240 lines
6.8 KiB
Markdown
240 lines
6.8 KiB
Markdown
---
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id: 5eb3e4b20aa93c437f9e9717
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title: Set of real numbers
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challengeType: 5
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forumTopicId: 385322
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dashedName: set-of-real-numbers
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---
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# --description--
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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:
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<ul>
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<li>[<i>a</i>, <i>b</i>]: {<i>x</i> | <i>a</i> ≤ <i>x</i> and <i>x</i> ≤ <i>b</i> }</li>
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<li>(<i>a</i>, <i>b</i>): {<i>x</i> | <i>a</i> < <i>x</i> and <i>x</i> < <i>b</i> }</li>
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<li>[<i>a</i>, <i>b</i>): {<i>x</i> | <i>a</i> ≤ <i>x</i> and <i>x</i> < <i>b</i> }</li>
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<li>(<i>a</i>, <i>b</i>]: {<i>x</i> | <i>a</i> < <i>x</i> and <i>x</i> ≤ <i>b</i> }</li>
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</ul>
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Note that if *a* = *b*, of the four only \[*a*, *a*] would be non-empty.
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**Task**
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<ul>
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<li>Devise a way to represent any set of real numbers, for the definition of "any" in the implementation notes below.</li>
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<li>Provide methods for these common set operations (<i>x</i> is a real number; <i>A</i> and <i>B</i> are sets):</li>
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<ul>
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<li>
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<i>x</i> ∈ <i>A</i>: determine if <i>x</i> is an element of <i>A</i><br>
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example: 1 is in [1, 2), while 2, 3, ... are not.
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</li>
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<li>
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<i>A</i> ∪ <i>B</i>: union of <i>A</i> and <i>B</i>, i.e. {<i>x</i> | <i>x</i> ∈ <i>A</i> or <i>x</i> ∈ <i>B</i>}<br>
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example: [0, 2) ∪ (1, 3) = [0, 3); [0, 1) ∪ (2, 3] = well, [0, 1) ∪ (2, 3]
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</li>
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<li>
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<i>A</i> ∩ <i>B</i>: intersection of <i>A</i> and <i>B</i>, i.e. {<i>x</i> | <i>x</i> ∈ <i>A</i> and <i>x</i> ∈ <i>B</i>}<br>
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example: [0, 2) ∩ (1, 3) = (1, 2); [0, 1) ∩ (2, 3] = empty set
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</li>
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<li>
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<i>A</i> - <i>B</i>: difference between <i>A</i> and <i>B</i>, also written as <i>A</i> \ <i>B</i>, i.e. {<i>x</i> | <i>x</i> ∈ <i>A</i> and <i>x</i> ∉ <i>B</i>}<br>
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example: [0, 2) − (1, 3) = [0, 1]
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</li>
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</ul>
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</ul>
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# --instructions--
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Write a function that takes 2 objects, a string and an array as parameters. The objects represents the set and have attributes: `low`, `high` and `rangeType`.
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The `rangeType` can have values 0, 1, 2 and 3 for `CLOSED`, `BOTH_OPEN`, `LEFT_OPEN` and `RIGHT_OPEN`, respectively. The function should implement a set using this information.
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The string represents the operation to be performed on the sets. It can be: `"union"`, `"intersect"` and `"subtract"` (difference).
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After performing the operation, the function should check if the values in the array are present in the resultant set and store a corresponding boolean value to an array. The function should return this array.
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# --hints--
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`realSet` should be a function.
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```js
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assert(typeof realSet == 'function');
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```
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`realSet({"low":0, "high":1, "rangeType":2}, {"low":0, "high":2, "rangeType":3}, "union", [1, 2, 3])` should return a array.
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```js
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assert(
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Array.isArray(
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realSet(
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{ low: 0, high: 1, rangeType: 2 },
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{ low: 0, high: 2, rangeType: 3 },
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'union',
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[1, 2, 3]
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)
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)
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);
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```
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`realSet({"low":0, "high":1, "rangeType":2}, {"low":0, "high":2, "rangeType":3}, "union", [1, 2, 3])` should return `[true, false, false]`.
