Definition:Bijection/Definition 5

Definition
A relation $f \subseteq S \times T$ is a bijection :
 * $(1): \quad$ for each $x \in S$ there exists one and only one $y \in T$ such that $\left({x, y}\right) \in f$
 * $(2): \quad$ for each $y \in T$ there exists one and only one $x \in S$ such that $\left({x, y}\right) \in f$.

Also known as
Hence the terminology:
 * If a bijection exists between two sets $S$ and $T$, then $S$ and $T$ are in one-to-one correspondence.

This must not be confused with one-to-one mapping, a less technical term for an injection.

Because of this confusion, it is recommended that the terms bijection and injection be used universally.

Also see

 * Equivalence of Definitions of Bijection


 * Injection is Bijection iff Inverse is Injection