DEF-R. Relation.
A 2-ary / binary relation on $X$ and $Y$ is a subset of the Cartesian product $X\times Y$.
A 3-ary / ternary relation on $X$, $Y$ and $Z$ is a subset of $X\times Y\times Z$.
4-ary / quaternary relations, 5-ary / 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 $R$ ($n\geq 2$), for any $x_1$, $\cdots$, $x_n$, 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, for any $x$ and $y$, we can also write
\[x\,R\,y \quad\text{for}\quad R(x,y).\]
REM-R-AR.
The arity of a relation is not an intrinsic property, as the arity of a tuple is not intrinsic (REM-TUP-AR). For example, a ternary relation on $X$, $Y$ and $Z$ could also be seen as a binary relation on $X\times Y$ and $Z$.
DEF-R-EMP. Empty Relation.
For any sets $X_1$, $\cdots$, $X_n$ ($n\geq 2$), $\varnothing$ is the empty relation on $X_1$, $\cdots$, $X_n$.
DEF-R-UNI. Universal Relation.
For any sets $X_1$, $\cdots$, $X_n$ ($n\geq 2$), $X_1\times\cdots\times X_n$ is the universal relation on $X_1$, $\cdots$, $X_n$.
DEF-R-DOM. Domain.
The domain of a binary relation $R$ is
\[\operatorname{dom}R = \left\{x\in\bigcup\bigcup R:(\exists y:x\,R\,y)\right\}.\]
DEF-R-RAN. Range.
The range of a binary relation $R$ is
\[\operatorname{ran}R = \left\{y\in\bigcup\bigcup R:(\exists x:x\,R\,y)\right\}.\]
DEF-R-FLD. Field.
The field of a binary relation $R$ is
\[\operatorname{field}R = \operatorname{dom}R\cup\operatorname{ran}R.\]