0) (fun x -> 0.0 < x && x < 1.0)
assert (d1 0.0)
assert (not (d1 0.5))
assert (d1 1.0)
assert (d1 2.0)
let d2 = difference (fun x -> 0.0 <= x && x < 3.0) (fun x -> 0.0 <= x && x <= 1.0)
assert (not (d2 0.0))
assert (not (d2 1.0))
assert (d2 2.0)
let d3 = difference (fun x -> 0.0 <= x && x <= Double.PositiveInfinity) (fun x -> 1.0 <= x && x <= 2.0)
assert (d3 0.0)
assert (not (d3 1.5))
assert (d3 3.0)
//test set intersection
let i1 = intersection (fun x -> 0.0 <= x && x < 2.0) (fun x -> 1.0 < x && x <= 2.0)
assert (not (i1 0.0))
assert (not (i1 1.0))
assert (not (i1 2.0))
0 // return an integer exit code
Go
Just the non-optional part:
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()
}
}
- 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.
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))
}
Haskell
{- Not so functional representation of R sets (with IEEE Double), in a strange way -}
import Data.List
import Data.Maybe
data BracketType = OpenSub | ClosedSub
deriving (Show, Enum, Eq, Ord)
data RealInterval = RealInterval {left :: BracketType, right :: BracketType,
lowerBound :: Double, upperBound :: Double}
deriving (Eq)
type RealSet = [RealInterval]
posInf = 1.0/0.0 :: Double -- IEEE tricks
negInf = (-1.0/0.0) :: Double
set_R = RealInterval ClosedSub ClosedSub negInf posInf :: RealInterval
emptySet = [] :: [RealInterval]
instance Show RealInterval where
show x@(RealInterval _ _ y y')
| y == y' && (left x == right x) && (left x == ClosedSub) = "{" ++ (show y) ++ "}"
| otherwise = [['(', '[']!!(fromEnum $ left x)] ++ (show $ lowerBound x) ++
"," ++ (show $ upperBound x) ++ [[')', ']']!!(fromEnum $ right x)]
showList [x] = shows x
showList (h:t) = shows h . (" U " ++) . showList t
showList [] = (++ "(/)") -- empty set
construct_interval :: Char -> Double -> Double -> Char -> RealInterval
construct_interval '(' x y ')' = RealInterval OpenSub OpenSub x y
construct_interval '(' x y ']' = RealInterval OpenSub ClosedSub x y
construct_interval '[' x y ')' = RealInterval ClosedSub OpenSub x y
construct_interval _ x y _ = RealInterval ClosedSub ClosedSub x y
set_is_empty :: RealSet -> Bool
set_is_empty rs = (rs == emptySet)
set_in :: Double -> RealSet -> Bool
set_in x [] = False
set_in x rs =
isJust (find (\s ->
((lowerBound s < x) && (x < upperBound s)) ||
(x == lowerBound s && left s == ClosedSub) ||
(x == upperBound s && right s == ClosedSub))
rs)
-- max, min for pairs (double, bracket)
max_p :: (Double, BracketType) -> (Double, BracketType) -> (Double, BracketType)
min_p :: (Double, BracketType) -> (Double, BracketType) -> (Double, BracketType)
max_p p1@(x, y) p2@(x', y')
| x == x' = (x, max y y') -- closed is stronger than open
| x < x' = p2
| otherwise = p1
min_p p1@(x, y) p2@(x', y')
| x == x' = (x, min y y')
| x < x' = p1
| otherwise = p2
simple_intersection :: RealInterval -> RealInterval -> [RealInterval]
simple_intersection ri1@(RealInterval l_ri1 r_ri1 x1 y1) ri2@(RealInterval l_ri2 r_ri2 x2 y2)
| (y1 < x2) || (y2 < x1) = emptySet
| (y1 == x2) && ((fromEnum r_ri1) + (fromEnum l_ri2) /= 2) = emptySet
| (y2 == x1) && ((fromEnum r_ri2) + (fromEnum l_ri1) /= 2) = emptySet
| otherwise = let lb = if x1 == x2 then (x1, min l_ri1 l_ri2) else max_p (x1, l_ri1) (x2, l_ri2) in
let rb = min_p (y1, right ri1) (y2, right ri2) in
[RealInterval (snd lb) (snd rb) (fst lb) (fst rb)]
simple_union :: RealInterval -> RealInterval -> [RealInterval]
simple_union ri1@(RealInterval l_ri1 r_ri1 x1 y1) ri2@(RealInterval l_ri2 r_ri2 x2 y2)
| (y1 < x2) || (y2 < x1) = [ri2, ri1]
| (y1 == x2) && ((fromEnum r_ri1) + (fromEnum l_ri2) /= 2) = [ri1, ri2]
