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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 (A=B; or x \leq y).

A relation is also a set of ordered pairs.

A relation from a set A to a set B is any subset of the Cartesian product A×B.

For example, if we let S be the set of all cities, and T the set of all U.S. States, we can define a relation R to be the the set of ordered pairs (s,t) for which the city s is in the state t.

See also total order.


As a relation \sim from a set S to a set T is formally viewed as a subset of the Cartesian product S \times T, the expression \left(s,t\right)\in \sim is a valid mathematical expression. However, such an expression can be cumbersome to write, and so we may adopt the alternate notation s \sim t.

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