Properties of injective functions
Hi.
problem:
Show that if $\displaystyle f:A\rightarrow B$ is injective and $\displaystyle E\subseteq A$, then $\displaystyle f^{-1}(f(E))=E$.
I worked with this for a while but was unable to come up with a good answer.
After a bit of searching, I fould an article on planetmath which answers this question, but I do not understand it completely.
Here's what it says:
Theorem. Suppose $\displaystyle f:A\rightarrow B$ is an injection. Then, for all $\displaystyle C\subseteq A$, it is the case that $\displaystyle f^{-1}(f(C))=C$.
Proof. It follows from the definition of $\displaystyle f^{-1}$ that $\displaystyle C\subseteq f^{-1}(f(C))$, whether or not $\displaystyle f$ happens to be injective.
Hence, all that need to be shown is that $\displaystyle f^{-1}(f(C))\subseteq C$.
Assume the contrary. Then there would exist $\displaystyle x\in f^{-1}(f(C))$ such that $\displaystyle x\notin C$.
By definition $\displaystyle x\in f^{-1}(f(C))$ means $\displaystyle f(x)\in f(C)$, so there exists $\displaystyle y\in A$ such that $\displaystyle f(x)=f(y)$.
Since $\displaystyle f$ is injective, one would have $\displaystyle x=y$, which is impossible because $\displaystyle y$ is supposed to belong to $\displaystyle C$ but $\displaystyle x$ is not supposed to belong to $\displaystyle C$.
I do not see why $\displaystyle y$ is supposed to belong to $\displaystyle C$. What am I missing here?
Thanks!