A relation is a subset of a Cartesian product.

A relation is often used to refer to a binary relation, which is a subset of $X \times Y$. An element $x \in X$ is said to be related to $y \in Y$ (denoted $xRy$) if the pair $(x,y) \in R \subset X \times X$.

A relation on $X$ is used to refer to a subset of $X \times X$.

A Function $F: X \rightarrow Y$ defines a relation, but not all relations correspond to functions.

Examples of relations

Total ordering – Partial ordering

Equivalence relation

http://mathworld.wolfram.com/Relation.html

The category of sets and relations is Self-dual, that is isomorphic to its Opposite category,