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I'm trying to show that $f(f^{-1}(E))= E$ is not necessarily true. I want to use a non-injective function as an example, but what I want to know is if I can consider $E$ to be a set which is not a subset of the range of $f$. If so, I'd like to know why. Thank you!

Dan
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  • If E is not a subset of the range, it must at least have non-empty intersection with the range. Otherwise, $f^{-1}(E)$ is empty and f(empty) is empty by definition. – DBFdalwayse Sep 11 '13 at 07:34

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Yes you can consider $E$ to be a set which is not a subset of the range of $f$. In fact $f(f^{-1}(E))\neq E$ if and only if $E$ is not a subset of the range. This is because $f(f^{-1}(E))=\text{range}(f)\cap E$ (can you prove it?).

walcher
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