Sets

A set is a collection of distinct elements.

The Compute Engine standard library includes definitions for common numeric sets. Checking if a value belongs to a set is done using the Element expression, or the \in (\in) command in LaTeX.

ce.expr(['Element', 3.14, 'NegativeIntegers']).evaluate().print();
// ➔ False

ce.parse("42 \\in \\Z").evaluate().print();
// ➔ True

Element and NotElement can also be used with a type name on the right hand side (e.g. integer, real, finite_real, number, any), in which case the check is done against the expression type.

ce.declare('x', 'finite_real');
ce.expr(['Element', 'x', 'real']).evaluate().print();
// ➔ True

ce.expr(['Element', 'x', 'integer']).evaluate().print();
// ➔ False

Checking if an element is in a set is equivalent to checking if the type of the element matches the type associated with the set.

const x = ce.expr(42);

x.type;
// ➔ "finite_integer"

x.type.matches("integer");
// ➔ true

x.isInteger;
// ➔ true

ce.expr(['Element', x, 'Integers']).evaluate().print();
// ➔ True

ce.parse("42 \\in \\Z").evaluate().print();
// ➔ True

Constants

Symbol	Notation		Definition
EmptySet	\varnothing or \emptyset	\varnothing or \emptyset	A set that has no elements
Numbers	\mathrm{Numbers}	\mathrm{Numbers}	Any number, real, imaginary, or complex
ComplexNumbers	\C	\C	Real or imaginary numbers
ExtendedComplexNumbers	\overline\C	\overline\C	Real or imaginary numbers, including +\infty, -\infty and \tilde\infty
ImaginaryNumbers	\imaginaryI\R	\imaginaryI\R	Complex numbers with a non-zero imaginary part and no real part
RealNumbers	\R	\R	Numbers that form the unique Dedekind-complete ordered field \left( \mathbb{R} ; + ; \cdot ; \lt \right), up to an isomorphism (does not include \pm\infty)
ExtendedRealNumbers	\overline\R	\overline\R	Real numbers extended to include \pm\infty
Integers	\Z	\Z	Whole numbers and their additive inverse \lbrace \ldots -3, -2, -1,0, 1, 2, 3\ldots\rbrace
ExtendedIntegers	\overline\Z	\overline\Z	Integers extended to include \pm\infty
RationalNumbers	\Q	\Q	Numbers which can be expressed as the quotient \nicefrac{p}{q} of two integers p, q \in \mathbb{Z}.
ExtendedRationalNumbers	\overline\Q	\overline\Q	Rational numbers extended to include \pm\infty
NegativeNumbers	\R_{<0}	\R_{<0}	Real numbers \lt 0
NonPositiveNumbers	\R_{\leq0}	\R_{\leq0}	Real numbers \leq 0
NonNegativeNumbers	\R_{\geq0}	\R_{\geq0}	Real numbers \geq 0
PositiveNumbers	\R_{>0}	\R_{>0}	Real numbers \gt 0
NegativeIntegers	\Z_{<0}	\Z_{<0}	Integers \lt 0, \lbrace \ldots -3, -2, -1\rbrace
NonPositiveIntegers	\Z_{\le0}	\Z_{\le0}	Integers \leq 0, \lbrace \ldots -3, -2, -1, 0\rbrace
NonNegativeIntegers	\N	\N	Integers \geq 0, \lbrace 0, 1, 2, 3\ldots\rbrace
PositiveIntegers	\N^*	\N^*	Integers \gt 0, \lbrace 1, 2, 3\ldots\rbrace

Union and Intersection accept any finite collections, including lists. The result is a Set, so duplicate elements are removed:

["Intersection", ["List", 1, 2], ["List", 2, 3]]
// ➔ ["Set", 2]

A MathJSON List is always a collection here, including when it has two elements. Interval notation is interpreted as an Interval while parsing LaTeX, before the set operation is constructed.

Functions

New sets can be defined using one of the following operators.

Function	Operation
CartesianProduct	\operatorname{A} \times \operatorname{B}	A.k.a the product set, the set direct product or cross product. Q173740
Complement	\operatorname{A}^\complement	The set of elements that are not in \operatorname{A}. If \operatorname{A} is a numeric type, the universe is assumed to be the set of all numbers. Q242767
Intersection	\operatorname{A} \cap \operatorname{B}	The set of elements that are in \operatorname{A} and in \operatorname{B} Q185837
Union	\operatorname{A} \cup \operatorname{B}	The set of elements that are in \operatorname{A} or in \operatorname{B} Q173740
Set	\lbrace 1, 2, 3 \rbrace	Set builder notation
SetMinus	\operatorname{A} \setminus \operatorname{B}	Q18192442
SymmetricDifference	\operatorname{A} \triangle \operatorname{B}	Disjunctive union = (\operatorname{A} \setminus \operatorname{B}) \cup (\operatorname{B} \setminus \operatorname{A}) Q1147242

Relations

To check the membership of an element in a set or the relationship between two sets using the following operators.

