Test #1 September 16, 2014

Transcription

1 CPSC 2070 Name: Test #1 September 16, questions, 100 points, 75 minutes. The weight of each question is in parentheses. Please use a pencil. Write neatly and legibly. If you don t have a pencil, ask Dr. Woodard for one. You may keep it for future tests. The last page is blank and may be detached and used as scratch paper. Put your name on it and turn it in with your test. There are multiple versions of this test. Please do your own work. 1. (10) Let q be the proposition I will do every exercise in this book and p be the proposition I will get an A in this course. Express each of these as a combination of p and q. (a) I will get an A in this course only if I do every exercise in this book. (b) I will get an A in this course and I will do every exercise in this book. (c) Either I will not get an A in this course or I will not do every exercise in this book. (d) For me to get an A in this course it is necessary and sufficient that I do every exercise in this book. p q p q Both p q and p q accepted p q 2. (10) Prove or disprove: (512) 1/2 = (the square root of 512) is irrational. Answer: (512) 1/2 = (2 9 ) 1/2 = (2 8 * 2) 1/2 = 2 4 (2) 1/2 = 16 (2) 1/2 16 is rational. (2) 1/2 is irrational. The product of a rational number and an irrational number is irrational. (How would you prove this?) Therefore, (512) 1/2 is irrational. 3. (15) What is the disjunctive normal form for: ( (x y) z ) Show how you arrived at your answer. x y z (x y) (x y) z ((x y) z) DNF T T T F T F T T F F F T <- (x y z) T F T T T F T F F T T F F T T T T F F T F T T F F F T F T F F F F F F T <- ( x y z)

2 4. (10) Use a Direct Proof to prove that: The sum of two rational numbers is a rational number. Let m and n be two rational numbers. Hence, m = a/b and n = c/d where a, b, c, d Z and b, d 0. Let p = m + n = a/b + c/d = (ad + bc)/bd Since a, b, c, d Z, (ad + bc) Z and bd Z Moreover, bd 0 Therefore, p = m + n is a rational number. 5. (15) Use a truth table to prove that ((p q) p) q is a tautology. p q p q p ((p q) p) ((p q) p) q T T T F F T T F T F F T F T T T F T F F F T F T 6. (10) Without using a truth table prove or disprove that: (p q) r (p r) (q r). (p q) r r (p q) Commutative property of (r p) (r p) Distributive property of over (p r) (q r) Commutative property of applied twice 7. (10) Let P(m,n) be the statement m + n = 12, where the universe of discourse for both variables is Z. State whether each of the following propositions is true or false. F P ( 4, 8) T n P(12 n,n) F m n P(m,n) T m n P(m,n) T n P(1,n) T P(15, 3) P(10,120) F_ m n P(m,n) T P (2,10) 8. (10) Determine whether each of these statements is True or False: T {x} {{x}} T φ {x} F φ {x} F {x} {x} T {x} {x} F {x} {x} T x {x} 9. (10) Determine which of the following statements is True or False: F is an irrational number T is a rational number F (5) 1/2 = (the square root of 5) is not a real number T is a rational number F The product of two irrational numbers is irrational. 2 * 2 = 2

3 CPSC 2070 Name: Test #1 September 16, questions, 100 points, 75 minutes. The weight of each question is in parentheses. Please use a pencil. Write neatly and legibly. If you don t have a pencil, ask Dr. Woodard for one. You may keep it for future tests. The last page is blank and may be detached and used as scratch paper. Put your name on it and turn it in with your test. There are multiple versions of this test. Please do your own work. 1. (10) Let q be the proposition I will do every exercise in this book and p be the proposition I will get an A in this course. Express each of these as a combination of p and q. (a) Either I will not get an A in this course or I will not do every exercise in this book. (b) For me to get an A in this course it is necessary and sufficient that I do every exercise in this book. (c) I will get an A in this course only if I do every exercise in this book. (d) I will get an A in this course and I will do every exercise in this book. Both p q and p q accepted p q p q.p q 2. (10) Prove or disprove: (1024) 1/2 = (the square root of 1024) is irrational. Answer: (1024) 1/2 = (2 10 ) 1/2 = 2 5 = 32 which is rational. Therefore, (1024) 1/2 is not irrational. 3. (15) What is the disjunctive normal form for: ( (a b) c ) Show how you arrived at your answer. a b c (a b) c (a b) c DNF T T T F F F T T F F T T <- (a b c) T F T T F T <- (a b c) T F F T T T <- (a b c) F T T T F T <- ( a b c) F T F T T T <- ( a b c) F F T F F F F F F F T T <- ( a b c)

