What is a relation in which each element of the first set is paired with exactly one element of the second set?

If every element of the first set is paired with exactly one element of the second set, it is called an injective (or one-to-one) function.

An example of such a relation is below.

Let f(x) and x be the set R (the set of all real numbers)

f(x)= x3, clearly this maps every element of the first set, x, to one and only one element of the second set, f(x), even though every element of the second set is not mapped to.