5) (cont.)

(b) You can graduate if you have completed the requirements of your course and you do

not owe money to the university and you do not have an overdue library loan.

(2 marks)

Let p be the proposition statement You can graduate,

q be the proposition statement You owe money to the university,

r be the proposition statement You have completed the requirements of your course,

and s be the proposition statement You have an overdue library book.

6) (a) Show that p q and (p q) (p q) are logically equivalent. (3 marks)

(b) Show that p (q r) and q (p r) are logically equivalent. (4 marks)

7) Show that each of these conditional statements is a tautology by using truth tables:

(a) (pq)q (2 marks)

(b) (pq) q (2 marks)

8) What is the cardinality of each of these sets: (4 marks)

(a) {x}

(b) {{x}}

(c) {x,{x}}

(d) {x,{x},{x,{x}}}

9) Find the power set of each of these sets, where x, y are distinct elements: (4 marks)

(a) {x, y}

(b) {x, y, z}

10) Let A={a,b,c,d} and B={x,y}. Find: (4 marks)

(a) A x B

(b) B x A

11) Let A = {1,2,3,4,5} and B={0,3,6}. Find: (4 marks)

(a) AB

(b) AB

(c) A-B

(d) B-A

12) Draw the Venn diagrams for each of these combinations of the sets A, B, and C:

(a) A(B-C) (2 marks)

(b) (AB)(AC) (3 marks)