Show that if $f$ is injective, then $f^{-1}(f(C))=C$

Question

Once again I am stuck on a question from Lay's Introduction to Analysis with Proof:

Suppose that $f : A \rightarrow B$ and let C $\subseteq$ A and $D \subseteq B$. Show that if $f$ is injective, then $f^{-1}(f(C))=C$

I just need to show that $f^{-1}(f(C) \subseteq C$ and $C \subseteq f^{-1}(f(C))$.

I have started with this but am unsure of where to go next;

Let $x \in f^{-1}(f(C))$. Then, $x \in \{f^{-1}(a) \mid a \in f(C)\}$.

Note that $x\in f^{-1}(A)$ just means that $f(x)\in A$. More details in my answer below. — Kim Jong Un, Sep 15 '14 at 05:44

score 1 · Answer 1 · answered Sep 15 '14 at 05:17

If $x\in f^{-1}(f(C))$ then $f(x)\in f(C)$. If $x$ is not in $C$, then there is some element $y\in C$ such that $x\neq y$ and $f(x)=f(y)$ but this violates injectiveness, so it must be that $x\in C$. Therefore, you have one direction of inclusion.

The reverse direction is always true regardless of injectiveness: suppose that $x\in C$, then $f(x)\in f(C)$ so that $x\in f^{-1}(f(C))$.

hrkrshnn · Accepted Answer · 2014-09-15T05:44:28.303

0

Let $f$ be a one-to-one function. If $x\in C$, then $f^{-1}(f(x))=\{x\}\Rightarrow f^{-1}({f(C)})=C$.

To prove the last implication, this about this. If $g(x)=x$ what is $g(C)$? $C$!

edited Sep 15 '14 at 05:44

answered Sep 15 '14 at 05:21

hrkrshnn

6,287

You have assumed the proof was true in your proof it seems ... – MathMajor Sep 15 '14 at 05:31
To use this proof don't we first need to prove that $f^{-1}(f(x))=x$ for injective $f$? – MathMajor Sep 15 '14 at 05:34
I think that is clear, since $f(x)$ maps to a unique element $f^{-1}(f(x))=x$. – hrkrshnn Sep 15 '14 at 05:35
Yes but $f^{-1}(x)$ only exists if $f$ is one-to-one does it not? – MathMajor Sep 15 '14 at 05:36
For $f^{-1}(x)$ to properly exist over the codomain, we need additional conditions. Here we are talking about $f^{-1}(f(x))$. – hrkrshnn Sep 15 '14 at 05:39

score 0 · Answer 3 · answered Sep 15 '14 at 05:51

0

Let $f^{-1}(f(C))=B$. Then there exists $x_b\in B$ and $x_c\in C$, such that, $f(x_b)=f(x_c)$. But $f$ is injective so, $x_b=x_c$. Hence $B=C$.

answered Sep 15 '14 at 05:51

nature1729

551

Show that if $f$ is injective, then $f^{-1}(f(C))=C$

3 Answers3