4 4. (10) Prove or disprove that: The quotient of two rational numbers is a rational number. Identify the method of proof that you used. Let m and n be two rational numbers. Hence, m = a/b and n = c/d where a, b, c, d Z and b, d 0. Assume p = m/n = (a/b)/(c/d) = (ad/bc). Since a, b, c, d Z, then both ad and bc Z. However, although by definition b 0, c may be 0. Hence, bc may be 0. Which means that p = (ad/bc) is not guaranteed to be rational. Proof by contradiction. 5. (15) Use a truth table to prove that ((a b) a) b is a tautology. a b (a b) a (a b) a) ((a b) a) b T T T F F T T F T F F T F T T T T T F F F T F T 6. (10) Without using a truth table prove or disprove that: (a b) c (a c) (b c). (a b) c c (a b) Commutative property of (c a) (c b) Distributive property of over (a c) (b c) Commutative property of applied twice 7. (10) Let P(m,n) be the statement m + n = 12, where the universe of discourse for both variables is Z. State whether each of the following propositions is true or false. T n P(1,n) T P(15, 3) P(10,120) F m n P(m,n) T P (2,10) T m n P(m,n) F P ( 4, 8) T n P(12 n,n) F m n P(m,n) 8. (10) Determine whether each of these statements is True or False: F {x} {x} T {x} {x} F {x} {x} T x {x} T φ {x} F φ {x} T {x} {{x}} 9. (10) Determine which of the following statements is True or False: F (7) 1/3 = (the cube root of 7) is not a real number F is an irrational number T is a rational number F is an irrational number F The product of two irrational numbers is irrational. 2 * 2 = 2

5 CPSC 2070 Name: Test #1 September 16, questions, 100 points, 75 minutes. The weight of each question is in parentheses. Please use a pencil. Write neatly and legibly. If you don t have a pencil, ask Dr. Woodard for one. You may keep it for future tests. The last page is blank and may be detached and used as a scratch sheet. Put your name on it and turn it in with your test. There are multiple versions of this test. Please do your own work. 1. (10) Let p be the proposition I will do every exercise in this book and q be the proposition I will get an A in this course. Express each of these as a combination of p and q. (a) I will get an A in this course only if I do every exercise in this book. (b) I will get an A in this course and I will do every exercise in this book. (c) Either I will not get an A in this course or I will not do every exercise in this book. (d) For me to get an A in this course it is necessary and sufficient that I do every exercise in this book. p q p q Both p q and p q accepted p q 2. (10) Prove or disprove: (2048) 1/2 = (the square root of 2048) is irrational. Answer: (2048) 1/2 = (2 11 ) 1/2 = (2 10 * 2) 1/2 = 2 5 (2) 1/2 = 32 (2) 1/2 32 is rational. (2) 1/2 is irrational. The product of a rational number and an irrational number is irrational. (How would you prove this?) Therefore, (512) 1/2 is irrational. 3. (15) What is the disjunctive normal form for: ( (p q) r ) Show how you arrived at your answer. p q r (p q) (p q) r ((p q) r) DNF T T T F T F T T F F F T <- (p q r) T F T T T F T F F T T F F T T T T F F T F T T F F F T F T F F F F F F T <- ( p q r)

6 4. (10) Use a Direct Proof to prove that: The difference between two rational numbers is a rational number. Let m and n be two rational numbers. Hence, m = a/b and n = c/d where a, b, c, d Z and b, d 0. Let p = m n = a/b c/d = (ad bc)/bd Since a, b, c, d Z, (ad bc) Z and bd Z Moreover, bd 0 Therefore, p = m n is a rational number. 5. (15) Use a truth table to prove that ((p q) p) q is a tautology. p q p q p ((p q) p) ((p q) p) q T T T F F T T F T F F T F T T T F T F F F T F T 6. (10) Without using a truth table prove or disprove that: (p q) r (p r) (q r). p q) r r (p q) Commutative property of (r p) (r p) Distributive property of over (p r) (q r) Commutative property of applied twice 7. (10) Let P(m,n) be the statement m + n = 12, where the universe of discourse for both variables is Z. State whether each of the following propositions is true or false. F_ P ( 4, 8) T P (2,10) F m n P(m,n) F_ m n P(m,n) T P(15, 3) P(10,120) T_ m n P(m,n) T n P(12 n,n) T n P(1,n) 8. (10) Determine whether each of these statements is True or False: T x {x} T {x} {x} F {x} {x} T {x} {{x}} T φ {x} F φ {x} F {x} {x} 9. (10) Determine which of the following statements is True or False: T is a rational number F is an irrational number T is a rational number F (5) 1/2 = (the square root of 5) is not a real number F The product of two irrational numbers is irrational. 2 * 2 = 2