Proof regarding inverse functions

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

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

Can I assume through this that $x \in A$ ? If so then how to correctly justify it?
December 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 $f: X \rightarrow Y$ . And two sets $A,B \subseteq X$ . I've got to figure out how are the following two sets related:
$f^{-1}(f(A))$ and $A$

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

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

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

Then $f^{-1}(f(A))=~?$
December 3rd 2012, 01:54 PM
MachinePL1993

Re: Proof regarding inverse functions

Then $f^{-1}(f(A))=\{1,2,3,4\}$

So is the following proof correct?

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

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

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

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

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

Is this right?
December 3rd 2012, 02:04 PM
Plato

Re: Proof regarding inverse functions

Quote:

Originally Posted by MachinePL1993

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

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