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.

1.    (20 Marks)    Construct the truth table for the following proposition

 pÚØq ® Øp

Solution.

 p    q
 Øp
 Øq
 pÚØq
 pÚØq ® Øp
 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 pÚq® r Û (p® r)Ù(q® r) Solution.

 p    q    r
 pÚq
 p® r
 q® r
 pÚq® r
 (p® r)Ù(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 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

 "x P(x) Ú\$y (P(y)ÙF(x,y)).

4.    (20 Marks)    Let P(x,y) denote x+y = 0, (x,y) Î \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 Î \mathbb R, there exists y0 = -x+1 such that

 x+y0 = x+(-x+1) = 1 ¹ 0.

Hence, \$x0"y P(x0,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) A-B

(d) A×B

(e) P(A)

Solution. (a) AÇB = {2}.

(b) AÈB = {1,2,Æ, {1}, 4}.

(c) A-B = {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}}.

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On 20 Apr 2001, 14:41.