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 : |
Reflexive but not Symmetric and Transitive Symmetric and Reflexive but not Transitive Equivalence relation Reflexive and Transitive but not Symmetric |
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 |