Explanation Proof: To prove that ∼ is an equivalence relation on A, we must prove that ∼ is reflexive, symmetric and transitive. Reflexive: If a ∈ A , a a = a 2 is a perfect square, so a ∼ a . Symmetric: If a , b , c ∈ A , a ∼ b and b ∼ c , then a b = n 2 and b c = m 2 are prefect squares. So, ( a b ) ( b c ) = ( n m ) 2 . To prove that a c is a perfect square, we prove that the prime powers occurring in the decomposition of a c as the product of primes is all even, that is, that a c = p 1 2 α 1 p 2 2 α 2 ....... p r 2 α r

3rd Edition

ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

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Chapter 4.3, Problem 7E

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To prove: That ∼ defines an equivalence relation on A where A be any subset of Z\{0} and, for a,b∈A, a∼b if ab is a perfect square (that is, ab=x2 for some integer x).

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Chapter 4.3, Problem 6E

Chapter 4.3, Problem 8E

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Chapter 4 Solutions

Discrete Mathematics with Graph Theory

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Ch. 4.1 - Prob. 1E Ch. 4.1 - 2. True or false? If false, give a counterexample....Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - 9. Suppose is a natural number and is the...Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.2 - True/False Questions Given two consecutive...Ch. 4.2 - Prob. 2TFQ Ch. 4.2 - Prob. 3TFQ Ch. 4.2 - Prob. 4TFQ Ch. 4.2 - Prob. 5TFQ Ch. 4.2 - Prob. 6TFQ Ch. 4.2 - Prob. 7TFQ Ch. 4.2 - Prob. 8TFQ Ch. 4.2 - Prob. 9TFQ Ch. 4.2 - Prob. 10TFQ Ch. 4.2 - Prob. 1E Ch. 4.2 - Prob. 2E Ch. 4.2 - Draw the Hasse diagrams for each of the following...Ch. 4.2 - List all minimal, minimum, maximal, and maximum...Ch. 4.2 - Let n be a natural number. 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To prove: That ∼ defines an equivalence relation on A where A be any subset of Z \ { 0 } and, for a , b ∈ A , a ∼ b if a b is a perfect square (that is, a b = x 2 for some integer x ).

Solution Summary: The author proves that sim defines an equivalence relation on A where A be any subset of Z.

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To prove: That ∼ defines an equivalence relation on A where A be any subset of Z\{0} and, for a,b∈A, a∼b if ab is a perfect square (that is, ab=x2 for some integer x).

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Chapter 4 Solutions