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$f(S\cap T) \neq f(S) \cap f(T)$

but

$f^{-1}(Q \cap R)=f^{-1}(Q) \cap f^{-1}(R)$

Can you explain it in simple terms, so I understand why and develop the intuition to see if a statement is true or false just by looking at it?

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    Please don’t get the idea that anybody can tell whether a formula or a statement is true or false just by looking at it. If you’ve *proved* it, then you may well see afterwards that the proof was easy; if, as in the case that $f(S\cap T)\ne f(S)\cap f(T)$, you’ve found an easy counterexample, then all will be clear after the fact. But *never* before the fact.2012-12-08
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    Formal proof for the second one is given [here](http://math.stackexchange.com/questions/228711/what-are-the-strategies-i-can-use-to-prove-f-1s-cap-t-f-1s-cap-f). Some counterexamples to the first one are given [here](http://math.stackexchange.com/questions/170725/do-we-have-always-fa-cap-b-fa-cap-fb).2013-10-01

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