Definition

A function is a relation between two sets that associates, to every element of the first set, exactly one element of the second set.

A relation associates elements of the first set to the elements of the second set, but it does not require all elements of the first set to participate in the relation, nor that the elements that do participate, are associated to exactly one element in the second set.

Unlike relations, which permit a single object of the source set to relate to multiple objects in the target set, the fundamental property of functions is that they disallow this. This restriction makes functions deterministic because for the same input, they always produce the same output (and not several, equally possible, outputs).

A function is a relation that associates each element $x$ of a set $X$ to a single element $y$ of a set $Y$.