Explanation Given information: I is the relation defined on ⁒ as follows: For all x, y ∈ R , x I y ⇔ x - y is an integer. Calculation: I = { ( m , n ) ∈ ⁒ × ⁒ | m − n is an integer } To prove: R is an equivalent relation. Reflexive: Let x ∈ ⁒ x − x = 0 x − x is an integer, since 0 is an integer and thus ( x , x ) ∈ I Symmetric: Let us assume that ( x , y ) ∈ I . By the definition of I : x − y is an integer . However the negation of an integer is also an integer. − ( x − y ) is an integer . Use distributive property, y − x is an integer . By the definition of I : ( y , x ) ∈ I Since ( x , y ) ∈ I implies ( y , x ) ∈ I , I is symmetric. Transitive: Let us assume that ( x , y ) ∈ I and ( y , z ) ∈ I . By the definition of I : ( x − y ) is an integer

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Chapter 8.3, Problem 28ES

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To prove:

The given relation is an equivalence relation and also find the distinct equivalence classes of given relation.

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Chapter 8.3, Problem 27ES

Chapter 8.3, Problem 29ES

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Chapter 8 Solutions

Discrete Mathematics With Applications

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To prove: The given relation is an equivalence relation and also find the distinct equivalence classes of given relation.

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To prove: The given relation is an equivalence relation and also find the distinct equivalence classes of given relation.

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Chapter 8 Solutions

To prove:

The given relation is an equivalence relation and also find the distinct equivalence classes of given relation.