| (y2 == x1) && ((fromEnum r_ri2) + (fromEnum l_ri1) /= 2) = [ri1, ri2]
| otherwise = let lb = if x1 == x2 then (x1, max l_ri1 l_ri2) else min_p (x1, l_ri1) (x2, l_ri2) in
let rb = max_p (y1, right ri1) (y2, right ri2) in
[RealInterval (snd lb) (snd rb) (fst lb) (fst rb)]
simple_complement :: RealInterval -> [RealInterval]
simple_complement ri1@(RealInterval l_ri1 r_ri1 x1 y1) =
[(RealInterval ClosedSub (inv l_ri1) negInf x1), (RealInterval (inv r_ri1) ClosedSub y1 posInf)]
where
inv OpenSub = ClosedSub
inv ClosedSub = OpenSub
set_sort :: RealSet -> RealSet
set_sort rs =
sortBy
(\s1 s2 ->
let (lp, rp) = ((lowerBound s1, left s1), (lowerBound s2, left s2)) in
if max_p lp rp == lp then GT else LT)
rs
set_simplify :: RealSet -> RealSet
set_simplify [] = emptySet
set_simplify rs =
concat (map make_empty (set_sort (foldl
(\acc ri1 -> (simple_union (head acc) ri1) ++ (tail acc))
[head sorted_rs]
sorted_rs)))
where
sorted_rs = set_sort rs
make_empty ri@(RealInterval lb rb x y)
| x >= y && (lb /= rb || rb /= ClosedSub) = emptySet
| otherwise = [ri]
-- set operations
set_complement :: RealSet -> RealSet
set_union :: RealSet -> RealSet -> RealSet
set_intersection :: RealSet -> RealSet -> RealSet
set_difference :: RealSet -> RealSet -> RealSet
set_measure :: RealSet -> Double
set_complement rs =
foldl set_intersection [set_R] (map simple_complement rs)
set_union rs1 rs2 =
set_simplify (rs1 ++ rs2)
set_intersection rs1 rs2 =
set_simplify $ concat [simple_intersection s1 s2 | s1 <- rs1, s2 <- rs2]
set_difference rs1 rs2 =
set_intersection (set_complement rs2) rs1
set_measure rs =
foldl (\acc x -> acc + (upperBound x) - (lowerBound x)) 0.0 rs
-- test
test = map (\x -> [x]) [construct_interval '(' 0 1 ']', construct_interval '[' 0 2 ')',
construct_interval '[' 0 2 ')', construct_interval '(' 1 2 ']',
construct_interval '[' 0 3 ')', construct_interval '(' 0 1 ')',
construct_interval '[' 0 3 ')', construct_interval '[' 0 1 ']']
restest = [set_union (test!!0) (test!!1), set_intersection (test!!2) (test!!3),
set_difference (test!!4) (test!!5), set_difference (test!!6) (test!!7)]
isintest s =
mapM_
(\x -> putStrLn ((show x) ++ " is in " ++ (show s) ++ " : " ++ (show (set_in x s))))
[0, 1, 2]
testA = [construct_interval '(' (sqrt (n + (1.0/6))) (sqrt (n + (5.0/6))) ')' | n <- [0..99]]
testB = [construct_interval '(' (n + (1.0/6)) (n + (5.0/6)) ')' | n <- [0..9]]
main =
putStrLn ("union " ++ (show (test!!0)) ++ " " ++ (show (test!!1)) ++ " = " ++ (show (restest!!0))) >>
putStrLn ("inter " ++ (show (test!!2)) ++ " " ++ (show (test!!3)) ++ " = " ++ (show (restest!!1))) >>
putStrLn ("diff " ++ (show (test!!4)) ++ " " ++ (show (test!!5)) ++ " = " ++ (show (restest!!2))) >>
putStrLn ("diff " ++ (show (test!!6)) ++ " " ++ (show (test!!7)) ++ " = " ++ (show (restest!!3))) >>
mapM_ isintest restest >>
putStrLn ("measure: " ++ (show (set_measure (set_difference testA testB))))
- Output:
union (0.0,1.0] [0.0,2.0) = [0.0,2.0)
inter [0.0,2.0) (1.0,2.0] = (1.0,2.0)
diff [0.0,3.0) (0.0,1.0) = {0.0} U [1.0,3.0)
diff [0.0,3.0) [0.0,1.0] = (1.0,3.0)
0.0 is in [0.0,2.0) : True
1.0 is in [0.0,2.0) : True
2.0 is in [0.0,2.0) : False
0.0 is in (1.0,2.0) : False
1.0 is in (1.0,2.0) : False
2.0 is in (1.0,2.0) : False
0.0 is in {0.0} U [1.0,3.0) : True
1.0 is in {0.0} U [1.0,3.0) : True
2.0 is in {0.0} U [1.0,3.0) : True
0.0 is in (1.0,3.0) : False
1.0 is in (1.0,3.0) : False
2.0 is in (1.0,3.0) : True
measure: 2.0758648411846696
Icon and Unicon
The following only works in Unicon. The code does a few crude simplifications of some representations, but more could be done.
