Equivalence of Definitions of Bijection/Definition 1 iff Definition 2

Theorem

The following definitions of the concept of Bijection are equivalent:

Definition 1

A mapping $f: S \to T$ is a bijection if and only if both:

$(1): \quad f$ is an injection

and:

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

Definition 2

A mapping $f: S \to T$ is a bijection if and only if:

$f$ has both a left inverse and a right inverse.

Proof

From Injection iff Left Inverse, $f$ is an injection if and only if $f$ has a left inverse mapping.

From Surjection iff Right Inverse, $f$ is a surjection if and only if $f$ has a right inverse mapping.

Putting these together, it follows that:

$f$ is both an injection and a surjection

if and only if:

$f$ has both a left inverse and a right inverse.

$\blacksquare$