Math 3336: Discrete Mathematics Practice Problems for Exam I

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1 Math 3336: Discrete Mathematics Practice Problems for Exam I The upcoming exam on Tuesday, February 26, will cover the material in Chapter 1 and Chapter 2*. You will be provided with a sheet containing the laws of logical equivalences and the rules of inference (and you can find it as page 3 of this practice exam). Do NOT print the one provided here. I will give each of you one during the exam. IMPORTANT! First try these problems as if it were the real exam; work by yourself without the text or your notes. This is supposed to be a gauge on what you need to work on to prepare for the exam. Answering these problems as you might handle homework problems won t necessarily give you much of a clue on what you need to work on. Instructions: Provide all steps necessary to solve the problem. Unless otherwise stated, your answer must be exact and reasonably simplified. Additionally, clearly indicate the value or expression that is your final answer. A. Propositional/Predicate logic 1. Find the truth table of the compound proposition (p q) (p r). 2. Give the converse, the contrapositive, and the inverse of the statement If it rains today, then I will drive to work. 3. Show that p (q r) and q (p r) are logically equivalent using the laws of logical equivalences. Be sure to cite each law whenever used. 4. Use the table of logical equivalences to simplify the compound proposition [(p q) p] q. Be sure to justify your answers. 5. Let P(m,n) be the statement m n, where the domain for both variables consists of all positive integers. [By m n, which we say as m divides n, we mean that n = km for some integer k.] Determine the truth values of each of these statements. (a) P(4,5) (b) P(2,4) (c) m n P(m,n) (d) m n P(m,n) (e) n m P(m,n) (f) n P(1,n) 6. Consider the compound proposition ( m n [P(m,n)]) ( n m[p(m,n)]) where both m and n are integers. Determine the truth value of the proposition if (a) P(m,n) is the statement m < n. (b) P(m,n) is the statement m n. 7. Suppose that the variable x represents students, F(x) means x is a freshman, and M(x) means x is a math major. For each of the three statements (a), (b), and (c), determine which of the symbolic statements are equivalent. (Note: Each statement may have multiple answers.) I. x [M(x) F(x)] II. x [M(x) F(x)] III. x [F(x) M(x)] IV. x [M(x) F(x)] V. x [F(x) M(x)] VI. x [ F(x) M(x)] VII. x [ (M(x) F(x))] VIII. x [ M(x) F(x)] IX. x [M(x) F(x)] X. x [M(x) F(x)] XI. x [F(x) M(x)] (a) Some freshmen are math majors. (b) Every math major is a freshman. (c) No math major is a freshman. Answer: Answer: Answer:

2 8. Determine whether the following argument is valid. p r q r () r If the argument if valid, provide a valid proof of the result (that is, use the laws of logical equivalences and the rules of inference to demonstrate that the conclusion is valid). If the argument is not valid, provide specific truth values of p, q, and r in which the premises are true, but the conclusion is false. 9. Express the following sentence using quantifiers : There is a student in this class who has taken some course in every department in the school of science. Then find the logical negation and translate into an English sentence. 10. Is the following argument valid? You can score well in the GRE only if you have good analytical skills. Every student who takes Discrete Math has good analytical skills or good memory. Maggie doesn t have good memory. Therefore, if Maggie takes Discrete Math, then Maggie will score well in the GRE. 11. Is the argument valid? No man is an island. Manhattan is an island. Therefore Manhattan is not a man. 12. Suppose P (x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose P (1, 3), P (2, 1), P (2, 2), P (2, 3), P (3, 1), P (3, 2) are true, and P (x, y) is false otherwise. Determine whether the following statements are true or false: [true/false] x yp (x, y) [true/false] x yp (x, y) [true/false] x y(p (x, y) P (y, x)) [true/false] y x(p (x, y) P (y, x)) [true/false] x y((x y) (P (x, y) P (y, x)) [true/false] y x((x y) P (x, y))

