Equivalence of Definitions of Injection/Definition 1 iff Definition 5

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Theorem

The following definitions of the concept of Injection are equivalent:

Definition 1

A mapping $f$ is an injection, or injective if and only if:

$\forall x_1, x_2 \in \Dom f: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$


That is, an injection is a mapping such that the output uniquely determines its input.

Definition 5

Let $f: S \to T$ be a mapping where $S \ne \O$.

Then $f$ is an injection if and only if:

$\exists g: T \to S: g \circ f = I_S$

where $g$ is a mapping.


That is, if and only if $f$ has a left inverse.


Proof

This is demonstrated in Injection iff Left Inverse.

$\blacksquare$