GATE | GATE-CS-2000 | Question 49

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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