Let R be the relation on the set R of real numbers defined by R = (a,…

The correct answer is Option (4) → reflexive and transitive but not symmetric

The condition factorises:

    \(a^{2} - ab = 0\)
    \(a(a - b) = 0\)

So \((a, b) \in R\) exactly when \(a = 0\) or \(a = b\).

Reflexive: \(a = a\) for every real a, so \((a, a) \in R\).

Not symmetric: \((0, 5) \in R\), since the first entry is 0. But \(5^{2} = 25\) and \(5 \times 0 = 0\), so \((5, 0) \notin R\).

Transitive: let \((a, b) \in R\) and \((b, c) \in R\).

    If \(a = 0\), then \((a, c) \in R\) at once.
    If \(a = b\), then \((b, c) \in R\) is the same as \((a, c) \in R\).

So R is reflexive and transitive but not symmetric.