1) List explicitly the members of each of the following sets. What is the cardinality of their power sets? (a) Sa = {d | d is a day of the week and d starts with the letter T or F } (b) Sb = {x | x N, x is a perfect square and 1401 < x < 2023} (c) Sc = {x + y | x, y N and x 5 , y 7 Z and 0 < x, y < 20} (d) Sd = {t | t is a triangle with four sides} (e) Se = {z | z {k, i, s, s} |z| = 2}} (f) Sf = {a, b, c, , {a, b}, a, b} \ {e, a, s, y} (g) Sg = {x | x Z, 50 x < 100 and x is a multiple of 5 or 11}

2. (6pts) Draw the Venn Diagram of the following: (a) (A B) C (b) (A B) C (c) (A B) (A C)

3) Let A, B and C be sets. Determine whether or not the following are valid. Justify your answer by using either set identities or membership tables. You can also use a counterexample to show that two sets are not equivalent. Notice, that the difference between two sets A and B can be denoted A \ B or A B. (a) A (B C) = (A C) B (b) (A B) (B A) = (A B) (c) (A B) (B A) = (A B)

d) (A C) (B C) = A B C (e) (B C) (A C) = A C = A (A C) (f) A (B C) = A (B C)

4) The symmetric difference of set A and B, denoted by AB, is the set containing elements in either A or B, but not in both A and B. Answer the followings: (a) Let A = {a, e, f, y, q, w} and B = {e, x, z, q, p, f }. Find AB. (b) Draw Venn Diagram of (A B) (A B) and (A B) (B A). Conclude that AB = (A B) (A B) and AB = (A B) (B A). (c) Using set identities and part (b) show that AB = (A B) (A B).

5) Determine whether f is a function or not. Justify your answer. Recall, that R is the set of all real numbers, Z is the set of all integers. (a) f : R R, f (x) = 1/x. (b) f : Z R, f (x) = x2 + 3. (c) f : R R, f (x) = x + 5. (d) f : Z R, f (x) = x2. (e) f : R Z, f (x) = dx