Relation

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 ($A = B$; or $x \le y$).

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.

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 $s \sim t$.