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:

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

$\blacksquare$


Sources