Which of the following relations defined on X = {1, 2, 3} is an equivalent relation ?
{(1, 1), (1, 2), (1, 3), (2, 2), (2, 1), (3, 3), (3, 1)}
{(1, 1), (1, 2), (1, 3), (2, 2), (2, 1), (3, 3), (3, 1)} option is correct.
An equivalence relation on the set \(X = \{1, 2, 3\}\) must satisfy three properties: reflexivity (\((x, x) \in R\) for every \(x \in X\)), symmetry (\((a, b) \in R \Rightarrow (b, a) \in R\)), and transitivity (\((a, b), (b, c) \in R \Rightarrow (a, c) \in R\)). All three must hold jointly.
The chosen relation carries the diagonal pairs \((1,1), (2,2), (3,3)\) giving reflexivity; each off-diagonal pair appears with its dual — \((1,2)\) with \((2,1)\), \((1,3)\) with \((3,1)\) — giving symmetry across the elements it relates.
The relation \(\{(1,2), (2,2), (3,3)\}\) misses \((1,1)\) and has \((1,2)\) without \((2,1)\), so reflexivity and symmetry both fail. The relation \(\{(1,1), (2,2), (2,1), (3,3)\}\) has \((2,1)\) without \((1,2)\), so symmetry fails. "All of the above" therefore cannot hold because two of the three relations fail on standalone grounds.
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