Discrete Structures Homework 1

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1 Discrete Structures Homework 1 Due: June 15. Section Determine whether these biconditionals are true or false. a) = 4 if and only if = 2 b) = 2 if and only if = 4 c) = 3 if and only if monkeys can fly d) 0 > 1 if and only if 2 > 1 32 Construct a truth table for each of these compound propositions. (3 pt) a) p p c) p (p q) e) (q p) (p q) Section Show that each of these conditional statements is a tautology. You will get bonus points if you show it without a truth table. a) [ p (p q)] q b) [(p q) (q r)] (p r) c) [p (p q)] q d) [(p q) (p r) (q r)] r EC Find a compound proposition (a formula) using the variables p, q and r such that changing the truth value of one variable always also changes the truth value of the compound proposition. Section Let C(x) be the statement x has a cat, let D(x) be the statement x has a dog, and let F (x) be the statement x has a ferret. Express each of these statements in terms of C(x), D(x), F (x), quantifiers, and logical connectivities. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets there is a student in your class who has this animal as pet. 16 Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a) x : x 2 = 2 b) x : x 2 = 1 c) x : x d) x : x 2 x

2 Section Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) z y x T (x, y, z) b) x y P (x, y) x y Q(x, y) c) x y (Q(x, y) Q(y, x)) d) y x z (T (x, y, z) Q(x, y)) 40 Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. (3 pt) a) x y : x = 1/y b) x y : y 2 x < 100 c) x y : x 2 y 3 EC Which of the following statements is right, which is wrong? Explain your answer. a) x F (x) x G(x) x (F (x) G(x)) b) x F (x) x G(x) x (F (x) G(x)) c) x F (x) x G(x) x (F (x) G(x)) d) x F (x) x G(x) x (F (x) G(x)) 2

3 Discrete Structures Homework 2 Due: June 22. Section Use rules of inference to show that the hypotheses If it does not rain or it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on., If the sailing race is held, then the trophy will be awarded, and The trophy was not awarded imply the conclusion It rained. Please use the following variables: r - It rains. f - It is foggy. s - The sailing race will be held. l - The life saving demonstration will go on. t - The trophy will be awarded. 28 Use rules of inference to show that if x ( P (x) Q(x) ) and x ( ( P (x) Q(x)) R(x) ) are true, then x ( R(x) P (x) ) is also true, where the domains of all quantifiers are the same. (3 pt) Section Use a direct proof to show that the product of two odd numbers is odd. 26 Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even. 32 Show that these statements about the real number x are equivalent: (i) x is rational (ii) x/2 is rational (iii) 3x 1 is rational. Section Show that the product of two of the numbers , , and is nonnegative. Is your proof constructive or nonconstructive? Hint. Ignore the value of the given numbers and don t try to find out if they are positive or negative. 20 Prove that given real number x there exist unique numbers n and ϵ such than x = n + ϵ, n is an integer, and 0 ϵ < 1 EC Prove the following statement: If a line L does not intersect a diagonal of a convex polygon P then L can intersect only one of the two subpolygons defined by that diagonal.

4 Discrete Structures Homework 3 Due: June 29. Section Determine whether these statements are true or false. (7 pt) a) { } b) {, { }} c) { } { } d) { } {{ }} e) { } {, { }} f) {{ }} {, { }} g) {{ }} {{ }, { }} 18 Find two sets A and B such that A B and A B. 24 Determine whether each of these sets is the power set of a set, where a and b are distinct elements. a) b) {, {a}} c) {, {a}, {, a}} d) {, {a}, {b}, {a, b}} Section Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find a) A B b) A B c) A B d) B A 18 Let A, B, and C be sets. Show that (3 pt) a) (A B) (A B C). c) (A B) C A C. e) (B A) (C A) = (B C) A. 50 Find i=1 A i and i=1 A i if for every positive integer i a) A i = {i, i + 1, i + 2,...} b) A i = {0, i} c) A i = (0, i), that is, the set of real numbers x with 0 < x < i. d) A i = (i, ), that is, the set of real numbers x with x > i. EC Show that (A 1 A 2 ) (B 1 B 2 ) = (A 1 B 1 ) (A 2 B 2 ).