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
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:
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'
With this in place, the required examples look like this:
('(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
Note that without the arguments these wind up being expressions. For example:
('(0,1]' union '[0,2)')has
(0&< *. 1&>:) +. 0&<: *. 2&>
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:
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 ]
The above examples work identically with this version, but also:
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
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:
1p_1 * _1 o. 0.5
0.166667
(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.)
(#~ 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
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:
zeros0toN=: ((>: # ])[:+/\1,$&4 2@<.)&.(6&*)
is a function to generate the values which correspond to the boundaries of the intervals we want:
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
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:
(+/_2 -~/\zA) - +/,0>.zA (<.&{: - >.&{.)"1/&(_2 ]\ ]) zB
2.07586
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:
intervalSet=: interval@('[',[,',',],')'"_)&":
A=: union/_2 intervalSet/\ zA
B=: union/_2 intervalSet/\ zB
diff=: A without B
We can replace the above sentence to compute the length of the difference with:
+/ ((2 (+/%#)\ edge diff) in diff) * 2 -~/\ edge diff
2.07588
(Note that this result is not exactly the same as the previous result. Determining why would be an interesting exercise in numerical analysis.)
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());
}
}
- 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
JavaScript
function realSet(set1, set2, op, values) {
const makeSet=(set0)=>{
let res = []
if(set0.rangeType===0){
for(let i=set0.low;i<=set0.high;i++)
res.push(i);
} else if (set0.rangeType===1) {
for(let i=set0.low+1;i<set0.high;i++)
res.push(i);
} else if(set0.rangeType===2){
for(let i=set0.low+1;i<=set0.high;i++)
res.push(i);
} else {
for(let i=set0.low;i<set0.high;i++)
res.push(i);
}
return res;
}
let res = [],finalSet=[];
set1 = makeSet(set1);
set2 = makeSet(set2);
if(op==="union")
finalSet = [...new Set([...set1,...set2])];
else if(op==="intersect") {
for(let i=0;i<set1.length;i++)
if(set1.indexOf(set2[i])!==-1)
finalSet.push(set2[i]);
} else {
for(let i=0;i<set2.length;i++)
if(set1.indexOf(set2[i])===-1)
finalSet.push(set2[i]);
for(let i=0;i<set1.length;i++)
if(set2.indexOf(set1[i])===-1)
finalSet.push(set1[i]);
}
for(let i=0;i<values.length;i++){
if(finalSet.indexOf(values[i])!==-1)
res.push(true);
else
res.push(false);
}
return res;
}
jq
Works with gojq, the Go implementation of jq provided `keys_unsorted` is replaced with `keys`
This entry focuses on functions that operate on "real sets" and not just intervals of the real number line, it being understood that a "real set" in the present context is a finite union of such intervals, in accordance with the problem description.
Since every "real set" in this sense can be represented in canonical form as a finite disjoint union of intervals, we will use the term RealSet to denote a canonical representation in jq of a "real set" as follows:
A RealSet is a jq array consisting of numbers and/or two-element arrays [a,b], where a and b are numbers with a < b, where:
- each number represents the closed interval containing that number;
- each array [a,b] represents the open interval from a to b exclusive;
- the items in the outer array are sorted in ascending order in the obvious way.
The jq values `infinite` and `-infinite` are also allowed, thus allowing infinite intervals to be represented.
Examples:
- [] representes the empty RealSet.
- 1,2 represents the RealSet consisting of the open interval from 1 to 2 exclusive.
- [1, [1,2], 2] represents the closed interval from 1 to 2 inclusive.
- [1,2] represents the union of the two closed intervals containing respectively 1 and 2.
- [-infinite, 0] represents the open interval consisting of the finite negative numbers.
- [infinite] represents the closed interval whose only element is positive inifinity.
