If R be a relation on the set of integers Z, given by $R = \{(a, b): (a - b)$ is a multiple of $3\}$, then R is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → an equivalence relation
Given relation:
$R = \{(a,b) : a - b \text{ is a multiple of } 3\}$ on $\mathbb{Z}$.
Reflexive:
$a - a = 0$, which is a multiple of 3 ⇒ reflexive.
Symmetric:
If $a - b$ is a multiple of 3, then $b - a = -(a - b)$ is also a multiple of 3 ⇒ symmetric.
Transitive:
If $a - b$ and $b - c$ are multiples of 3, then
$a - c = (a - b) + (b - c)$ is also a multiple of 3 ⇒ transitive.
Therefore, the relation is an equivalence relation.