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

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}