For clarity and to facilitate reuse, the RealSet function definitions are bundled together in a jq module, RealSet, available at Category:Jq/RealSet.jq. Here we summarize the key functions and illustrate their use.
1) To convert an arbitrary union of real intervals to a RealSet, use `RealSet/0`, e.g.
[ [1,5], [2,6] ] | RealSet #=> 1,6
2) Testing whether a RealSet is empty
Since the empty RealSet is just [], there is no real need to define a function for testing whether a RealSet is empty. To test whether an arbitrary union of real intervals is empty, use the idiom:
RealSet == []
For example:
[ [1,3], [0,1] ] | RealSet == [] #=> false
3) To check whether a specific number, $r, is in a RealSet,
one can use `containsNumber($r)`, and similarly to check whether an open interval is in a RealSet, one can use `containsOpenInterval($a; $b)` where $a < $b defines the open interval.
4) The basic binary operations on RealSets are:
add/1 intersection/1 minus/1
5) To compute the length of a RealSet: `RealSetLength/0`
This returns `infinite` if any component interval is infinite.
Tasks
include "realset" {search: "."};
def test_cases:
{ "(0, 1] ∪ [0, 2)": ( [ [0,1], 1] | add( [0, [0,2]] )),
"[0, 2) ∩ (1, 2]": ( [ 0, [0,2]] | intersection( [[1,2],2] ) ),
"[0, 3) − (0, 1)": ( [ 0, [0,3]] | minus( [[0,1]] ) ),
"[0, 3) − [0, 1]": ( [ 0, [0,3]] | minus( [0, [0,1], 1] ))
} ;
def keys_unsorted: keys; # for gojq
def tests($values):
"Checking containment of: \($values | join(" "))",
(keys_unsorted[] as $name
| "\($name) has length \(.[$name]|RealSetLength) and contains: \( [$values[] as $i | select(.[$name] | containsNumber($i) ) | $i] | join(" ") )" )
;
# A and B
def pi: 1 | atan * 4;
# For positive integers $n,
# we define B($n) to correspond to {x | 0 < x < $n and |sin(π x)| > 1/2}
def B($upper):
def x: 0.5 | asin / pi;
x as $x
| reduce range(0; $upper) as $i ([];
. + [ [$i + $x, $i + 1 - $x]]);
# |sin(π x²)| > 1/2
def A($upper):
B($upper * $upper) | map( map(sqrt) );
# The simple tests:
test_cases | tests([0,1,2]),
# A - B
"|A - B| = \(A(10) | minus( B(10) ) | RealSetLength)"- Output:
Checking containment of: 0 1 2 (0, 1] ∪ [0, 2) has length 2 and contains: 0 1 [0, 2) ∩ (1, 2] has length 1 and contains: [0, 3) − (0, 1) has length 2 and contains: 0 1 2 [0, 3) − [0, 1] has length 2 and contains: 2 |A - B| = 2.075864841184667
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()
- 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]: falseTesting with x = 1. Results:
(0, 1] ∪ [0, 2): true [0, 2) ∩ (1, 2]: false [0, 3) − (0, 1): true [0, 3) − [0, 1]: falseTesting with x = 2. Results:
(0, 1] ∪ [0, 2): false [0, 2) ∩ (1, 2]: false [0, 3) − (0, 1): true [0, 3) − [0, 1]: trueThe 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.
