Explanation Given information: E is the relation defined onZ as follows: For all m, n ∈ Z , m E n ⇔ 4 | ( m - n). Proof: (1). A relation will be equivalence relation if it is reflexive, symmetric and transitive. To proof: E is an equivalent relation A = Z E = { ( m , n ) ∈ A × A | 4 | ( m − n ) } Reflexive: Let x ∈ A = Z x − x = 0 = 0 ⋅ 4 By the definition of divides, 4 divides x − x and thus ( x , x ) ∈ E . Since ( x , x ) ∈ E for all x ∈ A and thus E is indeed reflexive. Symmetric: Let us assume that ( x , y ) ∈ E . by the definition of E : 4 | ( x − y ) By the definition of divides, there exists an integer k such that: x − y = 4 k Multiply each side by − 1: − ( x − y ) = − ( 4 k ) Use the distributive property: y − x = 4 ( − k ) By the definition of divides, 4 divides y − x as − k is an integer (because k is an integer). ( y , x ) ∈ E Since ( x , y ) ∈ E implies ( y , x ) ∈ E , E is symmetric. Transitive: Let us assume that ( x , y ) ∈ E and ( y , z ) ∈ E

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Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 8.3, Problem 20ES

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To prove:

(1) The given relation is an equivalence relation,

(2) Describe the distinct equivalence classes of each relation.

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Chapter 8.3, Problem 19ES

Chapter 8.3, Problem 21ES

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Chapter 8 Solutions

Discrete Mathematics With Applications

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To prove: (1) The given relation is an equivalence relation, (2) Describe the distinct equivalence classes of each relation.

Solution Summary: The author explains that an equivalence relation is reflexive, symmetric and transitive.

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To prove: (1) The given relation is an equivalence relation, (2) Describe the distinct equivalence classes of each relation.

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Chapter 8 Solutions

To prove:

(1) The given relation is an equivalence relation,

(2) Describe the distinct equivalence classes of each relation.