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```js
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assert.deepEqual(
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realSet(
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{ low: 0, high: 1, rangeType: 2 },
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{ low: 0, high: 2, rangeType: 3 },
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'union',
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[1, 2, 3]
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),
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[true, false, false]
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);
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```
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`realSet({"low":0, "high":2, "rangeType":3}, {"low":1, "high":2, "rangeType":2}, "intersect", [0, 1, 2])` should return `[false, false, false]`.
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```js
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assert.deepEqual(
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realSet(
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{ low: 0, high: 2, rangeType: 3 },
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{ low: 1, high: 2, rangeType: 2 },
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'intersect',
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[0, 1, 2]
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),
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[false, false, false]
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);
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```
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`realSet({"low":0, "high":3, "rangeType":3}, {"low":0, "high":1, "rangeType":1}, "subtract", [0, 1, 2])` should return `[true, true, true]`.
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```js
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assert.deepEqual(
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realSet(
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{ low: 0, high: 3, rangeType: 3 },
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{ low: 0, high: 1, rangeType: 1 },
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'subtract',
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[0, 1, 2]
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),
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[true, true, true]
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);
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```
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`realSet({"low":0, "high":3, "rangeType":3}, {"low":0, "high":1, "rangeType":0}, "subtract", [0, 1, 2])` should return `[false, false, true]`.
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```js
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assert.deepEqual(
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realSet(
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{ low: 0, high: 3, rangeType: 3 },
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{ low: 0, high: 1, rangeType: 0 },
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'subtract',
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[0, 1, 2]
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),
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[false, false, true]
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);
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```
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`realSet({"low":0, "high":33, "rangeType":1}, {"low":30, "high":31, "rangeType":0}, "intersect", [30, 31, 32])` should return `[true, true, false]`.
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```js
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assert.deepEqual(
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realSet(
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{ low: 0, high: 33, rangeType: 1 },
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{ low: 30, high: 31, rangeType: 0 },
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'intersect',
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[30, 31, 32]
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),
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[true, true, false]
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);
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```
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# --seed--
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## --seed-contents--
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```js
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function realSet(set1, set2, operation, values) {
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}
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```
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# --solutions--
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```js
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function realSet(set1, set2, operation, values) {
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const RangeType = {
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CLOSED: 0,
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BOTH_OPEN: 1,
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LEFT_OPEN: 2,
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RIGHT_OPEN: 3
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};
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function Predicate(test) {
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this.test = test;
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this.or = function(other) {
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return new Predicate(t => this.test(t) || other.test(t));
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};
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this.and = function(other) {
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return new Predicate(t => this.test(t) && other.test(t));
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};
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this.negate = function() {
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return new Predicate(t => !this.test(t));
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};
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}
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function RealSet(start, end, rangeType, predF) {
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this.low = start;
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this.high = end;
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if (predF) {
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this.predicate = new Predicate(predF);
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} else {
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this.predicate = new Predicate(d => {
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switch (rangeType) {
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case RangeType.CLOSED:
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return start <= d && d <= end;
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case RangeType.BOTH_OPEN:
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return start < d && d < end;
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case RangeType.LEFT_OPEN:
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return start < d && d <= end;
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case RangeType.RIGHT_OPEN:
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return start <= d && d < end;
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}
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});
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}
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this.contains = function(d) {
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return this.predicate.test(d);
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};
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this.union = function(other) {
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var low2 = Math.min(this.low, other.low);
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var high2 = Math.max(this.high, other.high);
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return new RealSet(low2, high2, null, d =>
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this.predicate.or(other.predicate).test(d)
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);
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};
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this.intersect = function(other) {
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var low2 = Math.min(this.low, other.low);
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var high2 = Math.max(this.high, other.high);
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return new RealSet(low2, high2, null, d =>
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this.predicate.and(other.predicate).test(d)
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);
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};
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this.subtract = function(other) {
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return new RealSet(this.low, this.high, null, d =>
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this.predicate.and(other.predicate.negate()).test(d)
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);
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};
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}
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set1 = new RealSet(set1.low, set1.high, set1.rangeType);
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set2 = new RealSet(set2.low, set2.high, set2.rangeType);
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var result = [];
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values.forEach(function(value) {
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result.push(set1[operation](set2).contains(value));
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});
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return result;
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}
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```
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