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.
$\blacksquare$