Definition:Bijection/Definition 1

Definition
A mapping $f: S \to T$ is a bijection both:
 * $(1): \quad f$ is an injection

and:
 * $(2): \quad f$ is a surjection.

That is, $f$ is a relation which is:
 * $(1): \quad$ left-total
 * $(2): \quad$ right-total
 * $(3): \quad$ functional (many-to-one)
 * $(4): \quad$ injective (one-to-many).

Also see

 * Equivalence of Definitions of Bijection