A binary relation on $X$ and $Y$ is a subset of the Cartesian product $X\times Y$.
A ternary relation on $X$, $Y$ and $Z$ is a subset of $X\times Y\times Z$.
quaternary relations, quinary relations, etc. are defined analogously.
An $n$-ary relation on $X$ ($n\geq 2$) is a subset of $X^n$.
Given an $n$-ary relation $R$ ($n\geq 2$), 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
\[x\,R\,y \quad\text{for}\quad R(x,y).\]
Examples.
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On sets $X_1$, $\cdots$, $X_n$ ($n\geq 2$), $\varnothing$ is the empty relation.
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On sets $X_1$, $\cdots$, $X_n$ ($n\geq 2$),
\[X_1\times\cdots\times X_n\]is the universal relation.
The domain of a binary relation $R$ is
\[\dom R = \left\{x:(\exists y:x\,R\,y)\right\}.\]This is a set:
\[\dom R \subseteq \bigcup\bigcup R.\]
The range of a binary relation $R$ is
\[\ran R = \left\{y:(\exists x:x\,R\,y)\right\}.\]This is a set:
\[\ran R \subseteq \bigcup\bigcup R.\]
The field of a binary relation $R$ is
\[\field R = \dom R\cup\ran R.\]