1. Let A = {{1},2,3} and B = {,1,{2},3}. Let P(A) be the power set of A, i.e., the set of all subsets of A.

1. a) Find A B.
2. b) FindP(A)\B.
3. c) Find A(AB).
4. d) Find a bijection f from A to B \ A. Present it in the 2-row form.
5. e) Find a function g from {0,1} to B\A such that f1(g(x)) = x+2 for x {0, 1}. Present it in the 2-row form.
6. f)DoesthereexistasurjectionfromA(AB)toP(A)\B?Why?

2.Let f: XY beafunctionandletAandBbesubsetsofX. We use the

notation f(A) to denote the image of A under f, that is, f(A) = {f(x): x A}.

1. a) Show that f(AB) = f(A)f(B) (hint: to show =, you need both and ).
2. b) Consider the statements: (1) f(AB)f(A)f(B); (2) f(A)f(B)f(AB). Which one is always true? (Hint: find an example or draw a diagram to help you decide)
3. c) Under which of the following additional conditions will both statements (1) and (2) be true: (i) f is surjective, or (ii) f is injective?