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On Z, the set of integers define a relation R as follows $a, b \in Z, aRb if 3 ∣ (2 a + b)$ then

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a

R is reflexive, symmetric but not transitive

b

R is reflexive, transitive but not symmetric

c

R is anti-symmetric

d

R is an equivalence relation

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detailed solution

Correct option is D

If a∈Z,aRa as3∣(2a+a)R is reflexive.Suppose a R b ⇒ 3∣(2a+b) ⇒ 3∣(3a+b−a) ⇒ 3|(b−a) ⇒3|(a−b) ⇒ 3|(3b+a−b)⇒3|(2b+a) ⇒ b~a. ⇒ Ris symmetric.Next, suppose a R b, b R c ⇒3|(2a+b),3|(2b+c)⇒3|(b−a),3|(c−b)⇒3∣(b−a+c−b)⇒3∣(c−a)⇒3∣(c−a+3a)⇒3∣(2a+c)∴aRcR is transitive. Hence, R is an equivalence relation

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