Solutions for Discrete Mathematics With Graph Theory With Discrete Math Workbook: Interactive Exercises (3rd Edition)
Problem 1TFQ:
If you want to prove a statement is true, it is enough to find 867 examples where it is true.Problem 2TFQ:
True/False Questions
2. If you want to prove a statement is false, it is enough to find one example...Problem 6TFQ:
The contrapositive of A Bis B A.Problem 7TFQ:
A Bis true if and only if its contrapositive is true.Problem 8TFQ:
True/False Questions
8. is a rational number.
Problem 9TFQ:
True/False Questions
9. 3.141 is a rational number.
Problem 10TFQ:
True/False Questions
10. If and are irrational numbers, then must be an irrational number.
Problem 11TFQ:
True/False Questions
11. The statement “Every real number is rational” can be proved false with a...Problem 12TFQ:
The statement There exists an irrational number that is not the square root of an integer can be...Problem 1E:
What is the hypothesis and what is the conclusion in each of the following implications?
The sum of...Problem 2E:
2. In each part of Exercise 1, what condition is necessary for what? What condition is sufficient...Problem 3E:
Exhibit a counterexample to each of the following statements. x2=4x=2. a and b integers and...Problem 4E:
Consider the following two statements: A: The square of every real number is positive. B: There...Problem 5E:
Determine whether the following implication is true. x is an even integer x+2 is an even integer.Problem 7E:
7. Answer Exercise 5 with replaced by .
Problem 8E:
Consider the statement A: If n is an integer, nn+1 is not an integer. Is A true or false? Either...Problem 9E:
9. Let be an integer greater than 1 and consider the statement “A: prime is necessary for to be...Problem 10E:
10. A theorem in calculus states that every differentiable function is continuous. State the...Problem 11E:
11. Let be an integer, . A certain mathematical theorem asserts that statements are equivalent.
(a)...Problem 12E:
Consider the assertions A: For every real number x, there exists an integer n such that nxn+1. B:...Problem 13E:
Answer Exercise 12 with A and B as follows. A: There exists a real number y such that yx for every...Problem 14E:
14. Answer true or false and supply a direct proof or a counterexample to each of the following...Problem 15E:
Prove that n an even integer n2+3n is an even integer. What is the converse of the implication in...Problem 16E:
16. (a) Let be an integer. Show that either or is even.
(b) Show that is even for any integer.
Problem 18E:
Prove that 2x24x+30 for any real number x.Problem 19E:
19. Let and be integers. By examining the four cases
i. both even,
ii. both odd,
iii. even, odd,...Problem 22E:
Prove that if n is an odd integer then there is an integer m such that n=4m+1 or n=4m+3. [Hint:...Problem 23E:
23. Prove that if is an odd integer, there is an integer such that or or or. (You may use the...Problem 24E:
24. Prove that there exists no smallest positive real number. [Hint: Find a proof by...Problem 26E:
26. (For students who have studied linear algebra) Suppose 0 is an eigenvalue of a matrix A. Prove...Problem 27E:
27. (a) Suppose and are integers such that . Prove that and .
(b) Prove that is not the square of a...Problem 28E:
Suppose a and b are integers such that a+b+ab=0. Prove that a=b=0 or a=b=2. Give a direct proof.Problem 30E:
30. Suppose that is a rational number and that is an irrational number. Prove that is irrational.
Browse All Chapters of This Textbook
Chapter 0 - Yes, There Are Proofs!Chapter 0.1 - Compund StatementsChapter 0.2 - Proofs In MathematicsChapter 1 - LogicChapter 1.1 - Truth TablesChapter 1.2 - The Algebra Of PropositionsChapter 1.3 - Logical ArgumentsChapter 2 - Sets And RelationsChapter 2.1 - SetsChapter 2.2 - Operations On Sets
Chapter 2.3 - Binary RelationsChapter 2.4 - Equivalence RelationsChapter 2.5 - Partial OrdersChapter 3 - FunctionsChapter 3.1 - Basic TerminologyChapter 3.2 - Inverses And CompositionChapter 3.3 - One-to-one Correspondence And The Cardinality Of A SetChapter 4 - The IntegersChapter 4.1 - The Division AlgorithmChapter 4.2 - Divisibility And The Euclidean AlgorithmChapter 4.3 - Prime NumbersChapter 4.4 - CongruenceChapter 4.5 - Applications Of CongruenceChapter 5 - Induction And RecursionChapter 5.1 - Mathematical InductionChapter 5.2 - Recursively Defined SequencesChapter 5.3 - Solving Recurrence Relations; The Characteristic PolynomialChapter 5.4 - Solving Recurrence Relations; Generating FunctionsChapter 6 - Principles Of CountingChapter 6.1 - The Principle Of Inclusion-exclusionChapter 6.2 - The Addition And Multiplication RulesChapter 6.3 - The Pigeonhole PrincipleChapter 7 - Permutations And CombinationsChapter 7.1 - PermutationsChapter 7.2 - CombinationsChapter 7.3 - Elementary ProbabilityChapter 7.4 - Probability TheoryChapter 7.5 - RepetitionsChapter 7.6 - DerangementsChapter 7.7 - The Binomial TheoremChapter 8 - AlgorithmsChapter 8.1 - What Is An Algorithm?Chapter 8.2 - ComplexityChapter 8.3 - Searching And SortingChapter 8.4 - Enumeration Of Permutations And CombinationsChapter 9 - GraphsChapter 9.1 - A Gentle IntroductionChapter 9.2 - Definitions And Basic PropertiesChapter 9.3 - IsomorphismChapter 10 - Paths And CircuitsChapter 10.1 - Eulerian CircuitsChapter 10.2 - Hamiltonian CyclesChapter 10.3 - The Adjacency MatrixChapter 10.4 - Shortest Path AlgorithmsChapter 11 - Applications Of Paths And CircuitsChapter 11.1 - The Chinese Postman ProblemChapter 11.2 - DigraphsChapter 11.3 - Rna ChainsChapter 11.4 - TournamentsChapter 11.5 - Scheduling ProblemsChapter 12 - TreesChapter 12.1 - Trees And Their PropertiesChapter 12.2 - Spanning TreesChapter 12.3 - Minimum Spanning Tree AlgorithmsChapter 12.4 - Acyclic Digraphs And Bellman's AlgorithmChapter 12.5 - Depth-first SearchChapter 12.6 - The One-way Street ProblemChapter 13 - Planar Graphs And ColoringsChapter 13.1 - Planar GraphsChapter 13.2 - Coloring GraphsChapter 13.3 - Circuit Testing And Facilities DesignChapter 14 - The Max Flow - Min Cut TheoremChapter 14.1 - Flows And CutsChapter 14.2 - Constructing Maximal FlowsChapter 14.3 - ApplicationsChapter 14.4 - Matchings
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