3 13. Use logical equivalences to show that [(p q) (p r) (q r)] r is a tautology. 14. For each of the given statements: 1 - Express each of the statements using quantifiers and propositional functions. 2 - Form the negation of the statement so that no negation is to the left of the quantifier. 3 - Express the negation in simple English. (Do not simply use the words it is not the case that... ). (a) Some drivers do not obey the speed limit. (b) All Swedish movies are serious. (c) No one can keep a secret. (d) No monkey can speak French. (e) There is someone in the class who does not have a good attitude. 15. Rewrite the following statments statements so that all negation symbols immediately precede predicates (that is, no negation is outside a quantifier or an expression involving logical connectives). Show al the steps in your derivation. (a) x yp (x, y) (b) y(q(y) x R(x, y)) (c) y( xr(x, y) xs(x, y)) (d) y( x zt (x, y, z) x zu(x, y, z)) 16. Prove these logical equivalences, assuming that the domain is nonempty. You will probably have to use a proof by cases on the two possible values of proposition yq(y) and yq(y) respectively. This proof will use word arguments. (a) x( yq(y) P (x)) yq(y) xp (x) (b) x( yq(y) P (x)) yq(y) xp (x) 17. For each of these arguments, determine whether the argument is correct or incorrect and explain why. (a) Everyone born in Ottawa has eaten a beaver tail. Susan has never eaten a beaver tail. Therefore Susan was not born in Ottawa. (b) A convertible car is fun to drive. Joe s car is not a convertible. Therefore, Joe s car is not fun to drive. (c) Emma likes all fine restaurants. Emma likes the restaurant Le Cordon Bleu. Therefore, Le Cordon Bleu is a fine restaurant. 1

4 18..Give a formal proof, using known rules of inference, to establish the conclusion of the argument (3rd statement) using the first 2 statements as premises, where the domain of all quantifiers is the same. Remember that a formal proof is a sequence of steps, each with a reason noted beside it; each step is either a premise, or is obtained from previous steps using inference rules. premise: x(p (x) Q(x)) premise: x(( P (x) Q(x)) R(x)) conclusion: x( R(x) P (x)) B. (Simple) Examples of proof techniques 1. Prove the following statements: (a) For every x > 0, if x is irrational, then x is irrational. (b) Between any two distinct rational numbers, there exist infinitely many rational numbers. 2. Prove that if m is a positive integer of the form 4k+3 for some non-negative integer k, then m is not the sum of the squares of two integers. 3. Prove that if a nonzero integer n is a perfect square (i.e., n = k 2, for some integer k) then n + 1 is not a perfect square. 4. If n is odd, then n 2 is odd is irrational. C. Sets (1) List the elements of the set {x x is a real number such that x 2 = 2 1 or x = 5}. (2) Show that (B A) (C A) = (B C) A. (3) What can you say about A, B if: a) A B = A; b) A B = B A. (4) List the elements of the set S = {x x Z and (x + 1)(2 x) 0}.

5 5. Show that, for any sets A, B, C, (A B) C = (A C) (B C). 6. Give an example to show that it not always true that (A B) C = A (B C). That is, find sets A, B, C such that the right side does not equal the left side. 7. Prove the following statements for any sets S, T, U. a. S T S T = T. b. S T S T = S. c. S U and T U = S T U. d. S T U and S T = = S U. 8. List all subsets of the set {, { }, 1 }. D. Basic Structures: Functions and Cardinality 1. a) Suppose A, B, and C are sets and f : A B and g : B C are functions such that g f : A C is injective. Can you conclude that both f and g are injective? Give a proof or a counterexample. b) Suppose A, B, and C are sets and f : A B and g : B C are functions such that g f : A C is surjective. Can you conclude that both f and g are surjective? Give a proof or a counterexample. 2. If f : X Y is injective and y Y, then f 1 ({y}) must be 1. Give a proof or a counterexample. 3. Suppose that f : X Y is a function, and that A X and B X. a) Prove that f(a B) f(a) f(b). b) Prove that f(a B) = f(a) f(b) given that f is an injective map. 4. For each of the following pairs of sets, say if they have the same cardinality or not. Give arguments (proofs) to justify your answers in each case. a) Z + and Z + Z +. b) Z + and (0, 1) = {x R 0 < x < 1}. 3

6 EQUIVALENCES AND IMPLICATION EQUIVALENCES Double negation law: ( p) p Identity laws: p F p, p T p Domination laws: p T T, p F F Negation laws: p p T, p p F Idempotent laws: p p p, p p p Commutative laws: q p, q p Associative laws: p (q r) () r, p (q r) () r Distributive laws: p (q r) () (p r), p (q r) () (p r) Absorption laws: p () p, p () p DeMorgan s laws: () p q, () p q q p (p q) 5. () p q 6. () (p r) p (q r) 7. (p r) (q r) () r 8. () (p r) p (q r) 9. (p r) (q r) () r 10. (p q) (q p) 11. p q 12. () ( p q) 13. ( ) p q RULES OF INFERENCE p (Addition) p (Simplification) p q (Conjunction) p q (Modus ponens) q p (Modus tollens) q r p r (Hypothetical syllogism) p q (Disjunctive syllogism) p r q r (Resolution)