// 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}")
}
- 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
Lua
function createSet(low,high,rt)
local l,h = tonumber(low), tonumber(high)
if l and h then
local t = {low=l, high=h}
if type(rt) == "string" then
if rt == "open" then
t.contains = function(d) return low< d and d< high end
elseif rt == "closed" then
t.contains = function(d) return low<=d and d<=high end
elseif rt == "left" then
t.contains = function(d) return low< d and d<=high end
elseif rt == "right" then
t.contains = function(d) return low<=d and d< high end
else
error("Unknown range type: "..rt)
end
elseif type(rt) == "function" then
t.contains = rt
else
error("Unable to find a range type or predicate")
end
t.union = function(o)
local l2 = math.min(l, o.low)
local h2 = math.min(h, o.high)
local p = function(d) return t.contains(d) or o.contains(d) end
return createSet(l2, h2, p)
end
t.intersect = function(o)
local l2 = math.min(l, o.low)
local h2 = math.min(h, o.high)
local p = function(d) return t.contains(d) and o.contains(d) end
return createSet(l2, h2, p)
end
t.subtract = function(o)
local l2 = math.min(l, o.low)
local h2 = math.min(h, o.high)
local p = function(d) return t.contains(d) and not o.contains(d) end
return createSet(l2, h2, p)
end
t.length = function()
if h <= l then return 0.0 end
local p = l
local count = 0
local interval = 0.00001
repeat
if t.contains(p) then count = count + 1 end
p = p + interval
until p>=high
return count * interval
end
t.empty = function()
if l == h then
return not t.contains(low)
end
return t.length() == 0.0
end
return t
else
error("Either '"..low.."' or '"..high.."' is not a number")
end
end
local a = createSet(0.0, 1.0, "left")
local b = createSet(0.0, 2.0, "right")
local c = createSet(1.0, 2.0, "left")
local d = createSet(0.0, 3.0, "right")
local e = createSet(0.0, 1.0, "open")
local f = createSet(0.0, 1.0, "closed")
local g = createSet(0.0, 0.0, "closed")
for i=0,2 do
print("(0, 1] union [0, 2) contains "..i.." is "..tostring(a.union(b).contains(i)))
print("[0, 2) intersect (1, 2] contains "..i.." is "..tostring(b.intersect(c).contains(i)))
print("[0, 3) - (0, 1) contains "..i.." is "..tostring(d.subtract(e).contains(i)))
print("[0, 3) - [0, 1] contains "..i.." is "..tostring(d.subtract(f).contains(i)))
print()
end
print("[0, 0] is empty is "..tostring(g.empty()))
print()
local aa = createSet(
0.0, 10.0,
function(x) return (0.0<x and x<10.0) and math.abs(math.sin(math.pi * x * x)) > 0.5 end
)
local bb = createSet(
0.0, 10.0,
function(x) return (0.0<x and x<10.0) and math.abs(math.sin(math.pi * x)) > 0.5 end
)
local cc = aa.subtract(bb)
print("Approx length of A - B is "..cc.length())
- Output:
(0, 1] union [0, 2) contains 0 is true [0, 2) intersect (1, 2] contains 0 is false [0, 3) - (0, 1) contains 0 is true [0, 3) - [0, 1] contains 0 is false (0, 1] union [0, 2) contains 1 is true [0, 2) intersect (1, 2] contains 1 is false [0, 3) - (0, 1) contains 1 is true [0, 3) - [0, 1] contains 1 is false (0, 1] union [0, 2) contains 2 is false [0, 2) intersect (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 /Wolfram Language
(* 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}]
- 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}
Nim
import math, strformat, sugar
type
RealPredicate = (float) -> bool
RangeType {.pure} = enum Closed, BothOpen, LeftOpen, RightOpen
RealSet = object
low, high: float
predicate: RealPredicate
proc initRealSet(slice: Slice[float]; rangeType: RangeType): RealSet =
result = RealSet(low: slice.a, high: slice.b)
result.predicate = case rangeType
of Closed: (x: float) => x in slice
of BothOpen: (x: float) => slice.a < x and x < slice.b
of LeftOpen: (x: float) => slice.a < x and x <= slice.b
of RightOpen: (x: float) => slice.a <= x and x < slice.b
proc contains(s: RealSet; val: float): bool =
## Defining "contains" makes operator "in" available.
s.predicate(val)
proc `+`(s1, s2: RealSet): RealSet =
RealSet(low: min(s1.low, s2.low), high: max(s1.high, s2.high),
predicate: (x:float) => s1.predicate(x) or s2.predicate(x))
proc `*`(s1, s2: RealSet): RealSet =
RealSet(low: max(s1.low, s2.low), high: min(s1.high, s2.high),
predicate: (x:float) => s1.predicate(x) and s2.predicate(x))
proc `-`(s1, s2: RealSet): RealSet =
RealSet(low: s1.low, high: s1.high,
predicate: (x:float) => s1.predicate(x) and not s2.predicate(x))
const Interval = 0.00001
proc length(s: RealSet): float =
if s.low.classify() in {fcInf, fcNegInf} or s.high.classify() in {fcInf, fcNegInf}: return Inf
if s.high <= s.low: return 0
var p = s.low
var count = 0.0
while p < s.high:
if s.predicate(p): count += 1
p += Interval
result = count * Interval
proc isEmpty(s: RealSet): bool =
if s.high == s.low: not s.predicate(s.low)
else: s.length == 0
when isMainModule:
let
a = initRealSet(0.0..1.0, LeftOpen)
b = initRealSet(0.0..2.0, RightOpen)
c = initRealSet(1.0..2.0, LeftOpen)
d = initRealSet(0.0..3.0, RightOpen)
e = initRealSet(0.0..1.0, BothOpen)
f = initRealSet(0.0..1.0, Closed)
g = initRealSet(0.0..0.0, Closed)
for n in 0..2:
let x = n.toFloat
echo &"{n} ∊ (0, 1] ∪ [0, 2) is {x in (a + b)}"
echo &"{n} ∊ [0, 2) ∩ (1, 2] is {x in (b * c)}"
echo &"{n} ∊ [0, 3) − (0, 1) is {x in (d - e)}"
echo &"{n} ∊ [0, 3) − [0, 1] is {x in (d - f)}\n"
echo &"[0, 0] is empty is {g.isEmpty()}.\n"
let
aa = RealSet(low: 0, high: 10,
predicate: (x: float) => 0 < x and x < 10 and abs(sin(PI * x * x)) > 0.5)
bb = RealSet(low: 0, high: 10,
predicate: (x: float) => 0 < x and x < 10 and abs(sin(PI * x)) > 0.5)
cc = aa - bb
echo &"Approximative length of A - B is {cc.length}."
