DISCRETE MATH: FINAL REVIEW DR. DANIEL FREEMAN 1) a. Does 3 = {3}? b. Is 3 {3}? c. Is 3 {3}? c. Is {3} {3}? c. Is {3} {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z x > 0} {x R x > 0}? 1. Chapter 1 review 2. Chapter 2 review 1) Construct a truth table for ( p q) (p q). 2) Construct a truth table to show that (p q) is logically equivalent to p q. What is the name for this law? 1
2 DR. DANIEL FREEMAN 3) Construct a truth table to determine if (p q) r is logically equivalent to (p r) (q r). 4) Use a truth table to determine if the following argument is logically valid. Write a sentence which justifies your conclusion. p q r r p q p r
DISCRETE MATH: FINAL REVIEW 3 You will be provided with the following information on the test. 2.1. Modus Ponens and Modus Tollens. The modus ponens argument form has the following form: If p then q. p q. Modus tollens has the following form: If p then q. q p. 2.2. Additional Valid Argument Forms: Rules of Inference. A rule of inference is a form of argument that is valid. Modus ponens and modus tollens are both rules of inference. Here are some more... Generalization p Elimination p q p q q Specialization p q p p Transitivity p q Proof by Division into Cases p q q r p q p r q r Conjunction p r q Contradiction Rule p c p q p
4 DR. DANIEL FREEMAN 5) Write a logical argument which determines what I ate for dinner. Number each step in your argument and cite which rule you use for each step. a. I did not have a coupon for buns or I did not have a coupon for hamburger. b. I had hamburgers or chicken for dinner. c. If I had hamburgers for dinner then I bought buns. d. If I did not have a coupon for buns then I did not buy buns. e. If I did not have a coupon for hamburger then I did not buy buns.
DISCRETE MATH: FINAL REVIEW 5 3. Chapter 3 review 1) a. Give an example of a universal conditional statement. b. Write the contrapositive of the example. c. Write the negation of the example. 2) Write the following statements symbolically using,,,,. Then write their negation. a. If x, y R then xy + 1 R. b. Every nonzero real number x has a multiplicative inverse y. c. Being divisible by 8 is not a necessary condition for an integer to be divisible by 4. d. If I studied hard then I will pass the test.
6 DR. DANIEL FREEMAN 2) Is it true or false that every real number bigger than 4 and less than 3 must be negative? Explain why. 4. Chapter 4 review 1) Prove using the definition of odd: For all integers n, if n is odd then ( 1) n = 1. 2) Prove using the definition of even: The product of any two even integers is divisible by 4.
DISCRETE MATH: FINAL REVIEW 7 3) Prove: For each integer n with 1 n 5, n 2 n + 11 is prime. 4) Show that:.123123123... is a rational number. 5) Prove using the definition of divides: For all integers a, b, and c, if a divides b and a divides c then a divides b c.
8 DR. DANIEL FREEMAN 6) Evaluate 60 div 8 and 60 mod 8. 7) Prove using the definition of mod: For every integer p, if pmod10 = 8 then pmod5 = 3.
9 a) How do you prove a statement by contradiction? DISCRETE MATH: FINAL REVIEW 9 9 b) Prove: There is no greatest integer. (for a more interesting problem, prove that there is no greatest prime number.) 9 a) How do you prove x D, if P (x) then Q(x) by contraposition? 9 b) Prove: For all integers a, b, and c, if a bc then a b.
10 DR. DANIEL FREEMAN 1) a. Compute: 5 i=2 i2 5. Chapter 5 review b. Compute: 1 i= 5 i c. Compute: 100 i=1 (i)2 (i 1) 2 d. Compute: 50 i=2 i(i 1) (i+1)(i+2)
DISCRETE MATH: FINAL REVIEW 11 3 a) What are the steps for proving For all integers n such that n a, P (n) is true using mathematical induction? 3 b) Prove: For all integers n 1, n 5i 4 = i=1 n(5n 3). 2
12 DR. DANIEL FREEMAN 6 a) What are the steps for proving For all integers n such that n a, P (n) is true using strong mathematical induction? 6 b) Prove: If s 0 = 12, s 1 = 29, and s k = 5s k 1 6s k 2 k 2, then s n = 5 3 n + 7 2 n for all integers n 0.
