Informally, a relation is a rule that describes how elements of a set relate, or interact, with elements of another set. Relations can include, but are not limited to, familial relations (Person A is Person B's mother; or Person A and Person B have the same last name), geographic relations (State A shares a border with State B), and numerical relations (; or ).
A relation is also a set of ordered tuples.
A relation from (or on) sets is any subset of the Cartesian product .
For example, if we let be the set of all cities, and the set of all U.S. States, we can define a relation to be the the set of ordered pairs for which the city is in the state .
See also total order.
If a set is a subset of a cartesian product of two sets A and B, it is called a binary relation on A and B. If a set is a subset of a cartesian square , then it is said to be a binary relation on S.
As a relation from a set to a set is formally viewed as a subset of the Cartesian product , the expression is a valid mathematical expression. However, such an expression can be cumbersome to write, and so we may adopt the alternate notation . Another possible notaton is .