What is the smallest change needed to make R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)} an equivalence relation on A={1,2,3}?

Correct answer: Add (1,3) and (3,1)

Answer: A. An equivalence relation must be reflexive, symmetric, and transitive. The given relation already contains all diagonal pairs, so it is reflexive. It also contains each reverse pair: (1,2) with (2,1), and (2,3) with (3,2), so it is symmetric. Transitivity now requires (1,3), because (1,2) and (2,3) are present. Once (1,3) is added, symmetry requires its reverse (3,1) as well. Adding both pairs makes every pair in A×A present, which certainly gives reflexivity, symmetry, and transitivity. Option B breaks symmetry, option C also breaks symmetry, and option D removes pairs rather than completing the needed connection. Memory cue: equivalence classes cannot overlap through a chain without joining their endpoints.