Let R be the relation on the set Z of all integers defined by \(R = \{(a, b) : a - b ext{ is a non-negative even integer}\}\). Then R is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → reflexive and transitive but not symmetric
Reflexive: \(a - a = 0\), and 0 is a non-negative even integer, so \((a, a) \in R\).
Not symmetric: \(4 - 2 = 2\), so \((4, 2) \in R\); but \(2 - 4 = -2\) is negative, so \((2, 4) otin R\).
Transitive: if \(a - b\) and \(b - c\) are non-negative even integers, then
\(a - c = (a - b) + (b - c)\)
is a sum of two non-negative even integers, which is again a non-negative even integer. So \((a, c) \in R\).
So R is reflexive and transitive but not symmetric.