CS2336 Discrete Mathematics

Transcription

1 CS2336 Discrete Mathematics Homework 1 Tutorial: March 20, 2014 Problems marked with * will be explained in the tutorial. 1. Determine which of the following statements are propositions and which are non-propositions. If a statement is a proposition, then decide the truth value of it. (a) 2008 is a leap year. (b) All people living in Taiwan can speak in Taiwanese. (c) No parking. (d) What is a polite society? (e) George Lucas is not an Asian. (f) If I have seen further, it is by standing on the shoulders of giants. 2. (*) Consider the expression (P Q) (P Q). In any way that you like, find an equivalent expression that is as short as possible. Prove that your expression is equivalent. 3. (*) Let M be the proposition you will get all pass. Let P be the proposition you work hard. Express each of the following statements using M, P, and logical connectives. (a) You will get all pass if you work hard. (b) You did not work hard but you will get all pass. (c) You will get all pass whether or not you work hard. 4. Write down the negation of the following statements. (a) a > 0 or b > 0. (b) Both Newton and Gauss are mathematicians. (c) If a 2 + b 2 = 0, then a = b = 0. (d) All people prefer taking trains than buses. 5. Explain why the following statements are correct. (a) (P Q) ( P Q) is a tautology. (b) P (P Q) is logically equivalent to P. (c) P (Q R) and (P Q) R are not logically equivalent. (d) The negation of P (Q R) is logically equivalent to ( P ) ( Q) ( R). 6. Which of the following statements are logically equivalent to P ( Q)? Why? (a) ( P ) ( Q) 1

2 (b) ( P ) Q (c) ( P ) Q (d) (P Q) 7. Construct the truth tables step by step of the following propositions. (a) P ( Q Q) (b) (P Q) (P Q) (c) P (Q R) 8. (*) What is wrong with this argument? Let H(x) be x is happy. Given that xh(x), we conclude that H(Lola). Therefore Lola is happy. 9. (*) What is wrong with this argument? Let S(x, y) be x is shorter than y. Given the premise ss(s, Max) it follows that S(Max, Max). Then by existential generalization it follows that xs(x, x), so that someone is shorter than himself. 10. (*) Identify the error or errors in this argument that supposedly shows that if x(p (x) Q(x)) is true then xp (x) xq(x) is true. (1) x(p (x) Q(x)) Premise. (2) P (c) Q(c) Universal instantiation from (1). (3) P (c) Simplification from (2). (4) xp (x) Universal generalization from (3). (5) Q(c) Simplification from (2). (6) xq(x) Universal generalization from (5). (7) x(p (x) Q(x)) Conjunction from (4) and (6). 11. Show that the following arguments are valid. (a) Premises: A (B C), A Conclusion: B (b) Premises: E F, E G, F Conclusion: G (c) Premises: G F, H F, G Conclusion: H (d) Premises: N V, V Conclusion: N (e) Premises: U C, L U, L Conclusion: C (f) Premises: (A B) (C D), A C, B Conclusion: D (g) Premises: (H J) (I K), L, L (H I) Conclusion: J K (h) Premises: ( P Q) R, R (S T ), S U, U T Conclusion: P 2

3 12. Show that each of the following arguments is invalid by providing a counterexample. Find an assignment of truth values for given primitive statements P, Q, R, and S such that all premises are true while the conclusion is false. (a) Premises: (((P Q) R) ( Q R) Conclusion: P (b) Premises: P Q, Q R, R S, S Q Conclusion: S (c) (*) Premises: P, P R, P (Q R), Q S Conclusion: S 13. Show that the following arguments are valid. (a) Premises: xa(x) Conclusion: xa(x) (b) Premises: xa(x) Conclusion: x(a(x) B(x)) (c) Premises: x(a(x) B(x)), x(a(x) C(x)) Conclusion: x(c(x) B(x)) (d) Premises: x( K(x) H(x)) y(l(y) L(y)) Conclusion: x H(x) 14. (*) Let P (x), Q(x), and R(x) be the following open statements. P (x) : x 2 7x + 10 = 0 Q(x) : x 2 2x 3 = 0 R(x) : x < 0 (a) Determine the truth or falsity of the following statements, where the universe is all integers. If a statement is false, provide a counterexample or explanation. i. x(p (x) R(x)) ii. x(q(x) R(x)) iii. x(q(x) R(x)) iv. x(p (x) R(x)) (b) Find the answers to part (a) when the universe consists of all positive integers. (c) Find the answers to part (a) when the universe contains only the integers 2 and There are four boxes, A, B, C, and D. There are cakes in some of them. We know that If there are cakes in at least one of A and B, then D contains cakes. If there are cakes in at least one of A and C, then at least one of B and D do not contain cakes. If there are cakes in at least one of A and D, then B contains cakes, too. What can we say about A? (a) There must be cakes in it. 3

4 (b) There must be no cake in it. (c) We cannot conclude anything about A. 16. (*) Four students (A, B, C and D) attended an exam. The following are what they said after the exam. A: I got the highest score. B: I got the highest score. C: My score is lower than A s. D: B s score is higher than C s. Only the one who got the highest score tells the truth, the others are all liars. List the names of the students according to their scores from highest to lowest. (Note: All students have different scores.) 17. (*, Challenging) Jane, Janice, Jack, Jasper, and Jim are the names of five high schools chums. Their last names in one order or another are Carter, Carver, Clark, Clayton, and Cramer. Jasper s mother is dead. In deference to a certain very wealthy aunt, Mr. and Mrs. Clayton agreed when they were first married that if they ever had a daughter they would name her Janice. Jane s father and mother have never met Jack s parents. The Cramer and Carter children have been teammates on several of the school s athletic teams. When Cramer heard that Carver was going to be out of town on the night of the school s Father and Son banquet, Cramer called Mrs. Carver and offered to adopt her son for the evening, but Jack s father had already asked him to go. The Clarks and Carters, who are very good friends, were delighted when their children began dating each other. What is the full name of each youngster? You can make the following assumptions: If person A dates person B, then A and B are of opposite gender. There are no mixed gender sports. Agreements are always kept. No one got divorced. 18. (*, Challenging) Albert Einstein once posed a brain teaser that he predicted only 2% of the worlds population would be able to solve. FACTS: (a) There are 5 houses in 5 different colours (red, white, green, yellow, blue). (b) In each house lives a person with a different nationality (British, Swedish, Danish, German, Norwegian). 4

5 (c) These 5 owners drink a certain beverage (tea, coffee, milk, beer, water), smoke a certain brand of cigarette (Pall Mall, Dunhill, Blend, Camel, Marlborough), and keep a certain pet (dog, birds, cats, horses, fish). (d) No owners have the same pet, brand of cigaratte, or drink. CLUES: (a) The Brit lives in a red house. (b) The Swede keeps a dog. (c) The Dane drinks tea. (d) The green house is on the left of the white house. (e) The green house owner drinks coffee. (f) The person who smokes Pall Mall keeps birds. (g) The owner of the yellow house smokes Dunhill. (h) The man living in the house right in the center drinks milk. (i) The Norwegian lives in the first house. (j) The man who smokes Blend lives next to the one who keeps cats. (k) The man who keeps horses lives next to the man who smokes Dunhill. (l) The owner who smokes Camel drinks beer. (m) The German smokes Marlborough. (n) The Norwegian lives next to the blue house. (o) The man who smokes Blend has a neighbour who drinks water. The question is, who keeps the fish? HINT: Try working it out with pencil and paper as a matrix 5