Discrete Mathematics Exam File Fall Exam #1

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1 Discrete Mathematics Exam File Fall 2015 Exam #1 1.) Which of the following quantified predicate statements are true? Justify your answers. a.) n Z, k Z, n + k = 0 b.) n Z, k Z, n + k = 0 2.) Prove that the difference of two even numbers is even. 3.) Construct a truth table for: (p q) q 4.) Consider the following statement: If it walks like a duck and it talks like a duck, then it is a duck. Let p: It walks like a duck. q: It talks like a duck. r: It is a duck. Write the statement symbolically. 5.) Evaluate each of the following. a.) b.) 17 mod 6 c.) -15 mod 8 d.) 5 (n 2 + 3) n=1 49 n=1 e.) n 6.) Construct a truth table to verify that (p q) ( p q). 7.) Find a general form for the sequence whose first five terms are: 4, 9, 16, 25, 36 8.) Prove that the square of an odd number is odd. Exam #2 1.) Use induction to prove the following. n i 2 = i=1 n (n + 1)(2 n + 1) 6 2.) Suppose our universe set is U = {a, b, c, d, e, f, g, h}, A = {a, c, e, g, h} and B = {a, d, f, h}. Find the following. a.) A B b.) A B' c.) A - B d.) A B e.) (A B)' 3.) Use Venn Diagrams to demonstrate that (A B)' = A' B'. 4.) Prove that (A B)' = A' B'. Do not use a Venn Diagram. 5.) Each number in the following Venn Diagram demonstrates how many elements of the universe set fall into the particular region shown. Using these numbers, verify that the inclusion-exclusion principle holds for A B C.

2 6.) You have a basket with apples in it. There are 23 green apples, 14 red apples and 8 plaid apples. You are going to take some apples out of the basket. a.) How many apples must you take out of the basket to be absolutely certain you have removed a matching pair of apples? b.) How many apples must you take out of the basket to be absolutely certain you have removed a matching pair of plaid apples? c.) How many apples must you take out of the basket to be absolutely certain you have removed four apples of the same color? 7.) Prove the following: If A B, then B' A'. Do not use a Venn Diagram. 8.) Use the pigeonhole principle to show that given any five positive integers, there must there be two whose difference is divisible by 4. 9.) Prove that for all integers a, b and c, if a divides b and b divides c, then a divides c. 10.) Use a proof by contradiction or contrapositive to show the following: Suppose n is a positive integer. If n 2 is even then n is even. Exam #2 1.) Use induction to prove the following. n i 3 i=1 = n2 (n + 1) ) Suppose our universe set is U = {a, b, c, d, e, f, g, h}, A = {a, c, g, h}, B = {d, f} and C = {b, e, f}. Find the following. a.) Do A, B and C form a partition of U? Explain your answer. b.) A B c.) C - B d.) A B' e.) (A C)'

3 3.) Use Venn Diagrams to prove or disprove that (A B)' = A' B'. Be sure to clearly state your conclusion. 4.) Prove that (A B)' = A' B'. Do not use a Venn Diagram. 5.) Suppose A B = 12, A B = 4, and A = 10. How many elements are in B? SHOW YOUR WORK!! 6.) You have a basket with apples in it. There are 12 green apples, six red apples and 11 plaid apples. You are going to take some apples out of the basket. a.) How many apples must you take out of the basket to be absolutely certain you have removed a matching pair of apples? b.) How many apples must you take out of the basket to be absolutely certain c.) you have removed a pair of plaid apples? How many apples must you take out of the basket to be absolutely certain you have removed seven apples of the same color? 7.) Prove the following: A (A B). Do not use a Venn Diagram. 8.) Use the pigeonhole principle to show that given any seven positive integers, there must there be three whose sum is divisible by 3. 9.) Prove that for all integers a, b and c, if a divides b and a divides c, then a divides b+c. 10.) Use a proof by contradiction or contrapositive to show the following: Suppose n is a positive integer. If n 2 is odd then n is odd. Exam #3 1.) Consider the following functions. f(x) = 3x - 4 g(x) = x 2 4 Do the following. a.) Find (f + g)(3). b.) Find the domain of g. c.) Determine whether or not f is 1-1. d.) If f is 1-1, find f -1. If not, write "Ha-Ha, I don't have to do this." e.) Find f f (5). f.) x 2 If the codomain for f is R, determine (with explanation) whether it is onto. If it is not, find the range. g.) Find f(x 2 + 1). h.) Find g(bob). i.) Determine whether or not g is ) Let A = R. Define the relation, R, on A by (a, b) R if and only if a * b > 0. a.) Is R reflexive? Explain. b.) Is R symmetric? Explain. c.) d.) e.) Is R transitive? Explain. Is R an equivalence relation? Explain your response. If R is an equivalence relation, list the equivalence classes. If it is not, write "Ha-Ha, I don't have to do this." 3.) Let A =Z. Define the relation, R, on A by (a, b) R if and only if a * b is even. a.) Is R reflexive? Explain. b.) Is R symmetric? Explain.

