Relation

Definition 1. A relation on sets $X_1$, $\cdots$, $X_n$ ($n \geq 1$) is a subset of the Cartesian product $X_1 \times \cdots \times X_n$; in this case it is an $n$-ary relation.

The set of all relations on $X_1$, $\cdots$, $X_n$ is

\[\rel(X_1, \cdots, X_n) = \powerset(X_1 \times \cdots \times X_n).\]

Let $R$ be a relation on $X_1$, $\cdots$, $X_n$. For any $x_1 \in X_1$, $\cdots$, $x_n \in X_n$, we can write

\[R(x_1, \cdots, x_n) \quad\text{for}\quad (x_1, \cdots, x_n) \in R.\]

If $R$ is a binary relation, we can also write

\[x \,R\, y \quad\text{for}\quad R(x, y).\]

An $n$-ary relation on a set $X$ ($n \geq 1$) is a subset of $X^n$.

Examples.

On any sets $X_1$, $\cdots$, $X_n$ ($n\geq 1$), $\empt$ is the empty relation (a relation that never holds).
On any sets $X_1$, $\cdots$, $X_n$ ($n \geq 1$),
\[X_1 \times \cdots \times X_n\]
is the universal relation (a relation that always holds).

Note. For Latinate names of different arities, see Arity (Wikipedia).

Definition 2. The domain of a binary relation $R$ is

\[\dom R = \left\{ x : (\exists y : x \,R\, y) \right\}.\]

This is a set by Separation Schema:

\[\dom R \subseteq \bigcup \bigcup R.\]

Definition 3. The image of a binary relation $R$ is

\[\im R = \left\{ y : (\exists x : x \,R\, y) \right\}.\]

This is a set by Separation Schema:

\[\im R \subseteq \bigcup \bigcup R.\]

Note. Image is also known as range.

Definition 4. The field of a binary relation $R$ is

\[\field R = \dom R \cup \im R.\]