Let R be the relation on the set R of real numbers defined by \(R = \{(a, b) : a - b \text{ is an integer}\}\). Then R is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → an equivalence relation
Reflexive: for every real a,
\(a - a = 0\), which is an integer.
Symmetric: if \(a - b = m\) is an integer, then
\(b - a = -m\), which is also an integer.
Transitive: if \(a - b = m\) and \(b - c = n\) are integers, then
\(a - c = (a - b) + (b - c) = m + n\), which is an integer.
So R is reflexive, symmetric and transitive, that is, an equivalence relation.