4 c.) d.) e.) Is R transitive? Explain. Is R an equivalence relation? Explain your response. If R is an equivalence relation, list the equivalence classes. If it is not, write "Ha-Ha, I don't have to do this." 4.) How many positive integers less than 1000 are multiples of 2, 3, or 5? 5.) What is the largest power of 5 that divides 1002!? 6.) The members of the Clark County Underwater Basket Weaving Club are as follows.8 Mr. & Mrs. Thomrangwithawatpong Mr. & Mrs. Ng Mr. & Mrs. Saltalamacchia Mr. & Mrs. Humperdink Mr. & Mrs. Holmes Mr. Watson All eleven members are present at the November meeting. a.) The members get in line for the lunch buffet. How many different lines are possible? b.) How many different lines are possible if each couple stands in line together? c.) How many different lines are possible if each couple stands in line together and Mr. Watson stands with Mr. & Mrs. Holmes (but not between them)? d.) For lunch they sit at a round table. In how many orders can they sit? e.) At the round table, in how many orders can they sit if couples must sit together? f.) During the meeting they put together a committee for the planning of their Lake Degray underwater basket weaving excursion. How many five person committees can be chosen? g.) How many five person committees can be chosen if there is a president, vice-president, secretary, treasurer and sergeant-at-arms? h.) How many five person committees can be chosen if at least three members must be female? i.) How many five person committees can be chosen if at least three members must be female and if there is a president, vice-president, secretary, treasurer and sergeant-at-arms? 7.) a.) How many different ways can three non-negative integers, x, y and z, add b.) up to 28? How many different ways can three positive integers, x, y and z, add up to 28? c.) How many different ways can three integers, x, y and z, add up to 28 if x > 0, y > 0 and z > 2? 8.) Use the binomial theorem to expand (2x - 3y) 5. Show all your work. 9.) Consider (x + 2y 2 ) 8. a.) Find the coefficient of the x 4 y 4 term. b.) Find the term (not just the coefficient) of the term with y 6 in it. 10.) Find the largest power of 12 that divides 1002!. 11.) In how many ways can you break 16 different people into four groups of 4 people, assuming the groups are distinct? 12.) Thirty runners run in a race. Trophies will be given to the top 8 runners (the trophies are distinct: first place, second place, etc.). How many ways can this be done?

13.) Consider the set S = {a, b, c, d, e, f, g}. a.) How many "words" of no more than five letters can be obtained from the letters in S? b.) How many five letter "words" can be obtained from S if no letter may be repeated?

5 13.) Consider the set S = {a, b, c, d, e, f, g}. a.) How many "words" of no more than five letters can be obtained from the letters in S? b.) How many five letter "words" can be obtained from S if no letter may be repeated? c.) How many five letter "words" can be obtained from S if at most one letter can appear twice? 14.) How many distinguishable lines can be obtained by arranging five identical red marbles, four identical green marbles and six identical plaid marbles in a line? Exam #4 1.) How many bags of 20 pieces of candy can one buy from a store that sells four types of candy? 2.) How many distinguishable arrangements of the letters in the word MISSISSIPPI are there? 3.) How many distinguishable four letter words can be made from the letters in the word PEOPLE? 4.) Show a solution to the following Tower of Hanoi puzzle. 5.) You flip a coin 14 times. What is the probability that you get heads exactly nine times? Show all necessary work and leave your answer as a decimal. 6.) Team A has a probability of 3/5 of winning any given game against Team B. Suppose the teams are playing a best of five tournament. a.) What is the probability Team A wins in three games? b.) What is the probability Team A wins in four games? c.) What is the probability Team A wins in five games? d.) What is the probability Team B wins the tournament? 7.) You roll a fair six-sided die four times. E1 is the event that the first two rolls sum to 7. E2 is the event that the last two rolls sum to 10. What is the probability of E1 or E2 happening? 8.) A game is played by pulling one of the following pieces of paper from a hat. The player wins the amount on the piece of paper. What is the expected value of the player's win? Show your work. $1 $1 $3 $3 $4 $4 $4 $5 $5 $6 $8 $10 9.) For each degree sequence below, in row A, draw a tree with that degree sequence and leave rows B and C blank. If no such tree exists, tell why no such tree exists.

