Definition 10

Let f be a mapping of A into B. The mapping f is said to be injective, or an injection, if any two distinct elements of A have distinct images under f. The mapping f is said to be surjective, or a surjection, if $f(\mathrm{A}) = \mathrm{B}$. If f is both injective and surjective, it is said to be bijective, or a bijection.

Instead of saying that $f$ is surjective, we sometimes say that $f$ is a mapping of A onto B, or that it is a parametric representation of B by means of A (in which case A is called the set of parameters of the representation, and the elements of A are called parameters). If $f$ is bijective, we sometimes say that f puts A and B in one-to-one correspondence. A bijection of A onto A is called a permutation of A.

Examples