Let F:X->Y be an arbitrary function over sets X and Y.(adsbygoogle = window.adsbygoogle || []).push({});

Why is F^{-1}(Y) = X always true?

Suppose B_{1}and B_{2}are some subsets of Y. Why is F^{-1}(B_{1}[tex]\bigcap[/tex] B_{2}) = F^{-1}(B_{1}) [tex]\bigcap[/tex] F^{-1}(B_{2}) always true?

These aren't homework questions. I'm just curious. I saw these statements the other day, and I already know that F(X) = Y is not necessarily true, and neither is (for some subsets A1 and A2 of X) F(A1 [tex]\bigcap[/tex] A2) = F(A1) [tex]\bigcap[/tex] F(A2).

Why would the other statements always be true? It seems to me that inverse functions are just like any other functions when it comes to the two theorems I already know. Why are they different?

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# Questions regarding function operations on sets.

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