> [!NOTE] Definition (Bijective function) > A given *function* $f$ is bijective if it is [[Injection|injective]] and [[Surjection|surjective]]. # Properties **Relation to invertibility**: We can prove that a function is bijective either by showing that it is both injective and surjective or by showing that that it is [[Bijection iff Invertible|invertible]]. # Applications **Permutation**: A [[Permutation of a Set|permutation]] of a set $A$ is a bijection from $A$ to itself. Note that the set of all permutations of $A$ forms a group under function composition known as the [[Symmetric Group|symmetric group]] of $A.$