Composite of Injections is Injection

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Theorem

A composite of injections is an injection.


That is:

If $f$ and $g$ are injections, then so is $f \circ g$.


Proof

Let $f$ and $g$ be injections.

Then:

\(\ds \map {f \circ g} x\) \(=\) \(\ds \map {f \circ g} y\)
\(\ds \leadsto \ \ \) \(\ds \map f {\map g x}\) \(=\) \(\ds \map f {\map g y}\) Definition of Composition of Mappings
\(\ds \leadsto \ \ \) \(\ds \map g x\) \(=\) \(\ds \map g y\) as $f$ is injective
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds y\) as $g$ is injective

$\blacksquare$


Sources