Let a relation \(R = \{(a, b) : \text{a is a factor of b},\ a, b \in \mathrm{N}\}\). Then, R is _____.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Reflexive and Transitive but NOT Symmetric.
Here \((a, b) \in R\) means a divides b.
Reflexive: yes. Every natural number divides itself, so \((a, a) \in R\) for all a.
Symmetric: no. 2 is a factor of 4, so \((2, 4) \in R\); but 4 is not a factor of 2, so \((4, 2) \notin R\).
Transitive: yes. If a divides b and b divides c, then b = ka and c = mb for natural numbers k, m:
\(c = mb = (mk)a\)
So a divides c, and \((a, c) \in R\).
Hence R is reflexive and transitive but not symmetric (so not an equivalence relation).