Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $f:A\rightarrow B$ is a surjection and $H\subseteq B$, how to show that $$ f(f^{-1}(H))=H $$

share|improve this question
    
Im not an expert or teacher in elementary set theory but from my experience i recall learning your definions well pays off well in trying to prove a short statement. I noticed that Asaf gave a correct answer in the meanwhile and i note that taking examples such as " let x be element of " occurs very often in elementary set theory proofs as well as " pulling an = into 2 times a C " if you know what i mean. You must master those skills for the future. –  mick Sep 2 '12 at 21:38
add comment

1 Answer

up vote 4 down vote accepted

Let $x\in H$, since $f$ is surjective there is some $a\in A$ such that $f(a)=x$, and this $a\in f^{-1}(H)$ by definition (since $x\in H$). Therefore $f(a)=x\in f(f^{-1}(H))$. Therefore $H\subseteq f(f^{-1}(H))$.

I leave you to do the second direction, pick some $x\in f(f^{-1}(H))$ and use the definition of $f(Y)$ and $f^{-1}(Z)$ for $Y\subseteq A, Z\subseteq B$, to conclude that $x\in H$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.