Injection iff Left Cancellable

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Theorem

A mapping $f$ is an injection if and only if $f$ is left cancellable.


Proof

From the definition: a mapping $f: Y \to Z$ is left cancellable if and only if:

$\forall X: \forall g_1: X \to Y, g_2: X \to Y: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$


Necessary Condition

Let $f: Y \to Z$ be an injection.

Let $X$ be a set

Let $g_1: X \to Y, g_2: X \to Y$ be mappings such that:

$f \circ g_1 = f \circ g_2$


Then $\forall x \in X$:

\(\displaystyle \map f {g_1 \paren x}\) \(=\) \(\displaystyle \map {f \circ g_1} x\) Definition of Composite Mapping
\(\displaystyle \) \(=\) \(\displaystyle \map {f \circ g_2} x\) By Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \map f {\map {g_2} x}\) Definition of Composite Mapping


As $f$ is an injection, $\map {g_1} x = \map {g_2} x$ and thus the condition for left cancellability holds.

$\Box$


Sufficient Condition

We use a Proof by Contraposition.

That is, we show that if $f: Y \to Z$ is not injective, then $f$ is not left cancellable.

Hence, suppose $f: Y \to Z$ is not injective.

Then:

$\exists y_1 \ne y_2 \in Y: f \left({y_1}\right) = f \left({y_2}\right)$

Let the two mappings $g_1: Y \to Y, g_2: Y \to Y$ be defined as follows:

$\forall y \in Y: g_1 \left({y}\right) = y$
$\forall y \in Y: g_2 \left({y}\right) = \begin{cases} y_2 & : y = y_1 \\ y & : y \ne y_1 \end{cases}$

Thus we have $g_1 \ne g_2$ such that $f \circ g_1 = f \circ g_2$.

That is, $f$ is not left cancellable.

From Rule of Transposition it follows that if $f$ is left cancellable, it is injective.

$\blacksquare$


Also see


Sources