Injection if Composite is Injection

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Theorem

Let $f$ and $g$ be mappings such that their composite $g \circ f$ is an injection.


Then $f$ is an injection.


Proof

Let $g \circ f$ be injective.

We need to show that $\map f a = \map f b \implies a = b$.

So suppose $\map f a = \map f b$.

Then:

\(\displaystyle \map {g \circ f} a\) \(=\) \(\displaystyle \map g {\map f a}\) Definition of Composition of Mappings
\(\displaystyle \) \(=\) \(\displaystyle \map g {\map f b}\) by hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \map {g \circ f} b\) Definition of Composition of Mappings

and so $a = b$ as $g \circ f$ is an injection.


So we have shown that:

$\map f a = \map f b \implies a = b$

Hence the result from the definition of injection.

$\blacksquare$


Also see


Sources