Relation

A relation from a set $A$ to a set $B$ is a set $R$ with

where $A\times B$ is the Cartesian product . If $(a,b)\in R$, one often writes $a\,R\,b$.

A relation on a set $A$ means a relation from $A$ to itself, i.e. a subset of $A\times A$. Special kinds of relations include equivalence relations , which encode “having the same type” in a precise sense.

Examples:

• The “less than or equal to” relation on $\mathbb{Z}$ is $R=\{(m,n)\in\mathbb{Z}\times\mathbb{Z}: m\le n\}$.
• For any set $A$, the equality relation on $A$ is $\{(a,a): a\in A\}\subseteq A\times A$.