Which of the following is not an equivalence relation on Z?
(a) a R b ⇔ a + b is an even integer
(b) a R b ⇔ a − b is an even integer
(c) a R b ⇔ a < b
(d) a R b ⇔ a = b

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Solution

(c) a R b ⇔ a < b

Clearly, R is not a symmetric relation. This is because if (a, b) is an element of relation R, then, a < b does not imply b < a $\forall$ a, b $\in$ Z.
Hence, R is not an equivalence relation on Z.

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Which of the following is not an equivalence relation on Z? (a) a R b ⇔ a + b is an even integer (b) a R b ⇔ a − b is an even integer (c) a R b ⇔ a < b (d) a R b ⇔ a = b

(c) a R b ⇔ a < b Clearly, R is not a symmetric relation. This is because if (a, b) is an element of relation R, then, a < b does not imply b < a ∀ a, b ∈ Z. Hence, R is not an equivalence relation on Z.

Which of the following is not an equivalence relation on Z?
(a) a R b ⇔ a + b is an even integer
(b) a R b ⇔ a − b is an even integer
(c) a R b ⇔ a < b
(d) a R b ⇔ a = b

(c) a R b ⇔ a < b

Clearly, R is not a symmetric relation. This is because if (a, b) is an element of relation R, then, a < b does not imply b < a $\forall$ a, b $\in$ Z.
Hence, R is not an equivalence relation on Z.