I don't know how to prove that $f(f⁻¹(Y))=Y$ iff $Y\subset Im(f)$
Thanks for your help!
Kind regards,
Phi.
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What are your thoughts? What have you tried? Can your prove that, for any $f$ and any $Y$, we have $f(f^{-1}(Y)) \subset Y$? – Ben Grossmann Feb 15 '17 at 17:12
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You just have to lay down the definitions of the objects at hand : first, do you see how $f(f^{-1}(Y))\subset Y$ no matter what ? If you do, the only thing you have to prove is that the reverse inclusion is equivalent to $Y\subset Im(f)$. With this, it should be clearer how to proceed – Maxime Ramzi Feb 15 '17 at 17:14
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Find many answers through: http://math.stackexchange.com/questions/359693/overview-of-basic-results-about-images-and-preimages – Asaf Karagila Feb 15 '17 at 17:15
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$\subseteq]$ $b\in f(f⁻¹(Y)) \rightarrow \exists a\in f⁻¹(Y)$ such that $f(a)=b$. So, $a\in f⁻¹(Y) \rightarrow f(a)\in Y$. So $f(f⁻¹(Y))\subset Y$ – Feb 15 '17 at 17:18
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$\supseteq]$ $b\in Y$ and by hyp. ($Y\subset Im(f)$) $\rightarrow \exists a\in f⁻¹(Y)$ such that $f(a)=b$. So $a\in f⁻¹(Y)$ $\rightarrow$ $b=f(a)\in f(f⁻¹(Y))$. Then $Y\subset f(f⁻¹(Y))$. – Feb 15 '17 at 17:25