functional relation

A functional relation is a relation where every element of the domain of definition is uniquely assigned, i.e. has at most one associated element in the codomain of definition.

Formally,
a binary relation $R$ over sets $X$ and $Y$ is functional if for all $x\in X$ and all $y_a,y_b\in Y$, if $(x,y_a)\in R$ and $(x,y_b)\in R$ then $y_a=y_b$.

i.e., no two elements of the codomain can have the same preimage.

If $f:S\to Y$ is a total functional relation, where $S\subset X$ such that $f:X\to Y$ is not total, we say $f:X\to Y$ is a partial function.