Let R b a relation on the set of integers Z such that R={$(a, b), a=2^kb, a, b, k\in z$}, then R is :

Equivalence relation

The correct answer is Option (3) → Equivalence relation

$R=a=2^kb$

(1) Reflexive

as $a=a=a=2^0a$ for every $a∈Z$

(2) Symmetric

for $(a,b)∈R⇒a=2^kb$

so $b=2^{-k}a$  $-k∈Z$

$⇒(b,a)∈R$

(3) Transitive

$(a,b)∈R, (b,c)∈R⇒a=2^{k_1}b,b=2^{k_2}c$

so $a = 2^{k_1+k_2}c$

so $(a,c)∈R$

Relation is equivalence relation