Student Number:_{} No._{} Marks _{}
1. You have 50 minutes for this exam.
2. This exam contains 5 questions and has a total of 100 marks.
3. Show all of your work. Your work must justify the answer that you
give.
1. (20 Marks) Construct the truth table for the following proposition

Solution.

2. (20 Marks)
Show that pÚq® r Û (p® r)Ù(q® r)
Solution.

Since pÚq® r and (p® r)Ù(q® r) have same
truth values, they are equivalent.
3. (20 Marks) Translate the following statement into the
logical expression.
Every student in York University has a computer or one of his friends has a computer.
Solution. Let the universal set D = the set of students in York University,
P(x): x has a computer.
F(x,y): x and y are friends.
The logical expression is

4. (20 Marks) Let P(x,y) denote x+y = 0, (x,y) Î \mathbb R^{2}.
(1) Express $x_{0}"y P(x_{0},y) in English.
(2) Determine the trueth value of $x_{0}"y P(x_{0},y).
Solution. (1) There exists a real number x_{0} such that for each real number y, x_{0}+y = 0 is true.
(2) For each x Î \mathbb R, there exists y_{0} = x+1 such that

Hence, $x_{0}"y P(x_{0},y) is false.
5. (20 Marks) Let A = {1,2,Æ} and B = {{1},2,4}. Find the following sets.
(a) AÇB
(b) AÈB
(c) AB
(d) A×B
(e) P(A)
Solution. (a) AÇB = {2}.
(b) AÈB = {1,2,Æ, {1}, 4}.
(c) AB = {1, Æ}.
(d) A×B = {(1, {1}), (1,2), (1,4), (2, {1}), (2,2), (2,4), (Æ, {1}), (Æ, 2), (Æ,4)}.
(e) P(A) = {Æ, {1}, {2}, {Æ}, {1,2}, {1, Æ}, {2,Æ}, {1, 2,Æ}}.