If R be a relation on the set of integers Z, given by \(R = \{(a, b): (a - b) \text{ is a multiple of } 3\}\), then R is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → an equivalence relation
Reflexive: \(a - a = 0 = 3 \times 0\), a multiple of 3, so \((a, a) \in R\) for every integer a.
Symmetric: if \(a - b = 3m\), then \(b - a = 3(-m)\), so \((b, a) \in R\).
Transitive: if \(a - b = 3m\) and \(b - c = 3n\), then
\(a - c = (a - b) + (b - c) = 3(m + n)\)
so \((a, c) \in R\).
R is reflexive, symmetric and transitive, so it is an equivalence relation.