Exam 1 Exam 1                              Friday, March 23, 2001



NAME:

Student Number:------            No.----           Marks ----






Instructions:

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


pq p


Solution.


p    q
p
q
pq
pq p
T    T
F
F
T
F
T    F
F
T
T
F
F    T
T
F
F
T
F    T
T
T
T
T



2.    (20 Marks)    Show that pq r (p r)(q r) Solution.


p    q    r
pq
p r
q r
pq r
(p r)(q r)
T    T    T
T
T
T
T
T
T    T    F
T
F
F
F
F
T    F    T
T
T
T
T
T
T    F    F
T
F
T
F
F
F    T    T
T
T
T
T
T
F    T    F
T
T
F
F
F
F    F    T
F
T
T
T
T
F    F    F
F
T
T
T
T

Since pq r and (p r)(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


"x P(x) $y (P(y)F(x,y)).


4.    (20 Marks)    Let P(x,y) denote x+y = 0, (x,y) \mathbb R2.

(1) Express $x0"y P(x0,y) in English.

(2) Determine the trueth value of $x0"y P(x0,y).

Solution. (1) There exists a real number x0 such that for each real number y, x0+y = 0 is true.

(2) For each x \mathbb R, there exists y0 = -x+1 such that


x+y0 = x+(-x+1) = 1 0.

Hence, $x0"y P(x0,y) is false.

5.    (20 Marks)    Let A = {1,2,} and B = {{1},2,4}. Find the following sets.

(a) AB

(b) AB

(c) A-B

(d) A×B

(e) P(A)

Solution. (a) AB = {2}.

(b) AB = {1,2,, {1}, 4}.

(c) A-B = {1, }.

(d) A×B = {(1, {1}), (1,2), (1,4), (2, {1}), (2,2), (2,4), (, {1}), (, 2), (,4)}.

(e) P(A) = {, {1}, {2}, {}, {1,2}, {1, }, {2,}, {1, 2,}}.


File translated from TEX by TTH, version 2.79.
On 20 Apr 2001, 14:41.