The relation R on the set \(A = \{x \in Z : 0 \le x \le 12\}\), given by \(R = \{(a, b) : |a - b| \text{ is a multiple of } 4\}\) is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Equivalence relation
Each of the three properties is checked for all a, b, c in A.
Reflexive: for every a,
\(|a - a| = 0 = 4 \times 0\)
which is a multiple of 4, so (a, a) ∈ R.
Symmetric: if (a, b) ∈ R, then
\(|b - a| = |a - b|\)
is the same multiple of 4, so (b, a) ∈ R.
Transitive: if (a, b) ∈ R and (b, c) ∈ R, then a − b = 4m and b − c = 4n for some integers m, n. Adding,
\(a - c = 4(m + n)\)
so |a − c| is a multiple of 4 and (a, c) ∈ R.
R is reflexive, symmetric and transitive, so it is an equivalence relation.
NCERT (2026-27 edition):
Types of Relations
