Is a unary function a binary relation

Question

$f: X \rightarrow Y$ is a unary function, however writing it in relation notation, we would write it $xRy$, which would be binary relation.

Is my assumption that a $n$-ary function, corresponds to a $(n+1)$-ary relation correct?

Answer 1

You are correct. To be precise, in set theory a function is usually defined as a special kind of relation.

Definition. A relation $f\subseteq X\times Y$ is called a function $f:X\to Y$ if for any $x\in X$ there is one and only one $y\in Y$ with $$(x,y)\in f \quad(\text{or, if you want } xfy).$$

This will generalize naturally to $n$-ary functions when you consider $X=X_1\times\cdots\times X_n$. Then you will have that an $n$-ary function is a special kind of $(n+1)$-ary relation on $X_1\times\cdots\times X_n\times Y$.