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.box(['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.box(['Element', 'x', 'real']).evaluate().print();
// ➔ True

ce.box(['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.box(42);

x.type;
// ➔ "finite_integer"

x.type.matches("integer");
// ➔ true

x.isInteger;
// ➔ true

ce.box(['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

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

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