A relation R is defined on the set of integers as xRy if f(x + y) is even. Which of the following statements is true?
(A) R is not an equivalence relation
(B) R is an equivalence relation having 1 equivalence class
(C) R is an equivalence relation having 2 equivalence classes
(D) R is an equivalence relation having 3 equivalence classes
Answer: (C)
Explanation: There are 2 2 equivalence classes.
1) All odd integers. (Reflexive as sum of two even is even, Symmetric and Transitive as + operation is Symmetric and Transitive)
2) All Even Integers. (Reflexive as sum of two odd is even, Symmetric and Transitive as + operation is Symmetric and Transitive)
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