6 A B C A B C If no such tree exists, in row B draw a simple graph with that degree sequence and leave row C blank. If no such simple graph exists, tell why no such simple graph exists. If no such simple graph exists, in row C draw a graph with that degree sequence. If no such graph exists, tell why no such graph exists. You should have no more than one graph for each degree sequence 1, 1, 1, 1, 2, 2, 3, 3 4, 4, 4, 4, 4 0, 0, 0, 1, 2, 3, 4, 5, 6, 7 0, 0, 1, 2, 3 Final Exam 1. Consider a group of 9 men and 13 women. a. How many five person committees can be chosen? b. How many five person committees can be chosen if the committee is to have at least four women? c. How many five person committees can be chosen if the committee has a president, vice-president, secretary and treasurer? 2. Consider the letters in the word MISSISSIPPI. a. How many distinguishable arrangements of the letters are there (using ALL the letters)? b. How many distinguishable arrangements are there if all letters are used and the Ps end up adjacent to each other? 3. In how many ways can four non-negative integers add up to 22? 4. In how many ways can 27 pennies be distributed among 5 children so that each child gets at least two pennies? 5. Construct a truth table for the following. ( p) (p q) 6. Use a Venn diagram to represent the set A B C. 7. Find a counterexample to show that the following statement is false. Clearly illustrate that it is a counter example. A ( B C) ( A B) C 8. Using an element-wise proof, prove either direction of the proof that (A B) C = (A C) (B C). You must use an element-wise proof for credit. 9. Prove 2 is irrational. 10. Complete exactly two of the following proofs. If you do more than two proofs, only the first two will be graded. a. If n is odd, then n 3 n is divisible by 4. b. If 3n is odd, then n is odd.

7 c. If n 2 is odd, then n is odd. 11. Prove the following statement is true for all integers n 1, using mathematical induction n 2 = n (n + 1)(2n + 1) Let R be a relation on R (the set of real numbers) given by the rule (x, y) R iff x - y is divisible by 3. Prove that R is an equivalence relation by proving it is reflexive, symmetric and transitive. 13. Use the Binomial Theorem for each of the following. a. Expand (x - 2y) 4. b. Determine the coefficient of x 4 y 7 in the expansion of (2x y) In how many ways can eight people be seated around a circular table? 15. Let our "universe" be U={a, b, c, d, e, f, g, h, i, j, k}. Let A={a, b, c, d}, B={a, e, i}, and C={a, b, e, i}. Evaluate each of the following. a. Evaluate (A C) B. b. Evaluate A B C. c. Evaluate (B C) - A. 16. Suppose you have a class of 23 students. a. Explain why it must be true that at least two students must share the same birth month. b. How many more students must be present to be guaranteed that three share the same birth month? c. How many more students must be present to be guaranteed that seven share the same birth month? 17. Consider the sequences that follow. For each, if it is possible, draw a tree with that degree sequence. If that is not possible, tell why and draw a simple graph with that degree sequence. If that is not possible, tell why and draw a graph with that degree sequence. If that is not possible, tell why. a. 1, 1, 1, 2, 3 b. 3, 4, 4, 4, 4, 5 c. 1, 2, 2, 3, 4, 4, 5

Discrete Structures. Fall Homework3

Discrete Structures. Fall Homework3 Discrete Structures Fall 2015 Homework3 Chapter 5 1. Section 5.1 page 329 Problems: 3,5,7,9,11,15 3. Let P(n) be the statement that 1 2 + 2 2 + +n 2 = n(n + 1)(2n + 1)/6 for the positive integer n. a)