- Output:
0 ∊ (0, 1] ∪ [0, 2) is true 0 ∊ [0, 2) ∩ (1, 2] is false 0 ∊ [0, 3) − (0, 1) is true 0 ∊ [0, 3) − [0, 1] is false 1 ∊ (0, 1] ∪ [0, 2) is true 1 ∊ [0, 2) ∩ (1, 2] is false 1 ∊ [0, 3) − (0, 1) is true 1 ∊ [0, 3) − [0, 1] is false 2 ∊ (0, 1] ∪ [0, 2) is false 2 ∊ [0, 2) ∩ (1, 2] is false 2 ∊ [0, 3) − (0, 1) is true 2 ∊ [0, 3) − [0, 1] is true [0, 0] is empty is false. Approximative length of A - B is 2.07587.
PARI/GP
Define some sets and use built-in functions:
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))Output:
[0, 1] [] [0, 1, 2] [2]
Perl
use utf8;
# numbers used as boundaries to real sets. Each has 3 components:
# the real value x;
# a +/-1 indicating if it's x + ϵ or x - ϵ
# a 0/1 indicating if it's the left border or right border
# 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 || $a->[1] <=> 0
}
package RealSet;
use Carp;
use overload (
'""' => \&_str,
'|' => \&_or,
'&' => \&_and,
'~' => \&_neg,
'-' => \&_diff,
'bool' => \¬_empty, # set is true if not empty
);
my %pm = qw/ [ -1 ( 1 ) -1 ] 1 /;
sub range {
my ($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";
}
# 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)
}
# "|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";
output
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
Phix
with javascript_semantics enum ID,ARGS function cf(sequence f, atom x) return call_func(f[ID],deep_copy(f[ARGS])&x) end function function Union(sequence a, b, atom x) return cf(a,x) or cf(b,x) end function function Inter(sequence a, b, atom x) return cf(a,x) and cf(b,x) end function function Diffr(sequence a, b, atom x) return cf(a,x) and not cf(b,x) end function function OpOp(atom a, b, x) return a < x and x < b end function function ClCl(atom a, b, x) return a <= x and x <= b end function function OpCl(atom a, b, x) return a < x and x <= b end function function ClOp(atom a, b, x) return a <= x and x < b end function -- expected -- ---- desc ----, 0 1 2, --------------- set method --------------- constant s = {{"(0,1] u [0,2)", {1,1,0}, {Union,{{OpCl,{0,1}},{ClOp,{0,2}}}}}, {"[0,2) n (1,2]", {0,0,0}, {Inter,{{ClOp,{0,2}},{OpCl,{1,2}}}}}, {"[0,3) - (0,1)", {1,1,1}, {Diffr,{{ClOp,{0,3}},{OpOp,{0,1}}}}}, {"[0,3) - [0,1]", {0,0,1}, {Diffr,{{ClOp,{0,3}},{ClCl,{0,1}}}}}} for i=1 to length(s) do sequence {desc, expect, method} = s[i] for x=0 to 2 do bool r = cf(method,x) string error = iff(r!=expect[x+1]?"error":"") printf(1,"%d in %s : %t %s\n", {x, desc, r, error}) end for printf(1,"\n") end for
- Output:
0 in (0,1] u [0,2) : true 1 in (0,1] u [0,2) : true 2 in (0,1] u [0,2) : false 0 in [0,2) n (1,2] : false 1 in [0,2) n (1,2] : false 2 in [0,2) n (1,2] : false 0 in [0,3) - (0,1) : true 1 in [0,3) - (0,1) : true 2 in [0,3) - (0,1) : true 0 in [0,3) - [0,1] : false 1 in [0,3) - [0,1] : false 2 in [0,3) - [0,1] : true
Extra credit - also translated from Go, but with an extended loop and crude summation, inspired by Java/Kotlin.
