3rd Edition

ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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Textbook Question

Chapter 0.1, Problem 8TFQ

The double implication “ 2 2 = 5 if and only if 3 2 = 9 ” is true.

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Chapter 0.1, Problem 7TFQ

Chapter 0.1, Problem 9TFQ

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7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)

Chapter 0 Solutions

Discrete Mathematics with Graph Theory

Ch. 0.1 - pandq is false if pandq are false.Ch. 0.1 - If pandq is false, then both pandq are false.Ch. 0.1 - True/False Questions 3. It is possible for both “”...Ch. 0.1 - True/False Questions 4. It is possible for both “”...Ch. 0.1 - The implication If 22=5, then 32=9 is true.Ch. 0.1 - The negation of a=b=0 is ab0.Ch. 0.1 - The converse of the implication in Question 5 is...Ch. 0.1 - True/False Questions 8. The double implication “...Ch. 0.1 - It is possible for both an implication and its...Ch. 0.1 - The statement Some frogs have red toes makes use...

Ch. 0.1 - The negation of an existential quantifier is its...Ch. 0.1 - Classify each of the following statements as...Ch. 0.1 - Classify each of the following statements as...Ch. 0.1 - 3. Rewrite each of the following statements so...Ch. 0.1 - 4. Determine whether each of the following...Ch. 0.1 - Write down the negation of each of the following...Ch. 0.1 - 6. Write down the converse and contrapositive of...Ch. 0.1 - Rewrite each of the following statements using the...Ch. 0.1 - Is it possible for both an implication and its...Ch. 0.1 - On page 4 of the text, we stated as more or less...Ch. 0.2 - If you want to prove a statement is true, it is...Ch. 0.2 - True/False Questions 2. If you want to prove a...Ch. 0.2 - The sentence A is a sufficient condition for Bis...Ch. 0.2 - True/False Questions 4. If A B, BC, CD, and CA...Ch. 0.2 - True/False Questions 5. If A B, BC, CD, and CA...Ch. 0.2 - The contrapositive of A Bis B A.Ch. 0.2 - A Bis true if and only if its contrapositive is...Ch. 0.2 - True/False Questions 8. is a rational number. Ch. 0.2 - True/False Questions 9. 3.141 is a rational...Ch. 0.2 - True/False Questions 10. If and are irrational...Ch. 0.2 - True/False Questions 11. The statement “Every...Ch. 0.2 - The statement There exists an irrational number...Ch. 0.2 - What is the hypothesis and what is the conclusion...Ch. 0.2 - 2. In each part of Exercise 1, what condition is...Ch. 0.2 - Exhibit a counterexample to each of the following...Ch. 0.2 - Consider the following two statements: A: The...Ch. 0.2 - Determine whether the following implication is...Ch. 0.2 - State the converse of the implication in Exercise...Ch. 0.2 - 7. Answer Exercise 5 with replaced by . Ch. 0.2 - Consider the statement A: If n is an integer, nn+1...Ch. 0.2 - 9. Let be an integer greater than 1 and consider...Ch. 0.2 - 10. A theorem in calculus states that every...Ch. 0.2 - 11. Let be an integer, . A certain mathematical...Ch. 0.2 - Consider the assertions A: For every real number...Ch. 0.2 - Answer Exercise 12 with A and B as follows. A:...Ch. 0.2 - 14. Answer true or false and supply a direct proof...Ch. 0.2 - Prove that n an even integer n2+3n is an even...Ch. 0.2 - 16. (a) Let be an integer. Show that either or...Ch. 0.2 - 17. Provide a direct proof that is odd for all...Ch. 0.2 - Prove that 2x24x+30 for any real number x.Ch. 0.2 - 19. Let and be integers. By examining the four...Ch. 0.2 - Let n be an integer. Prove that n2 is even if and...Ch. 0.2 - Prob. 21E Ch. 0.2 - Prove that if n is an odd integer then there is an...Ch. 0.2 - 23. Prove that if is an odd integer, there is an...Ch. 0.2 - 24. Prove that there exists no smallest positive...Ch. 0.2 - 25. Let be the product of positive integers and ....Ch. 0.2 - 26. (For students who have studied linear algebra)...Ch. 0.2 - 27. (a) Suppose and are integers such that . Prove...Ch. 0.2 - Suppose a and b are integers such that a+b+ab=0....Ch. 0.2 - Suppose a is an irrational number. Prove that 1a...Ch. 0.2 - 30. Suppose that is a rational number and that is...Ch. 0.2 - Prob. 31E Ch. 0.2 - 32. Find a proof or exhibit a counterexample to...Ch. 0.2 - Prob. 33E Ch. 0.2 - Prob. 34E Ch. 0.2 - Prob. 35E Ch. 0.2 - Prob. 36E Ch. 0.2 - Prob. 37E Ch. 0.2 - Prove that there exist irrational numbers a and b...Ch. 0 - State, with a reason, whether each of the...Ch. 0 - Prob. 2RE Ch. 0 - 3. Write down the converse, the contrapositive and...Ch. 0 - Prob. 4RE Ch. 0 - Prob. 5RE Ch. 0 - Prob. 6RE Ch. 0 - Prob. 7RE Ch. 0 - Prob. 8RE Ch. 0 - 9. Let be an integer. Prove that is odd if and...Ch. 0 - Give a direct proof of the fact that a25a+6 is...Ch. 0 - Prob. 11RE Ch. 0 - Prob. 12RE Ch. 0 - 13. Prove, by way of contradiction, that if is a...Ch. 0 - Prob. 14RE Ch. 0 - Prob. 15RE Ch. 0 - Prob. 16RE Ch. 0 - Prob. 17RE Ch. 0 - Prob. 18RE Ch. 0 - Each of the integers 31, 331, 3331, 33331, 333331,...

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