# Functional relation

A binary relation $R$ on a set $A$ satisfying $R^{-1}\circ R\subseteq\Delta$, where $R^{-1}$ is the transposed relation, $\Delta$ is the diagonal of $A$ and $\circ$ denotes composition. This means that $(a,b)\in R$ and $(a,c)\in R$ imply that $b=c$, that is, for each $a\in A$ there is at most one $b\in A$ such that $(a,b)\in R$. Thus, $R$ determines a function (perhaps not defined everywhere) on $A$. When it satisfies $R^{-1}\circ R=\Delta$ this function is well-defined everywhere and is one-to-one.
A functional relation is more generally defined as a binary relation $R\subset A\times B$ between sets $A$ and $B$ such that $(a,b)\in R$ and $(a,c)\in R$ imply $b=c$.