Equivalence of Definitions of Injection/Definition 1 iff Definition 6
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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 6
Let $f: S \to T$ be a mapping where $S \ne \O$.
Then $f$ is an injection if and only if $f$ is left cancellable:
- $\forall X: \forall g_1, g_2: X \to S: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$
where $g_1$ and $g_2$ are arbitrary mappings from an arbitrary set $X$ to the domain $S$ of $f$.
Proof
This is demonstrated in Injection iff Left Cancellable.
$\blacksquare$