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**dubito** Let $\displaystyle f:A \rightarrow B$ and $\displaystyle g: B \rightarrow C$ be functions.

Show that if $\displaystyle g \circ f$ is injective, then $\displaystyle f$ is injective.

Here is what I did.

$\displaystyle Proof$. Spse. $\displaystyle g \circ f$ and $\displaystyle f$ is not injective. Then $\displaystyle \exists x_1,x_2 \in f \ni f(x_1)=f(x_2)$ but $\displaystyle x_1 \neq x_2$.

But then $\displaystyle x_1,x_2 \in g \circ f$ would imply $\displaystyle x_1 \neq x_2$ thus $\displaystyle g \circ f$ is not injective. A contradiction.

Therefore if $\displaystyle g \circ f$ is injective, then $\displaystyle f$ is injective.

Is this OK? for some reason I feel like I am missing something or that my logic is incorrect.

Thanks

dubito