Discrete Mathematics Exam File Spring Exam #1

Transcription

1 Discrete Mathematics Exam File Spring 2008 Exam #1 1.) Consider the sequence a n = 2n + 3. a.) Write out the first five terms of the sequence. b.) Determine a recursive formula for the sequence. 2.) Consider the sequence a 1 = 3, a n = a n-1 + 2n + 1, n > 1. a.) Write out the first five terms of the sequence. b.) Determine a closed form formula for the sequence. 3.) Consider the conditional p q. a.) Use a truth table to show whether or not the conditional and its contrapositive are logically equivalent. b.) Use a truth table to show whether or not the conditional and its converse are logically equivalent. 4.) Let Z denote the set of integers. 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 5.) For each of the statements in #4, write the negation of the statement. 6.) For each of the following, identify the reasoning as either valid (modus ponens), valid (modus tollens), invalid (converse fallacy) or invalid (inverse fallacy). a.) If Bob Gibson pitched the game, then the Cardinals won. The Cardinals did not win the game. Therefore, Bob Gibson did not pitch the game. b.) If today is Tuesday, then Bob is in Gurdon. Bob is in Gurdon. Therefore, today is Tuesday. c.) If calculus is a prerequisite for the course, then Mary will not take the course. Calculus is not a prerequisite for the course. Therefore, Mary took the course. d.) If the baseball team scores no runs then they will not win the game. The baseball team scored no runs. Therefore they did not win the game. e.) If today is raining then it is Saturday. Today is raining. Therefore it is Saturday. 7.) For each symbolic logic statement in the left column, identify the statement from the right column to which it is logically equivalent. p q A (p q) (p r ) p t B q p p (q r ) C q p (q p) D q p p q E q p p q F p 8.) There are three paths from Town A to Town B. Only one of the paths is free from land mines. The North Path has a sign on it that says "There are land mines here." The Central Path has a sign that says "There are mines on the north path." The South Path has a sign that says "The south path is clear." One and

2 only one of the signs is false. Which path is clear of land mines? Explain your answer. N NORTH PATH There Are Land Mines Here Town A CENTRAL PATH There Are Mines On The North Path SOUTH PATH The South Path Is Clear Town B 9.) Use a truth table to check if the following symbolic logic statements are logically equivalent; state your conclusion using complete sentence(s). p ( p q) and p ( p q) 10.) Use a truth table to determine if the following is a valid argument; depending on your approach, you may have to add columns. Give your conclusion using complete sentences. p r q ( ) r q p 11.) For D = { 3, 4, 7, 8, 9, 11, 13}, which of the following are true? Justify your answers. a.) x D, if x is odd then, x > 7. b.) x D, if x > 13, then x is even. 12.) Determine the truth value of [( q r) p] [ r ( s q)] if p, r, and s are all true and q is false. Justify your answer. 13.) Suppose that P and Q are logical statements made up of propositions p, q and r, along with some combinations of,, and. Suppose P Q. Prove that P Q is a tautology. Exam #2 1.) Compute the following: a.) 23 mod 5 b.) 6 mod 9 c.) -35 mod 8 d.) (8n 3-4n n + 9) mod 4 e.) mod 4 2.) Disprove the following statement by giving a counterexample. The difference of any two odd integers is odd. 3.) If 12 eggs are each to be dyed a single solid color and five colors are available, what can you say about the number of eggs of the same color?

3 4.) Prove that the sum of a rational number and an irrational number is irrational. 5.) n ( n + 1) Prove that, for every natural number, n, L + n =. 2 6.) Prove that any three integers contain a pair whose sum is even. 7.) Use the contrapositive to prove the following. Let n be a natural number. If n 2 is even, then n is even. 8.) Use proof by contradiction to show that, if n is an odd integer, then n 2 + n is even. 9.) Show that 3 is irrational. 10.) Suppose a, b and c are integers. Show that if a b and a c, then, for any integers, m and n, a (mb + nc). Exam #3 1.) Let U = {a, b, c, d, e, f, g, h, i, j}, A = {a, b, c, d}, B = {a, e, i}, C = {a, b, e, i}. [Note: U denotes the universe for this problem.] a.) Evaluate ( B C ) A. b.) Evaluate ( A B) ' C. c.) Evaluate (A C) B. 2.) Evaluate {1, 2} x {a, b, c}. 3.) Let A={a, 1, 2}. Find (A), that is, find the power set of A. 4.) Using appropriate set notation, describe the shaded region represented in the following Venn diagram. 5.) Without using Venn Diagrams, prove ( A B) ' = A ' B '. 6.) Give an element-wise proof that for sets A, B, C, ( A B) ( A C) A ( B C). 7.) Find a counterexample to the following statement A U ( B C) = ( A U B) C. 8.) As in the book, let n(s) denote the number of elements in a set S. Suppose n (A) = 23, n( A I B) = 5, and n( A U B) = 42. Find n(b). 9.) Given A = { 2k : k Z} and B = { 2k + 1: k Z}. Determine, with justification, if {A, B} is a partition of Z. 10.) Perform the following conversions. a.) ten to base 8 b.) 1AB301 hex to base 10 c.) two to hexadecimal d.) BEEF hex to binary e.) ten to base 7

