Let F:X->Y be an arbitrary function over sets X and Y. Why is F-1(Y) = X always true? Suppose B1 and B2 are some subsets of Y. Why is F-1(B1 [tex]\bigcap[/tex] B2) = F-1(B1) [tex]\bigcap[/tex] F-1(B2) 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?