Function	Notation
Element	x \in \operatorname{A}	x \in \operatorname{A}
NotElement	x \not\in \operatorname{A}	x \not\in \operatorname{A}
NotSubset	\operatorname{A} \nsubset \operatorname{B}	\operatorname{A} \nsubset \operatorname{B}
NotSuperset	\operatorname{A} \nsupset \operatorname{B}	\operatorname{A} \nsupset \operatorname{B}
Subset	\operatorname{A} \subset \operatorname{B} \operatorname{A} \subsetneq \operatorname{B} \operatorname{A} \varsubsetneqq \operatorname{B}	\operatorname{A} \subset \operatorname{B} \operatorname{A} \subsetneq \operatorname{B} \operatorname{A} \varsubsetneqq \operatorname{B}
SubsetEqual	\operatorname{A} \subseteq \operatorname{B}	\operatorname{A} \subseteq \operatorname{B}
Superset	\operatorname{A} \supset \operatorname{B} \operatorname{A} \supsetneq \operatorname{B} \operatorname{A} \varsupsetneq \operatorname{B}	\operatorname{A} \supset \operatorname{B} \operatorname{A} \supsetneq \operatorname{B} \operatorname{A} \varsupsetneq \operatorname{B}
SupersetEqual	\operatorname{A} \supseteq \operatorname{B}	\operatorname{A} \supseteq \operatorname{B}

Intervals

An interval is a set of real numbers that contains all numbers between two endpoints. Intervals can be open (excluding endpoints), closed (including endpoints), or half-open (including one endpoint but not the other).

Interval Notation

The Compute Engine supports both American and ISO/European interval notation:

Notation	LaTeX	MathJSON	Description
[a, b]	[a, b]	["Interval", a, b]	Closed interval (both endpoints included)
(a, b)	(a, b)	["Interval", ["Open", a], ["Open", b]]	Open interval (both endpoints excluded)
[a, b)	[a, b)	["Interval", a, ["Open", b]]	Half-open (closed-open)
(a, b]	(a, b]	["Interval", ["Open", a], b]	Half-open (open-closed)
]a, b[	]a, b[	["Interval", ["Open", a], ["Open", b]]	Open interval (ISO notation)

The Open wrapper indicates that an endpoint is excluded from the interval.

Delimiter Variants

All interval notations support LaTeX delimiter sizing commands:

• Explicit bracket commands: \lbrack, \rbrack, \lparen, \rparen
• Sizing prefixes: \left/\right, \bigl/\bigr, \Bigl/\Bigr, \biggl/\biggr, \Biggl/\Biggr
• Spacing commands: \mathopen/\mathclose

// All of these parse to the same Interval expression:
ce.parse('[3, 4)').json;
ce.parse('\\lbrack 3, 4\\rparen').json;
ce.parse('\\left[ 3, 4 \\right)').json;
ce.parse('\\bigl[ 3, 4 \\bigr)').json;
ce.parse('\\mathopen\\lbrack 3, 4\\mathclose\\rparen').json;
// → ["Interval", 3, ["Open", 4]]

ce.parse('[0, 1)').json;
// ➔ ["Interval", 0, ["Open", 1]]

ce.parse('(-\\infty, 0]').json;
// ➔ ["Interval", ["Open", ["Negate", "PositiveInfinity"]], 0]

Contextual Interval Parsing

When bracket notation appears in a set context (such as with \in, \cup, \cap, \subset, etc.), the Compute Engine automatically interprets it as an interval:

// In set context: [0, 1] becomes an Interval
ce.parse('x \\in [0, 1]').json;
// ➔ ["Element", "x", ["Interval", 0, 1]]

ce.parse('[0, 1] \\cup [2, 3]').json;
// ➔ ["Union", ["Interval", 0, 1], ["Interval", 2, 3]]

// Standalone: [0, 1] remains a List for backward compatibility
ce.parse('[0, 1]').json;
// ➔ ["List", 0, 1]

Interval Serialization

Intervals are serialized using American notation with explicit LaTeX bracket commands:

ce.expr(['Interval', 0, ['Open', 1]]).latex;
// ➔ "\\lbrack0, 1\\rparen"

ce.expr(['Interval', ['Open', 0], ['Open', 1]]).latex;
// ➔ "\\lparen0, 1\\rparen"