Answer Key For Discrete Mathematics Seventh Edition

Answer Key For Discrete Mathematics Seventh Edition Answer Key for Discrete Mathematics Seventh Edition This document serves as an answer key for the exercises found in the seventh edition of Discrete Mathematics by Kenneth H Rosen It provides solutions and explanations for a selection of problems from each chapter aiming to facilitate student learning and understanding of the concepts covered The answer key is structured according to the books chapter organization For each chapter a concise overview of the key concepts and definitions is provided followed by a breakdown of selected exercises with their solutions presented in a clear and detailed manner Please note This answer key is not intended to be a comprehensive solution manual It focuses on providing guidance and explanations for a representative selection of exercises encouraging students to engage in independent problemsolving and critical thinking Chapter 1 The Foundations Logic and Proofs 11 Propositional Logic This section introduces the basic building blocks of logic propositions truth values logical connectives and truth tables 12 Predicates and Quantifiers Here predicates quantifiers universal and existential and their logical relationships are explored 13 Methods of Proof Various proof techniques including direct proofs proofs by contradiction and proofs by mathematical induction are introduced and illustrated Selected Exercises and Solutions Exercise 111 Question Construct a truth table for the proposition p q p q Solution p q p q p q p q p q p q T T F F T F F T F F T F T T F T T F T F F 2 F F T T T F F Exercise 125 Question Express the following statement using predicates and quantifiers Every computer science student has taken a course in discrete mathematics Solution Let Cx represent x is a computer science student and Dx represent x has taken a course in discrete mathematics The statement can be expressed as xCx Dx Exercise 133 Question Prove by contradiction that if n is an integer and n is even then n is even Solution Assume for the sake of contradiction that n is odd Then n can be written as n 2k 1 for some integer k Squaring both sides we get n 2k 1 4k 4k 1 22k 2k 1 This shows that n is odd contradicting our initial assumption that n is even Therefore our assumption that n is odd must be false meaning n must be even Chapter 2 Basic Structures Sets Functions Sequences and Sums 21 Sets This chapter introduces the concept of sets various set operations and basic set properties 22 Functions Properties of functions including injectivity surjectivity and bijectivity are explored in detail 23 Sequences and Summations Sequences summation notation and various summation properties are discussed Selected Exercises and Solutions Exercise 217 Question Let A a b c B x y and C 1 2 Find A B C Solution A B C a x 1 a x 2 a y 1 a y 2 b x 1 b x 2 b y 1 b y 2 c x 1 c x 2 c y 1 c y 2 Exercise 229 Question Determine whether the function f R R defined by fx x is injective surjective or bijective Solution The function fx x is injective because for any two distinct real numbers x and x fx fx It is also surjective because for any real number y there exists a real number x such that fx y namely x y Since f is both injective and surjective it is bijective 3 Exercise 235 Question Evaluate the sum i110 i 3 Solution i110 i 3 1 3 2 3 3 3 10 3 1 4 9 100 10 3 385 30 415 Chapter 3 Counting 31 The Basics of Counting Introduces fundamental counting principles including the sum rule product rule and the pigeonhole principle 32 Permutations and Combinations Explores the concepts of permutations and combinations and their applications in counting arrangements and selections 33 Binomial Coefficients and Identities Introduces binomial coefficients Pascals Identity and other important binomial identities Selected Exercises and Solutions Exercise 313 Question A restaurant offers 5 appetizers 10 main courses and 4 desserts How many different meals can be ordered if a meal consists of one appetizer one main course and one dessert Solution By the product rule there are 5 10 4 200 different meals possible Exercise 321 Question How many ways can we arrange the letters in the word APPLE Solution There are 5 letters with the letter P repeating twice Therefore the number of arrangements is 5 2 60 Exercise 337 Question Use Pascals Identity to prove that for any positive integer n k0n n choose k 2n Solution Base case For n 1 1 choose 0 1 choose 1 1 1 2 2 holds Inductive step Assume that for some positive integer k j0k k choose j 2k We need to show that j0k1 k1 choose j 2k1 Using Pascals Identity j0k1 k1 choose j k1 choose 0 j1k k1 choose j k1 choose k1 1 j1k k choose j1 k choose j 1 1 j0k1 k choose j j1k k choose j 1 2 j0k k choose j 2 2k by the inductive hypothesis 2k1 By mathematical induction the formula holds for all positive integers n 4 Chapter 4 Induction and Recursion 41 Mathematical Induction Introduces the principle of mathematical induction and its applications in proving statements about integers 42 Recursive Definitions and Structural Induction Covers recursive definitions and structural induction providing tools for defining and proving properties of recursively defined structures 43 Recursive Algorithms Explores the design and analysis of recursive algorithms illustrating their effectiveness in solving various problems Selected Exercises and Solutions Exercise 413 Question Prove that for all positive integers n 1 2 3 n nn12 Solution Base case For n 1 the equation holds 1 1112 Inductive step Assume that for some positive integer k 1 2 3 k kk12 We need to show that 1 2 3 k1 k1k22 Using the inductive hypothesis 1 2 3 k1 1 2 3 k k1 kk12 k1 k1k22 By mathematical induction the formula holds for all positive integers n Exercise 425 Question Define the Fibonacci sequence recursively Solution The Fibonacci sequence Fn is defined as F0 0 F1 1 Fn Fn1 Fn2 for n 2 Exercise 433 Question Write a recursive algorithm to compute the greatest common divisor GCD of two positive integers Solution function gcda b if b 0 return a else 5 return gcdb a mod b Conclusion This answer key provides a foundation for understanding the concepts and problemsolving techniques presented in Discrete Mathematics Seventh Edition By working through the provided solutions and engaging in independent problemsolving students can gain a deeper understanding of the essential concepts in this field Remember this is just a starting point The true learning comes from actively engaging with the material exploring different approaches and seeking clarification when necessary Embrace the challenge and let the world of discrete mathematics open up to you