Let B be the set of all books in a library, and let R be the relation on B defined by \(R = \{(x, y) : x \text{ and } y \text{ have the same number of pages}\}\). Then R is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → an equivalence relation
Let \(n(x)\) be the number of pages of book x. Then \((x, y) \in R\) means \(n(x) = n(y)\).
Reflexive: \(n(x) = n(x)\) for every book x, so \((x, x) \in R\).
Symmetric: if \(n(x) = n(y)\), then \(n(y) = n(x)\), so \((y, x) \in R\).
Transitive: if \(n(x) = n(y)\) and \(n(y) = n(z)\), then \(n(x) = n(z)\), so \((x, z) \in R\).
So R is reflexive, symmetric and transitive, that is, an equivalence relation.