Part 1:

We let f: A - B be a function. For any subset Y C B, we define the inverse image of Y under f (denoted by f-1(Y)) as the collection of elements in the domain A that maps to elements in f(Y). That is, f - 1 (Y) = {x E Alf (x) EY}. Prove the following statements (a) Us f-1(f(U)) for any subset U C A. Give an example which U c f-1 (f(U)) (b) f(f-1(V)) EV for any subset V C B. Give an example which f (f-1(V)) c V (c) f(UaEl Xa) = VaElf(Xa) and f-1(UQEl Ya) = Vaelf-1(Ya). Here, Xa is subset of A and Ya is subset of B for all a E I (d) f(naEl Xa) S naElf (Xa) and f-1(nQEI Ya) = naelf-1(Ya). (*Note: In (c) and (d), I is called index set and Xa = {xx EXa for some a El} and Xa = {x|x EXa for all a El} aEl aEl (*Note 2: Here, A C B means that A is proper subset of B in the sense that A C B but A + B)We let f: X - Y be a function, prove that f is injective if and only if f (A n B) = f(A) n f (B) for all A, B C X. (Hint: To prove " - " (i.e. f(An B) = f(A) n f(B) implies f is injective) part, you can consider "proof by contradiction" and derive a contradiction by considering suitable choices of A and B)