## FANDOM

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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 $t$.

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.
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 $\sim(x,y)$.