Consider the relation R in the set Z of integers which is given by R = {(a, b) : 2 divides a - b}. Choose the correct option for the relation R:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) - Equivalence relation
for every $a∈z$
$a-a=0$ (divisible by z)
$⇒(a,a)∈R$ (Reflexive)
$(a,b)∈R$ ⇒ $a - b$ divisible by z ⇒ $b - a$ divisible by z
$⇒(b,a)∈R$ (Symmetric)
$(a,b)∈R,(b,c)∈R$
$a - b$ divisible by z, $b - c$ divisible by z
$a - b + b - c$ divisible by z
$⇒(a,c)∈R$ (Transitive)
⇒ R is equivalence Relation