Let \(A = \{2, 3, 4, \ldots, 10\}\) and let R be the relation on A given by \(R = \{(a, b) : a \text{ and } b \text{ have a common factor greater than } 1\}\). Then R is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → reflexive and symmetric but not transitive
Reflexive: every a in A is at least 2, so a itself is a common factor of a and a that is greater than 1. Hence \((a, a) \in R\).
Symmetric: a common factor of a and b is also a common factor of b and a.
Not transitive: take 2, 6 and 3.
2 and 6 have the common factor 2, so \((2, 6) \in R\)
6 and 3 have the common factor 3, so \((6, 3) \in R\)
2 and 3 have no common factor greater than 1, so \((2, 3) \notin R\)
So R is reflexive and symmetric but not transitive.