Discussion
Let \(X = \{a_1, a_2, a_3, \ldots, a_n\}\) be a set consisting of \(n\) elements. The relation \(R = \{(a_1, a_1), (a_2, a_2), (a_3, a_3), \ldots, (a_n, a_n)\}\) on the set \(X\) is
Answer: D
The relation is reflexive, symmetric and transitive.
\(R\) contains exactly the pairs \((a_k, a_k)\), so it is the identity relation on \(X\). It is reflexive because every \((a_k, a_k)\) is in \(R\). It is symmetric because the only pairs are of the form \((a, a)\), and reversing them gives the same pair. It is transitive because \((a, b)\) and \((b, c)\) in \(R\) force \(a = b = c\), so \((a, c) = (a, a)\) is in \(R\).
The identity relation is therefore an equivalence relation, and every option that denies one of the three properties is wrong.
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