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Can anyone please help me to solve this question?

Prove that $f(f^{-1} (f(A) ) )=f(A)$.

I have proved the first part: Let $y\in f(f^{-1} (f(A) ) )$. Then there exists $x\in f^{-1} (f(A) )$ such that $f(x)=y$. By definition this means that $y=f(x)\in f(A)$.

How do I prove the other part?

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  • $\begingroup$ Can you not approach it the same way? $\endgroup$ – Clayton Aug 9 '13 at 18:28
  • $\begingroup$ please can you help me in proving it. $\endgroup$ – sertar Aug 9 '13 at 18:34
  • $\begingroup$ I can't shake the feeling that this question was asked sometime in the last two days. $\endgroup$ – Asaf Karagila Aug 9 '13 at 18:38
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Let $x \in f(A)$. Then there is $y \in A$ such that $f(y)=x$. Since $y \in A$, $f(y) \in f(A)$ and so $y \in f^{-1}(f(A))$. Therefore, $f(y) \in f(f^{-1}(f(A)))$.

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