function aspxx(atom x) return abs(sin(PI*x*x))>0.5 end function function aspx(atom x) return abs(sin(PI*x)) >0.5 end function constant A = {Inter,{{OpOp,{0,10}},{aspxx,{}}}}, B = {Inter,{{OpOp,{0,10}},{aspx,{}}}}, C = {Diffr,{A,B}} atom x = 0, step = 0.00001, count = 0 while x<=10 do count += cf(C,x) x += step end while printf(1,"Approximate length of A-B: %.5f\n",{count*step})
- Output:
Approximate length of A-B: 2.07587
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))
- 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
;; 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)))
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
Raku
(formerly Perl 6)
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;
- 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
REXX
no error checking, no ∞
/*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)
- output is the same as the next REXX version (below).
has error checking, ∞ support
/*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)
- 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
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
Test case:
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}"
- 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:
(with Rational suffix.)
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}"
- 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
Rust
This implementation defines a RealSet as either:
- A
RangeSet, which contains all numbers from a start to an end, inclusive, or exclusive. When thecontains()method is called on a range set, it will return whether a number is between its bounds. - A
CompositeSetwhich represents aSetOperation(union, intersection, or difference) between two otherRealSets, which themselves can be composite or not. When thecontains()method is called on a composite set for a given number, it will recursively call contains() on its component sets to check whether they contain the number. Depending on the operation, this will define whether the number is contained in this set.
Since we use Rust's f64, which is a standard IEEE 754 double-precision floating-point number, we get correct behavior at infinity for free.
#[derive(Debug)]
enum SetOperation {
Union,
Intersection,
Difference,
}
#[derive(Debug, PartialEq)]
enum RangeType {
Inclusive,
Exclusive,
}
#[derive(Debug)]
struct CompositeSet<'a> {
operation: SetOperation,
a: &'a RealSet<'a>,
b: &'a RealSet<'a>,
}
#[derive(Debug)]
struct RangeSet {
range_types: (RangeType, RangeType),
start: f64,
end: f64,
}
#[derive(Debug)]
enum RealSet<'a> {
RangeSet(RangeSet),
CompositeSet(CompositeSet<'a>),
}
impl RangeSet {
fn compare_start(&self, n: f64) -> bool {
if self.range_types.0 == RangeType::Inclusive {
self.start <= n
} else {
self.start < n
}
}
fn compare_end(&self, n: f64) -> bool {
if self.range_types.1 == RangeType::Inclusive {
n <= self.end
} else {
n < self.end
}
}
}
impl<'a> RealSet<'a> {
fn new(start_type: RangeType, start: f64, end: f64, end_type: RangeType) -> Self {
RealSet::RangeSet(RangeSet {
range_types: (start_type, end_type),
start,
end,
})
}
fn operation(&'a self, other: &'a Self, operation: SetOperation) -> Self {
RealSet::CompositeSet(CompositeSet {
operation,
a: self,
b: other,
})
}
fn union(&'a self, other: &'a Self) -> Self {
self.operation(other, SetOperation::Union)
}
fn intersection(&'a self, other: &'a Self) -> Self {
self.operation(other, SetOperation::Intersection)
}
fn difference(&'a self, other: &'a Self) -> Self {
self.operation(other, SetOperation::Difference)
}
fn contains(&self, n: f64) -> bool {
if let RealSet::RangeSet(range) = self {
range.compare_start(n) && range.compare_end(n)
} else if let RealSet::CompositeSet(range) = self {
match range.operation {
SetOperation::Union => range.a.contains(n) || range.b.contains(n),
SetOperation::Intersection => range.a.contains(n) && range.b.contains(n),
SetOperation::Difference => range.a.contains(n) && !range.b.contains(n),
}
} else {
unimplemented!();
}
}
}
fn make_contains_phrase(does_contain: bool) -> &'static str {
if does_contain {
"contains"
} else {
"does not contain"
}
}
use RangeType::*;
fn main() {
for (set_name, set) in [
(
"(0, 1] ∪ [0, 2)",
RealSet::new(Exclusive, 0.0, 1.0, Inclusive)
.union(&RealSet::new(Inclusive, 0.0, 2.0, Exclusive)),
),
(
"[0, 2) ∩ (1, 2]",
RealSet::new(Inclusive, 0.0, 2.0, Exclusive)
.intersection(&RealSet::new(Exclusive, 1.0, 2.0, Inclusive)),
),
(
"[0, 3) − (0, 1)",
RealSet::new(Inclusive, 0.0, 3.0, Exclusive)
.difference(&RealSet::new(Exclusive, 0.0, 1.0, Exclusive)),
),
(
"[0, 3) − [0, 1]",
RealSet::new(Inclusive, 0.0, 3.0, Exclusive)
.difference(&RealSet::new(Inclusive, 0.0, 1.0, Inclusive)),
),
] {
println!("Set {}", set_name);
for i in [0.0, 1.0, 2.0] {
println!("- {} {}", make_contains_phrase(set.contains(i)), i);
}
}
}
Set (0, 1] ∪ [0, 2) - contains 0 - contains 1 - does not contain 2 Set [0, 2) ∩ (1, 2] - does not contain 0 - does not contain 1 - does not contain 2 Set [0, 3) − (0, 1) - contains 0 - contains 1 - contains 2 Set [0, 3) − [0, 1] - does not contain 0 - does not contain 1 - contains 2
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.
