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Relation

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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 tuples.

A relation from (or on) sets S_1,\ldots,S_n is any subset of the Cartesian product S_1 \times \ldots \times S_n.

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.

Binary relation

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 S^2, then it is said to be a binary relation on S.

Notation

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. Another possible notaton is \sim(x,y).

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