4 Exam #4 1.) Given A = { 2k : k Z} and B = { 2k + 1: k Z}. Determine, with justification, if {A, B} is a partition of Z. 2.) In a small state, a license plate consists of two letters from {A, R, K, N, S} followed by 3 or 4 digits. (Repetitions of letters or digits are allowed.) How many license plates are possible? 3.) Consider the set B = {0,1,2,3,4,5,6}. a.) How many 5-digit numbers use distinct digits from B? b.) How many of these are odd? 4.) How many batting orders for a baseball team (9 players) are possible from a roster of 20 players? 5.) A club of 10 men and 8 women is forming a 5-person steering committee. Of these, how many are possible for each of these situations? a.) The committee contains exactly 3 men. b.) at least 3 men. c.) The committee contains at least one man. 6.) Consider the following definitions. A = {a, b, c, d, e} B = {1, 3, 4, 5, 6} C = {x, y, z, w} D = {Bob, Mary, Dave, Jane, Joseph, Martha} f is a relation defined by {(a, 1), (b, 4), (c, 3), (d, 5), (e, 4)} g is a relation defined by {(x, a), (y, b), (y, c), (z, b), (w, d)} h is a relation defined by {(Bob, a), (Mary, b), (Dave, b), (Jane, c), (Joseph, d), (Martha, e} Each relation is defined with one of A, B, C or D as its domain and one as its codomain. a.) Fill in the chart. f g h Is it a function? Yes or No b.) c.) What is its domain? What is its codomain? Is it one-to-one? Yes or No Is it onto? Yes or No Which composition(s) would be defined? For any of f, g or h that are invertible, give the set of ordered pairs which define the inverse function. If none are invertible, say so. 7.) Consider the set A = {1, 2, 3}. Draw the arrow diagram for the relation R = {(a, b} a, b (A), a b}. 8.) Suppose A is a finite set. Prove that R is an equivalence relation on (A) if, for a, b (A), (a, b) R if and only if n(a) = n(b). 9.) Use the Binomial Theorem to expand (2x + y) ) What is the coefficient of x 8 y 5 in the expansion of (2x - y) 13?

5 Final Exam 1.) 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.) (A C) B. b.) (A B) ' C. c.) (B C) - A. d.) Do A, B and C form a partition of U? Why or why not? 2.) Consider the conditional p q. a.) Use a truth table to show whether or not the conditional and its contrapositive are logically equivalent. b.) Use a truth table to show whether or not the conditional and its converse are logically equivalent. 3.) Prove that, for any integer n, the number n 2 + n is even. 4.) Perform the following conversions. a.) 1235 ten to base 8 b.) 1AB01 hex to base 10 c.) BAA0 hex to binary d.) 2345 ten to base 6 e.) Suppose we are doing work with various number bases. You are given the number but the base was accidentally erased. What can you say about the base? 5.) Compute the following: a.) 47 mod 5 b.) 6 mod 6 c.) -52 mod 7 d.) (8n 3-24n n + 9) mod 8 e.) mod 4 6.) Consider the following definitions. A = {a, b, c, d, e} B = {1, 3, 4, 5, 6} C = {x, y, z, w} D = {Bob, Mary, Dave, Joseph, Martha} f is a relation defined by {(x, a), (y, b), (z, b), (w, d)} g is a relation defined by {(Bob, a), (Mary, b), (Dave, c), (Joseph, d), (Martha, e} h is a relation defined by {(a, 1), (b, 4), (c, 3), (d, 5), (e, 4)} Each relation is defined with one of A, B, C or D as its domain and one as its codomain. a.) Fill in the chart. f g h Is it a function? Yes or No What is its domain? What is its codomain? Is it one-to-one? Yes or No Is it onto? Yes or No b.) Which composition(s) would be defined? c.) For any of f, g or h that are invertible, give the set of ordered pairs which define the inverse function. If none are invertible, say so. 7.) n ( n + 1) Use induction to prove that for every natural number, n, L + n =. 2

6 8.) Prove that for all integers n > 4, if n is a perfect square, then n - 1 is not a prime number. 9.) Use the contrapositive to prove the following. For all integers n, if n 2 is even, then n is even. 10.) Coins are randomly removed from a piggy bank containing 12 pennies, eight nickels, ten dimes, and three quarters. How many coins must be removed to guarantee: a.) three pennies? b.) three coins of the same kind? c.) three quarters? d.) a pair of pennies and a pair of dimes? 11.) Consider (S) (the power set of S) if S = {1, 2, 3}. Define a relation on (S) as follows. A relates to B if and only if A B Determine whether or not this defines an equivalence relation (S). 12.) Define an invertible function between A = {a, b, c, d, e} and B = {red, yellow, green, blue, brown}. Explain how this proves the two sets have the same number of elements. 13.) Prove that (A B) C (A C) (B C). 14.) Prove that ( A B) ' = A ' B '. 15.) Use a counterexample to disprove that the difference of the squares of any two odd numbers is odd. 16.) a.) Five children order one small soft drink each from a stand that offers seven kinds of soft drinks. How many outcomes are possible? b.) To save time, the chaperone decides to order some selection of five small soft drinks from the seven kinds available, without consulting the children or deciding who will get which drink. How many ways may the chaperone place the order?