Equivalence of Definitions of Bijection/Definition 1 iff Definition 2
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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
- $f$ has both a left inverse and a right inverse.
$\blacksquare$