Department of Mathematics and Statistics Math B: Discrete Mathematics and Its Applications Test 1: October 19, 2006

Transcription

1 Department of Mathematics and Statistics Math B: Discrete Mathematics and Its Applications est 1: October 19, 2006 Name: (Last/amily) (irst) Student No. Instructions: 1. his test has 8 questions and you have to answer 8 questions and show all your work and reasons to your answers. 2. Put answers and rough work on the question paper, using the back pages if necessary. 3. You may use non-programmable and non-graphing calculators. 4. his test is for 80 Minutes. Question Maximum Marks Marks Obtained otal 75 (est paper A) 1

2 1.(4 Marks) Construct a truth table for (p q) ( p q) p q p q p q (p q) ( p q) 2.(6 Marks) Evaluate the expressions. a) = b) = c) = (15 Marks) Determine if these system specifications are consistent: he book is stored in the library or it is borrowed. he book is stored in the library. If the book is stored in the library, then it is borrowed. Let s be he book is stored in the library and b he book is borrowed. hen the statements are: s b, s, s b. s b s b s s b We see that on the first row of the table all the values are true. So system specifications are consistent. 2

3 4. (4 Marks) Show that (p q) p is a tautology by using truth table. p q p q (p q) p We see that (p q) p. 5. (7 Marks) Show that (p r) (p r) and (p q) r are logically equivalent. applying p q= p q, (p q)= p q and the distributive law: (p q) r= (p q) r=( p q) r= =( p r) ( q r)= (p r) (p r). So they are logically equivalent. Also truth table can be used to prove the equivalence. 6. (8 Marks) Determine the truth value of each of these statements if the domain consists of all integers. a) n(n+1>n) b) n(3n=5n) c) n(n=-n) d) n(n 2 >n) a) is true, since 1>0 is all time true. b) b) is true, since 3x0=5x0=0. c) is true, since 0=-0. d) d) is false, since 0 2 >0 is false. 3

4 7. (15 Marks) ranslate these statements into logical expressions using predicates, quantifiers, and logical connectives. Let the domain consist of all students in your class. a) Someone in your class can speak rench. b) Everyone in your class is smart. c) here is a student in your class who was not born in Canada. d) A student in your class has been in a spaceship. e) No student in your class has taken a course in logic programming. Let the domain D be all students in the/your class and variable x D. So the statements are: a) x (x), where (x) is x can speak rench b) x S(x), where S(x) is x is smart c) x C(x), where C(x) is x is born in Canada d) x Ω(x), where Ω (x) is x has been in a spaceship e) x L(x)= x L(x), where L(x) is x has taken a course in logic programming. 8. (16 Marks) Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. hen negate the expressions. a) he sum of two positive integers is positive. b) he difference of two positive integers is not necessarily positive. c) he sum of the squares of two integers is greater than or equal to the square of their sum. d) he absolute value of the product of two integers is the product of their absolute values. D=Z. he expressions: a) x y((x>0) (y>0) (x+y>0)) or x>0 y>0 (x+y>0). b) x y((x>0) (y>0) (x-y>0)) or x>0 y>0 (x-y>0) in an other word x y((x>0) (y>0) (x-y=<0)) or x>0 y>0 (x-y=<0). c) x y(x 2 + y 2 ( x + y) 2 ) d) x y( xy = x y ). 4

5 he negations: a) x y((x>0) (y>0) (x+y>0))= x y((x>0) (y>0) (x+y=<0)), x>0 y>0 (x+y>0)= x>0 y>0 (x+y=<0). b) ( x y((x>0) (y>0) (x-y>0)))= x y((x>0) (y>0) (x-y>0)), x>0 y>0 (x-y=<0)= x>0 y>0 (x-y>0). c) x y(x 2 + y 2 ( x + y) 2 )= x y(x 2 + y 2 < ( x + y) 2 ). d) x y( xy = x y )= x y( xy x y ). 5