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
}
Basic problems:
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]"
}
}
Extra credit:
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]"
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
Wren
import "./dynamic" for Enum
var RangeType = Enum.create("RangeType", ["CLOSED", "BOTH_OPEN", "LEFT_OPEN", "RIGHT_OPEN"])
class RealSet {
construct new(start, end, pred) {
_low = start
_high = end
_pred = (pred == RangeType.CLOSED) ? Fn.new { |d| d >= _low && d <= _high } :
(pred == RangeType.BOTH_OPEN) ? Fn.new { |d| d > _low && d < _high } :
(pred == RangeType.LEFT_OPEN) ? Fn.new { |d| d > _low && d <= _high } :
(pred == RangeType.RIGHT_OPEN) ? Fn.new { |d| d >= _low && d < _high } : pred
}
low { _low }
high { _high }
pred { _pred }
contains(d) { _pred.call(d) }
union(other) {
if (!other.type == RealSet) Fiber.abort("Argument must be a RealSet")
var low2 = _low.min(other.low)
var high2 = _high.max(other.high)
return RealSet.new(low2, high2) { |d| _pred.call(d) || other.pred.call(d) }
}
intersect(other) {
if (!other.type == RealSet) Fiber.abort("Argument must be a RealSet")
var low2 = _low.max(other.low)
var high2 = _high.min(other.high)
return RealSet.new(low2, high2) { |d| _pred.call(d) && other.pred.call(d) }
}
subtract(other) {
if (!other.type == RealSet) Fiber.abort("Argument must be a RealSet")
return RealSet.new(_low, _high) { |d| _pred.call(d) && !other.pred.call(d) }
}
length {
if (_low.isInfinity || _high.isInfinity) return -1 // error value
if (_high <= _low) return 0
var p = _low
var count = 0
var interval = 0.00001
while (true) {
if (_pred.call(p)) count = count + 1
p = p + interval
if (p >= _high) break
}
return count * interval
}
isEmpty { (_high == _low) ? !_pred.call(_low) : length == 0 }
}
var a = RealSet.new(0, 1, RangeType.LEFT_OPEN)
var b = RealSet.new(0, 2, RangeType.RIGHT_OPEN)
var c = RealSet.new(1, 2, RangeType.LEFT_OPEN)
var d = RealSet.new(0, 3, RangeType.RIGHT_OPEN)
var e = RealSet.new(0, 1, RangeType.BOTH_OPEN)
var f = RealSet.new(0, 1, RangeType.CLOSED)
var g = RealSet.new(0, 0, RangeType.CLOSED)
for (i in 0..2) {
System.print("(0, 1] ∪ [0, 2) contains %(i) is %(a.union(b).contains(i))")
System.print("[0, 2) ∩ (1, 2] contains %(i) is %(b.intersect(c).contains(i))")
System.print("[0, 3) − (0, 1) contains %(i) is %(d.subtract(e).contains(i))")
System.print("[0, 3) − [0, 1] contains %(i) is %(d.subtract(f).contains(i))\n")
}
System.print("[0, 0] is empty is %(g.isEmpty)\n")
var aa = RealSet.new(0, 10) { |x| (0 < x && x < 10) && ((Num.pi * x * x).sin.abs > 0.5) }
var bb = RealSet.new(0, 10) { |x| (0 < x && x < 10) && ((Num.pi * x).sin.abs > 0.5) }
var cc = aa.subtract(bb)
System.print("Approx length of A - B is %(cc.length)")
- 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
zkl
No ∞
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) }) }
}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.
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__);- 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!
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