DISCRETE MATH: FINAL REVIEW 13 6. Chapter 6 review 1) Let B = {n Z n = 21r + 10 for some r Z} and C = {m Z m = 7s + 3 for some s Z}. Prove that B C. 2) Let A = {a, b, c, d, e}, B = {d, e, f, g} and C = {b, c, d, f}. What is (A B) C?
14 DR. DANIEL FREEMAN (1) Commutative Laws: For all sets A and B, A B = B A and A B = B A. (2) Associative Laws: For all sets A, B, and C, (A B) C = A (B C) and (A B) C = A (B C). (3) Distributive Laws: For all sets A, B, and C, A (B C) = (A B) (A C) and A (B C) = (A B) (A C). (4) Identity Laws: For all sets A, A = A and A =. (5) Complement Laws: For all sets A, A A c = U and A A c =. (6) Double Complement Law: For all sets A, (A c ) c = A. (7) Idempotent Laws: For all sets A, A A = A and A A = A. (8) Universal Bound Laws: For all sets A, A U = U and A =. (9) De Morgan s Laws: For all sets A and B, (A B) c = A c B c and (A B) c = A c B c. (10) Absorption Laws: For all sets A and B, A (A B) = A and A (A B) = A. (11) Complements of U and : U c = and c = U (12) Set Difference Law: For all sets A and B, A B = A B c
3) Prove the second part of the Distributive Law DISCRETE MATH: FINAL REVIEW 15
16 DR. DANIEL FREEMAN 4) Prove the second part of De Morgan s Law.
DISCRETE MATH: FINAL REVIEW 17 5) Prove or give a counterexample to the statement: For all sets A,B, and C, A (B C) = (A B) (A C)
18 DR. DANIEL FREEMAN 6) Prove or give a counterexample to the statement: For all sets A,B, and C, A (B C) = (A B) (A C)
DISCRETE MATH: FINAL REVIEW 19 7) Prove using the given set theory laws that for all sets A,B, and C, A (B C) = (A B) (A C)
20 DR. DANIEL FREEMAN 7. Chapter 7 review 9) Prove or give a counterexample to the statement: For all functions f : X Y, and all sets C, D Y, f 1 (C D) = f 1 (C) f 1 (D)
DISCRETE MATH: FINAL REVIEW 21 10) Prove or give a counterexample to the statement: For all functions f : X Y, and all sets C Y f(f 1 (C) = C 11) Prove or give a counterexample to the statement: For all functions f : X Y, and all sets A X f 1 (f(a)) = A
22 DR. DANIEL FREEMAN 8. Chapter 9 review 12) Prove that the number of r permutations of a set of n elements is P (n, r) = n! (n r)!. 13) Prove that if X is a set of n elements then the number of subsets of X with r elements is ( ) n n! = r r!(n r)!
DISCRETE MATH: FINAL REVIEW 23 14) Prove that if X is a set of n elements then the number of subsets of X is N(P(X)) = 2 n 15) Use problem 13) and 14) to prove that n n! r!(n r)! = 2n r=0
24 DR. DANIEL FREEMAN 14) How many 3 digit numbers contain the digit 1? Justify your answer. 15) How many 3 digit numbers contain the digit 0? Justify your answer. 16) How many 3 digit numbers contain the digit 0 or the digit 1?
DISCRETE MATH: FINAL REVIEW 25 17) In a standard 52 card deck, there are 4 suits of 13 cards each. a. How many cards do you need to draw to be guaranteed of drawing at least 2 of the same suit? b. How many cards do you need to draw to be guaranteed of drawing at least 3 of the same suit? 18) A group of 15 workers are supervised by 5 managers. Each worker is assigned exactly one manager, and no manager supervises more than 4 workers. Show that at least 3 managers supervise 3 or more workers.
26 DR. DANIEL FREEMAN 19) How many ways are there to choose 5 people from a group of 14 to work as a team? 20) Suppose that there are 6 men and 8 women in the group. How many ways are there to choose a team consisting of 2 men and 3 women. 21) Suppose that there are 6 men and 8 women in the group. How many ways are there to choose a team consisting of 5 people such that at least one is a man.