(a) Explanation Given information: For a , b ∈ Z , define a ∼ b if and only if a 2 − b 2 is divisible by 3. Proof: To prove that ∼ defines an equivalence relation on Z, we will have to prove that the relation ∼ is reflexive, symmetric and transitive. Reflexive: For any a ∈ Z , we have a ∼ a because a 2 − a 2 = 0 is divisible by 3. Symmetric: If any a , b ∈ Z and a ∼ b , then a 2 − b 2 = 3 k for some integer k . Then, b 2 − a 2 = − 3 k . We see that b 2 − a 2 is also divisible by 3. Thus, b ∼ a . Transitive: If any a , b , c ∈ Z , a ∼ b and b ∼ c (b) To determine The value of 0 ¯ . (c) To determine The value of 1 ¯ .

3rd Edition

ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

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Chapter 4, Problem 15RE

(a)

To determine

To prove: That ∼ defines an equivalence relation on Z.

(b)

To determine

The value of 0¯.

(c)

To determine

The value of 1¯.

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Chapter 4, Problem 14RE

Chapter 4, Problem 16RE

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Chapter 4 Solutions

Discrete Mathematics with Graph Theory

Ch. 4.1 - 227 Ch. 4.1 - The Well-Ordering Principle states that any...Ch. 4.1 - Prob. 3TFQ Ch. 4.1 - Prob. 4TFQ Ch. 4.1 - Prob. 5TFQ Ch. 4.1 - Prob. 6TFQ Ch. 4.1 - Prob. 7TFQ Ch. 4.1 - Prob. 8TFQ Ch. 4.1 - Prob. 9TFQ Ch. 4.1 - Prob. 10TFQ

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To prove: That ∼ defines an equivalence relation on Z.

Solution Summary: The author proves that sim defines an equivalence relation on Z. The relation is reflexive, symmetric and transitive.

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To prove: That ∼ defines an equivalence relation on Z.

The value of 0¯.

The value of 1¯.

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Chapter 4 Solutions