# Proof regarding inverse functions

• Dec 3rd 2012, 01:16 PM
MachinePL1993
Proof regarding inverse functions
Hello,
I've got to solve a following problem.

f is a function like this $\displaystyle f: X \rightarrow Y$. And two sets $\displaystyle A,B \subseteq X$. I've got to figure out how are the following two sets related:
$\displaystyle f^{-1}(f(A))$ and $\displaystyle A$

I got to here:
$\displaystyle x \in f^{-1}(f(A)) \Leftrightarrow f(x) \in f(A)$

Can I assume through this that $\displaystyle x \in A$? If so then how to correctly justify it?
• Dec 3rd 2012, 01:26 PM
Plato
Re: Proof regarding inverse functions
Quote:

Originally Posted by MachinePL1993
Hello,
I've got to solve a following problem.

f is a function like this $\displaystyle f: X \rightarrow Y$. And two sets $\displaystyle A,B \subseteq X$. I've got to figure out how are the following two sets related:
$\displaystyle f^{-1}(f(A))$ and $\displaystyle A$

I got to here:
$\displaystyle x \in f^{-1}(f(A)) \Leftrightarrow f(x) \in f(A)$

Can I assume through this that $\displaystyle x \in A$? If so then how to correctly justify it?

Let $\displaystyle f=\{(1,a),~(2,a),~(3,b),~(4,b)\}$ and $\displaystyle A=\{1,3\}$

Then $\displaystyle f^{-1}(f(A))=~?$
• Dec 3rd 2012, 01:54 PM
MachinePL1993
Re: Proof regarding inverse functions
Then $\displaystyle f^{-1}(f(A))=\{1,2,3,4\}$

So is the following proof correct?

So let's assume that set $\displaystyle C$ is defined like this $\displaystyle C=\{f(x)|x \in A\}$, this means that $\displaystyle C \subseteq Y$ and $\displaystyle C=f(A)$.

So let's take any $\displaystyle x$ such that $\displaystyle x \in A$, then $\displaystyle f(x) \in C$

Since $\displaystyle C=f(A)$, this means that $\displaystyle f^{-1}(f(A))=f^{-1}(C)$

So I have proven that $\displaystyle x \in A \rightarrow x \in f^{-1}(f(A))$, in other words $\displaystyle A \subseteq f^{-1}(f(A))$

But $\displaystyle f^{-1}(f(A))$ is not a subset of $\displaystyle A$ which can be proven by a counterexample like the one above.

Is this right?
• Dec 3rd 2012, 02:04 PM
Plato
Re: Proof regarding inverse functions
Quote:

Originally Posted by MachinePL1993
So is the following proof correct?
Since $\displaystyle C=f(A)$, this means that $\displaystyle f^{-1}(f(A))=f^{-1}(C)$
So I have proven that $\displaystyle x \in A \rightarrow x \in f^{-1}(f(A))$, in other words $\displaystyle f^{-1}(f(A)) \subseteq A$
But $\displaystyle f^{-1}(f(A))$ is not a subset of $\displaystyle A$ which can be proven by a counterexample like the one above.

That is too much. Try to prove
$\displaystyle A\subseteq f^{-1}(f(A))$