# Relation

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Relations are nonnegative ordered pairs.A relation consists of a domain and range.A domain is a set of all first coordinates(x) of a relation, while a range is the set of all second coordinates(y).

E.G. Relation is {(1,10),(2,20),(3,30),(4,40)}

Its domain is {1,2,3,4}

Its range is {10,20,30,40}

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

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.

## NotationEdit

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 .