How to show that the composite function of two injective functions is injective

Question

I have 2 injective functions,$X:A \to B$ and $Y:B \to C$.

How do I show that $Y \circ X$ is injective.

In the form of a proof.

Answer 1

Well you may start by being sure what does injectivity means:

$X:A \to B$ is injective if every time $X(a)=X(a')$ then $a=a'$.

To see that $Y \circ X:A \to C$ is injective, you have to see that every time that $Y \circ X(a)=Y \circ X(a')$ then $a=a'$.

So take $a,a' \in A$ such that $Y \circ X(a) = Y \circ X(a')$. This means that $Y(X(a))=Y(X(a'))$. Now we use the injectivity of $Y$: Since $Y:B\to C$ is injective, any time $Y(b)=Y(b')$ then $b=b'$. Think of $X(a)$ and $X(a')$ as $b$ and $b'$ respectively.

Then $Y(X(a))=Y(X(a'))$ implies $X(a)=X(a')$. Now try to use injectivity of $X$ to show that $a=a'$