# Sets

A set is a collection of distinct elements.

A domain, such as Integer Boolean, is a set used to represent the possible values of an expression.

## Constants

Symbol Notation Definition
EmptySet $$\varnothing$$ or $$\emptyset$$

The domains also define a number of sets.

## Functions

New sets can be defined using a set expression. A set expression is an expression with one of the following head functions.

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

## Relations

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