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To prove: The equality 1 1 ⋅ 3 + 1 3 ⋅ 5 + ⋅ ⋅ ⋅ + 1 ( 2 n − 1 ) ( 2 n + 1 ) = n 2 n + 1 holds for all positive integers n .

To prove: The equality 1 1 ⋅ 3 + 1 3 ⋅ 5 + ⋅ ⋅ ⋅ + 1 ( 2 n − 1 ) ( 2 n + 1 ) = n 2 n + 1 holds for all positive integers n .

Discrete Mathematics

Discrete Mathematics

5th Edition

ISBN: 9780134689562

Author: Dossey, John A.

Publisher: Pearson,

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 2, Problem 64SE

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To prove: The equality 11⋅3+13⋅5+⋅⋅⋅+1(2n−1)(2n+1)=n2n+1 holds for all positive integers n.

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Chapter 2, Problem 63SE

Chapter 2, Problem 65SE

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Chapter 2 Solutions

Discrete Mathematics

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - In Exercises 5–8, compute A × B for each of the...Ch. 2.1 - In Exercises 5–8, compute A × B for each of the...Ch. 2.1 - In Exercises 5–8, compute A × B for each of the...Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Give an example of sets for which , but A ≠ B. Ch. 2.1 - Give an example of sets for which , but A ≠ B. Ch. 2.1 - Give an example of sets for which , but A ≠ B. Ch. 2.1 - Give an example of sets for which (A − B) − C ≠ A...Ch. 2.1 - Use Theorems 2.1 and 2.2 as in Example 2.4 to...Ch. 2.1 - Use Theorems 2.1 and 2.2 as in Example 2.4 to...Ch. 2.1 - Use Theorems 2.1 and 2.2 as in Example 2.4 to...Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Use Theorems 2.1 and 2.2 as in Example 2.4 to...Ch. 2.1 - If A is a set containing m elements and B is a set...Ch. 2.1 - Under what conditions is A − B = B − A? Ch. 2.1 - Under what conditions is A ⋃ B = A? Ch. 2.1 - Under what conditions is A ⋂ B = A? Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.1 - Prob. 37E Ch. 2.1 - Prove the set equalities in Exercises...Ch. 2.1 - Prob. 39E Ch. 2.1 - Prove that (A × C) ⋃ (B × D) ⊆ (A ⋃ B) × (C ⋃ D). Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - In Exercises 1-12, determine which of the...Ch. 2.2 - Prob. 6E Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - Prob. 8E Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - In Exercises 1–12, determine which of the...Ch. 2.2 - Prob. 12E Ch. 2.2 - In Exercises 13-18, show that the given relation R...Ch. 2.2 - In Exercises 13-18, show that the given relation R...Ch. 2.2 - In Exercises 13-18, show that the given relation R...Ch. 2.2 - In Exercises 13-18, show that the given relation R...Ch. 2.2 - Prob. 17E Ch. 2.2 - In Exercises 13–18, show that the given relation R...Ch. 2.2 - Prob. 19E Ch. 2.2 - Write the equivalence relation on {1, 2, 3, 4, 5,...Ch. 2.2 - Prob. 21E Ch. 2.2 - Prob. 22E Ch. 2.2 - Prob. 23E Ch. 2.2 - Let R1 and R2 be equivalence relations on sets S1...Ch. 2.2 - Determine the number of relations on a set S...Ch. 2.2 - Prob. 26E Ch. 2.2 - Prob. 27E Ch. 2.2 - How many partitions are there of a set containing...Ch. 2.2 - Prob. 29E Ch. 2.2 - Prob. 30E Ch. 2.2 - Prob. 31E Ch. 2.2 - Prob. 33E Ch. 2.3 - In Exercises 1–8, determine whether the given...Ch. 2.3 - Prob. 2E Ch. 2.3 - Prob. 3E Ch. 2.3 - Prob. 4E Ch. 2.3 - Prob. 5E Ch. 2.3 - Prob. 6E Ch. 2.3 - Prob. 7E Ch. 2.3 - Prob. 8E Ch. 2.3 - Prob. 9E Ch. 2.3 - Prob. 10E Ch. 2.3 - Prob. 11E Ch. 2.3 - Prob. 12E Ch. 2.3 - Prob. 13E Ch. 2.3 - Prob. 14E Ch. 2.3 - Prob. 15E Ch. 2.3 - Prob. 16E Ch. 2.3 - Prob. 17E Ch. 2.3 - Prob. 18E Ch. 2.3 - Prob. 19E Ch. 2.3 - Prob. 20E Ch. 2.3 - Prob. 21E Ch. 2.3 - Prob. 22E Ch. 2.3 - Prob. 23E Ch. 2.3 - Prob. 24E Ch. 2.3 - Prob. 25E Ch. 2.3 - Prob. 26E Ch. 2.3 - Prob. 27E Ch. 2.3 - Consider the “divides” relation on the set of...Ch. 2.3 - Prob. 29E Ch. 2.3 - Prob. 30E Ch. 2.3 - Prob. 31E Ch. 2.3 - Prob. 32E Ch. 2.3 - Prob. 33E Ch. 2.3 - Prob. 34E Ch. 2.3 - Prob. 35E Ch. 2.3 - Prob. 37E Ch. 2.3 - Prob. 38E Ch. 2.3 - Prob. 39E Ch. 2.3 - Prob. 40E Ch. 2.3 - Prob. 41E Ch. 2.3 - Prob. 42E Ch. 2.4 - In Exercises 1–4, determine which of the given...Ch. 2.4 - In Exercises 1–4, determine which of the given...Ch. 2.4 - In Exercises 1–4, determine which of the given...Ch. 2.4 - In Exercises 1–4, determine which of the given...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - In Exercises 5–12, determine whether the given g...Ch. 2.4 - Prob. 13E Ch. 2.4 - Prob. 14E Ch. 2.4 - Prob. 15E Ch. 2.4 - Prob. 16E Ch. 2.4 - Prob. 17E Ch. 2.4 - Prob. 18E Ch. 2.4 - Prob. 19E Ch. 2.4 - Prob. 20E Ch. 2.4 - Prob. 21E Ch. 2.4 - Prob. 22E Ch. 2.4 - Prob. 23E Ch. 2.4 - Prob. 24E Ch. 2.4 - Prob. 25E Ch. 2.4 - Prob. 26E Ch. 2.4 - Prob. 27E Ch. 2.4 - Prob. 28E Ch. 2.4 - Prob. 29E Ch. 2.4 - Prob. 30E Ch. 2.4 - Prob. 31E Ch. 2.4 - Prob. 32E Ch. 2.4 - Prob. 33E Ch. 2.4 - Prob. 34E Ch. 2.4 - Prob. 35E Ch. 2.4 - Prob. 36E Ch. 2.4 - Prob. 37E Ch. 2.4 - Prob. 38E Ch. 2.4 - Prob. 39E Ch. 2.4 - Determine formulas for the functions gf and fg in...Ch. 2.4 - Prob. 41E Ch. 2.4 - Prob. 42E Ch. 2.4 - Prob. 43E Ch. 2.4 - Prob. 44E Ch. 2.4 - In Exercises 45–52, Z denotes the set of integers....Ch. 2.4 - In Exercises 45–52, Z denotes the set of integers....Ch. 2.4 - In Exercises 45–52, Z denotes the set of integers....Ch. 2.4 - In Exercises 45–52, Z denotes the set of integers....Ch. 2.4 - Prob. 49E Ch. 2.4 - In Exercises 45–52, Z denotes the set of integers....Ch. 2.4 - In Exercises 45–52, Z denotes the set of integers....Ch. 2.4 - Prob. 52E Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - In Exercises 53–60, X denotes the set of real...Ch. 2.4 - Find a subset Y of the set of real numbers X such...Ch. 2.4 - Find a subset Y of the set of real numbers X such...Ch. 2.4 - Prob. 63E Ch. 2.4 - If X has m elements and Y has n elements, how many...Ch. 2.4 - Prob. 65E Ch. 2.4 - Prob. 66E Ch. 2.4 - Prob. 67E Ch. 2.4 - Prob. 68E Ch. 2.4 - Prob. 69E Ch. 2.4 - Prob. 70E Ch. 2.5 - Compute the Fibonacci numbers F1 through F10. Ch. 2.5 - Suppose that a number xn is defined recursively by...Ch. 2.5 - Prob. 3E Ch. 2.5 - Prob. 4E Ch. 2.5 - Prob. 5E Ch. 2.5 - Prob. 6E Ch. 2.5 - Prob. 7E Ch. 2.5 - Prob. 8E Ch. 2.5 - Prob. 9E Ch. 2.5 - In Exercises 7–10, determine what is wrong with...Ch. 2.5 - In Exercises 11–26, prove each of the given...Ch. 2.5 - In Exercises 11–26, prove each of the given...Ch. 2.5 - Prob. 13E Ch. 2.5 - Prob. 14E Ch. 2.5 - Prob. 15E Ch. 2.5 - Prob. 16E Ch. 2.5 - Prob. 17E Ch. 2.5 - In Exercises 11–26, prove each of the given...Ch. 2.5 - Prob. 19E Ch. 2.5 - Prob. 20E Ch. 2.5 - Prob. 21E Ch. 2.5 - Prob. 22E Ch. 2.5 - Prob. 23E Ch. 2.5 - Prob. 24E Ch. 2.5 - Prob. 25E Ch. 2.5 - Prob. 26E Ch. 2.5 - A sequence s0, s1, s2,… is called a geometric...Ch. 2.5 - A sequence, s0, s1, s2,… is called an arithmetic...Ch. 2.6 - Prob. 1E Ch. 2.6 - Prob. 2E Ch. 2.6 - Prob. 3E Ch. 2.6 - Evaluate the numbers in Exercises 1–12. 4. C(12,...Ch. 2.6 - Evaluate the numbers in Exercises 1–12. 5. C(11,...Ch. 2.6 - Prob. 6E Ch. 2.6 - Prob. 7E Ch. 2.6 - Evaluate the numbers in Exercises 1–12. 8. C(13,...Ch. 2.6 - Evaluate the numbers in Exercises 1–12. 9. C(n,...Ch. 2.6 - Prob. 10E Ch. 2.6 - Prob. 11E Ch. 2.6 - Evaluate the numbers in Exercises 1–12. 12. Ch. 2.6 - Prob. 13E Ch. 2.6 - How many nonempty subsets of the set {a, e, i, o,...Ch. 2.6 - At Avanti’s, a pizza can be ordered with any...Ch. 2.6 - If a test consists of 12 questions to be answered...Ch. 2.6 - Prob. 17E Ch. 2.6 - Jennifer’s grandmother has told her that she can...Ch. 2.6 - Prob. 19E Ch. 2.6 - Prob. 20E Ch. 2.6 - Prob. 21E Ch. 2.6 - Prob. 22E Ch. 2.6 - Prob. 23E Ch. 2.6 - Prob. 24E Ch. 2.6 - Prob. 25E Ch. 2.6 - Prob. 26E Ch. 2.6 - Prob. 27E Ch. 2.6 - Prob. 28E Ch. 2.6 - Prove each of the statements in Exercises 29–40 by...Ch. 2.6 - Prob. 30E Ch. 2.6 - Prob. 31E Ch. 2.6 - Prove each of the statements in Exercises 29–40 by...Ch. 2.6 - Prob. 33E Ch. 2.6 - Prove each of the statements in Exercises 29–40 by...Ch. 2.6 - Prob. 35E Ch. 2.6 - Prob. 36E Ch. 2 - Prob. 1SE Ch. 2 - Prob. 2SE Ch. 2 - Prob. 3SE Ch. 2 - Prob. 4SE Ch. 2 - Prob. 5SE Ch. 2 - Prob. 6SE Ch. 2 - Prob. 7SE Ch. 2 - Prob. 8SE Ch. 2 - Prob. 9SE Ch. 2 - Draw Venn diagrams depicting the sets in Exercises...Ch. 2 - Prob. 11SE Ch. 2 - Prob. 12SE Ch. 2 - Prob. 13SE Ch. 2 - Prob. 14SE Ch. 2 - Prob. 15SE Ch. 2 - Prob. 16SE Ch. 2 - Prob. 17SE Ch. 2 - Prob. 18SE Ch. 2 - Prob. 19SE Ch. 2 - Prob. 20SE Ch. 2 - Prob. 21SE Ch. 2 - Prob. 22SE Ch. 2 - Prob. 23SE Ch. 2 - Prob. 24SE Ch. 2 - Prob. 25SE Ch. 2 - Prob. 26SE Ch. 2 - Prob. 27SE Ch. 2 - Prob. 28SE Ch. 2 - Prob. 29SE Ch. 2 - Prob. 30SE Ch. 2 - Prob. 31SE Ch. 2 - Prob. 32SE Ch. 2 - Prob. 33SE Ch. 2 - Prob. 34SE Ch. 2 - Prob. 35SE Ch. 2 - How many equivalence relations on S = {a, b, c}...Ch. 2 - Prob. 37SE Ch. 2 - Prob. 38SE Ch. 2 - Prob. 39SE Ch. 2 - Prob. 40SE Ch. 2 - Prob. 41SE Ch. 2 - Prob. 42SE Ch. 2 - Prob. 43SE Ch. 2 - Prob. 44SE Ch. 2 - Prob. 45SE Ch. 2 - Prob. 46SE Ch. 2 - Prob. 47SE Ch. 2 - Prob. 49SE Ch. 2 - Prob. 50SE Ch. 2 - Prob. 51SE Ch. 2 - Prob. 52SE Ch. 2 - Prob. 53SE Ch. 2 - Prob. 54SE Ch. 2 - Prob. 55SE Ch. 2 - Prob. 56SE Ch. 2 - Prob. 57SE Ch. 2 - Prob. 58SE Ch. 2 - Prob. 59SE Ch. 2 - Prob. 60SE Ch. 2 - Prob. 61SE Ch. 2 - Prob. 62SE Ch. 2 - Prob. 63SE Ch. 2 - Prob. 64SE Ch. 2 - Prove the results in Exercises 63–72 by...Ch. 2 - Prob. 66SE Ch. 2 - Prob. 67SE Ch. 2 - Prob. 68SE Ch. 2 - Prob. 69SE Ch. 2 - Prob. 70SE Ch. 2 - Prob. 71SE Ch. 2 - Prob. 72SE Ch. 2 - Prob. 1CP Ch. 2 - Prob. 6CP Ch. 2 - Prob. 7CP Ch. 2 - Prob. 12CP

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