Explanation Given information: Define Q on the set ⁒ × ⁒ as follows: For all ( w , x ) , ( y , z ) ∈ ⁒ × ⁒ , ( w , x ) Q ( y , z ) ⇔ x = z . Proof: Q = { ( w , x ) , ( y , z ) ∈ ⁒ × ⁒ | x = z } To prove: Q is an equivalence relation. Reflexive: Let ( x , y ) ∈ ⁒ × ⁒ Since y = y , ( ( x , y ) , ( x , y ) ) ∈ Q Since ( ( x , y ) , ( x , y ) ) ∈ Q for all ( x , y ) ∈ ⁒ × ⁒ , and thus Q is reflexive . Symmetric: Let us assume that ( ( w , x ) , ( y , z ) ) ∈ Q . By the definition of Q : x = z However, x = z is equivalent with z = x. ( ( y , z ) , ( w , x ) ) ∈ Q Since ( ( w , x ) , ( y , z ) ) ∈ Q implies ( ( y , z ) , ( w , x ) ) ∈ Q , Q is symmetric

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Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 8.3, Problem 30ES

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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 29ES

Chapter 8.3, Problem 31ES

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

Solution Summary: The author explains that the given relation is an equivalence relation and also find the distinct enumerations of the 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.