5 Discrete Structures Homework 4 Due: July 6. Section Determine whether each of these functions from Z to Z is one-to-one. a) f(n) = n 1 b) f(n) = n c) f(n) = n 3 d) f(n) = n 2 14 Determine whether f : Z Z Z is onto if (3 pt) a) f(m, n) = 2m n c) f(m, n) = m + n + 1 e) f(m, n) = m Determine whether each of these functions is a bijection from R to R. a) f(x) = 3x + 4 b) f(x) = 3x c) f(x) = x+1 x+2 d) f(x) = x EC Given a function f : X Y. Define a set F which represents f. Note. To represent the image of an element x X, you can use f(x). Hint. If you draw the function in a coordinate system, how would you define the set of points drawn? Section 2.4 (3 pt) 16 Find the solution to each of these recurrence relations with the given initial conditions. Use an iterative approach such as that used in Example 10. a) a n = a n 1, a 0 = 5 c) a n = a n 1 n, a 0 = 4 e) a n = (n + 1)a n 1, a 0 = 2 f) a n = 2na n 1, a 0 = 3 34 Compute each of these double sums. 3 2 a) (i j) b) c) d) i=1 j=1 3 i=0 j=0 3 i=1 j=0 2 2 (3i + 2j) 2 j i=0 j=0 3 i 2 j 3

6 Discrete Structures Homework 5 Due: July 13. Section Let P (n) be the statement that n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What is the statement P (1)? b) Show that P (1). c) What is the inductive hypothesis? d) What do you need to prove the inductive step e) Complete the inductive step, identifying where you use the inductive hypothesis. 6 Prove that 1 1! + 2 2! + + n n! = (n + 1)! 1 whenever n is a positive integer. 32 Prove that 3 divides n 3 + 2n whenever n is a positive integer. 40 Prove that if A 1, A 2,..., A n and B are sets, then (A 1 A 2 A n ) B = (A 1 B) (A 2 B) (A n B). Section Give a recursive definition of S m (n) = m + n, the sum of the integer m and the nonnegative integer n. 26 Give a recursive definition of (6 pt) a) the set of odd positive integers. b) the set of positive integer powers of 3. c) the set of polynomials with integer coefficients. For example: 5x 3 2x or 7x 4 8x 3 + x EC Give a recursive definition of the rational numbers Q. Use as Basis Step only 0 Q and 1 Q.

7 Discrete Structures Homework 6 Due: July 20. Section A particular brand of shirts comes in 12 colours, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made. 56 The name of a variable in the C programming language is a string that contains uppercase letters, lowercase letters, digits, or underscores. Further, the first character in the sting must be a letter, either uppercase or lowercase, or an underscore. If the name of a variable is determined by its first eight characters, how many different variables can be named in C? (Note that the name of a variable my contain fewer than eight characters.) Section A bowl contains 10 red and 10 blue balls. A woman selects balls at random without locking at them. a) How many balls must she select to be sure of having at least three balls of the same color? b) How many balls must she select to be sure of having at least three blue balls? 34 Assuming that no one has more than 1,000,000 hairs on the head of any person an that the population of New York City was 8,008,278 in 2010, show there had to be at least nine people in New York City in 2010 with the same number of Hairs on their heads. Section A coin is flipped eight times where each flip comes up either heads or tails. How many possible outcomes a) are there in total? b) contain exactly 3 heads? c) contain at least 3 heads? d) contain the same number of heads and tails? 22 How many Permutations of the letters ABCDEF GH contain (6 pt) a) the string ED? b) the string CDE? c) the strings BA and F GH? d) the string AB, DE, and GH? e) the string CAB and BED? f) the string BCA and ABF?

8 Section What is the coefficient of x 8 y 9 in the expansion of (3x + 2y) 17? 12 The row of Pascal s triangle containing the binomial coefficients ( 10 k ), 0 k 10, is: Use Pascal s identity to produce the row immediately following this row in Pascal s triangle. 2

9 Discrete Structures Homework 7 Due: July 27. Section What is the probability that a card selected at random from a standard deck of 52 cards is an ace or a heart? 16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? 20 What is the probability that a five-card poker hand contains a royal flush, this is, the 10, jack queen king, and ace of one suit? 24 Find the probability of winning the lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding a) 30. b) 36. c) 42. d) What is the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth price, respectively, in a drawing if 50 people enter a contest and a) no one can win more than one price. b) winning more than one price is allowed. Section What is the probability of these events when we randomly select a permutation of {1, 2, 3} (3 pt) a) 1 precedes 3. b) 3 precedes 1. c) 3 precedes 1 and 3 precedes What is the probability of these events when we randomly select a permutation of the 26 lowercase letters of the English alphabet? (6 pt) a) The first 13 of the permutation letters are in alphabetical order. b) a is the first letter of the permutation and z is the last letter. c) a and z are next to each other in the permutation. d) a and z are not next to each other in the permutation. e) a and z are separated by at least 23 letters in the permutation. f) z precedes both a and b in the permutation.

10 16 Show that if E and F are independent events then E and F are also independent 34 Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p. a) the probability of no successes b) the probability of at least one success c) the probability of at most one success d) the probability of at least two successes EC Assume, you chose the answer to this question randomly. How likely is it that your answer is correct. (3 pt) a) 50 % b) 25 % c) 33 % d) 25 % Hint. Do not take this question too serious ;-) 2