CSE 2315 - Discrete Structures Homework 1- Fall 2010 Due Date: Sept. 16 2010, 3:30 pm Statements, Truth Values, and Tautologies 1. Which of the following are statements? a) 3 + 7 = 10 b) All cars are blue. c) It will rain on Wednesday. d) There are more insects than mammals. e) Is the sun shining right now? f) x 3 < 7 g) 7 4 = 9 h) Is John at home? 2. Construct the truth tables for the following expressions and determine if they are tautologies or contradictions. a) A B A b) ((A B) C) C c) A (C B) C C A d) (A (A B) (B B) e) A B C f) A B (B A) 3. Let M, R, S, F, and C be the following statements: M R S F C Mary likes dogs Cats run away from dogs Cats are smaller than dogs Dogs are fast Dogs chase cats 2010 Manfred Huber Page 1
Translate the following compound statements into symbolic notation. a) Cats are smaller than dogs and run away from them. b) If cats run away from dogs then dogs chase them. c) Only if dogs chase cats do cats run away from dogs. d) Dogs are fast and cats run away from dogs if and only if dogs chase cats. e) Mary likes dogs only if dogs do not chase cats. f) If cats are smaller than dogs and dogs are fast then cats run away from dogs and dogs chase them. Translate the following symbolic statements into English. g) M R C h) R F C i) F M R C j) F (R C) M k) F C l) S R C M Proofs using Propositional Logic For all your propositional logic proofs you can use only the rules given in the following tables. In addition you are allowed to apply the deduction method. All other rules have to be proven first. Equivalence Rules Rule Name Expression Equivalent Expression Commutativity (comm) P Q Q P P Q Q P Associativity (ass) (P Q) R P (Q R) (P Q) R P (Q R) Distributivity (dis) (P Q) R (P R) (Q R) (P Q) R (P R) (Q R) De Morgan s Laws (De Morgan) (P Q) P Q (P Q) P Q Implication (imp) P Q P Q Double negation (dn) (P ) P 2010 Manfred Huber Page 2
Inference Rules Rule Name From Can Derive Conjunction (con) P, Q P Q Simplification (sim) P Q P, Q Modus ponens (mp) P, P Q Q Modus tollens (mt) P Q, Q P Addition (add) P P Q For all proofs the steps have to be annotated such as to indicate the rule and which elements of the proof sequence it was applied to. 4. Use propositional calculus to prove that the following arguments are valid. a) (A C) C A b) (A (B C)) (B C) A c) (C (A B)) (C B) (C A) B d) (A (B C)) (A B) (A C) (A B C) e) (A (B C)) ((A B) (C D)) A D f) A (B C) (D B E) (C F A) D E 5. Translate the following arguments into symbolic notation and use propositional logic to prove their validity. Your translation has to include a description of all the statement symbols used. a) Only if John and Mary met yesterday could Mary have given John her house keys. Since the book was taken from Mary s house today by either John or Mary without breaking the door, someone with a house key must have taken it. To meet, John and Mary would have had to be in the same place but yesterday Mary was in a different place than John. Therefore Mary must have taken the book herself. b) Jane has a cat and a dog and no other pets and finds a mouse in front of her door. Only one of her pets could have left the mouse in front of the door and only a pet that catches mice could have left it. To catch a mouse the pet would have had to leave the apartment but Jane s cat did not leave the apartment all dat. Therefore Jane s dog must have left the mouse in front of her door. Quantifiers, Predicates, and Validity 6. Give the truth values of the following formulas for the interpretation where the domain consists of all integers greater than 0 and G(x, y) is x > y, O(x) is x is odd, and T (x) is x is a multiple of 3. 2010 Manfred Huber Page 3
a) ( x)( y)(o(x) G(x, y)) b) ( x)( y)(o(x) G(x, y)) c) ( x)(( y)(t (y) G(y, x)) G(x, 1)) d) ( x)(( y)(o(y) ( z)g(x, z))) e) ( x)(t (x) O(x)) ( y)o(y) f) ( x)(t (x) ( y)g(x, y)) g) ( x)( y)(g(x, y) ( z)g(y, z)) h) ( x)( y)(g(y, x) O(y)) 7. Identify all instances of free and bound variables in the following formulas. a) P (x) ( x)(q(x, y) ( z)p (z)) b) ( x)(( y)p (x, y) Q(y) R(x, y)) c) ( x)( y)(( z)(p (x, z) Q(z, y)) (R(x, y) S(x, z))) d) ( x)(( y)p (x, y) Q(x) ( y)(r(y, x) Q(y))) e) ( x)q(x, y) P ( z)( y)r(y, z) S(z) f) ( y)(q(y) ( x)(q(x, y) P )) g) (P ( y)q(x, y)) ( x)t (x, y) h) ( y)(( x)p (x, y) S Q(x, z) ( z)r(y, z)) 8. For each of the following formulas find one interpretation which makes them true and one which makes them false. a) ( x)(( y)p (x, y) Q(x)) b) ( x)( y)(p (y) Q(x, y)) c) ( x)(p (x) ( y)(p (y) (y = a) Q(y, x))) d) ( x)(p (a) ( y)(p (y) Q(a, y)) R(x)) e) ( x)( y)(p (x, y) R(y, x)) f) ( x)( y)(((y = a) (y = x) Q(x, y)) ( z)(r(x, z) (z = a))) 9. Let the domain be the universal domain and let the predicates and constants have the following interpretation. T (x, y) x is taller than y R(x) x is a race car driver F (x, y) x drives faster than y W (x) x wins the race b Bill j John 2010 Manfred Huber Page 4
Translate the following compound statements into symbolic representation. a) If Bill drives faster than John then John does not win the race. b) The fastest driver wins the race. c) No more than one driver can win the race. d) Taller race car drivers do not drive faster. e) John is not the tallest race car diver and he wins the race only if he drives faster than all other dirvers. f) Sometimes the winner of the race is not the fastest and not the shortest driver. g) The winner of the race has to be a race car driver and can not be the slowest race car driver. h) Bill wins the race only if he is taller than John and drives faster than any other race car driver besides John. Translate the following formulas into English. i) ( x)(r(x) (F (b, x) (b = x))) j) ( x)( y)(r(x) W (x) (R(y) F (y, x) (y = x))) k) ( y)( x)(r(y) W (y) T (x, y) F (x, y)) l) ( x)(r(x) W (x) ( y)((y x) T (y, x))) m) ( x)( y)(r(x) (R(y) (T (y, x) F (y, x)))) n) ( x)(r(x) F (x, j) ( y)(r(y) T (x, y) (x = y))) 2010 Manfred Huber Page 5