On A = {1,2,3}, let R = {(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}. Which pair or pairs must be added to make R an equivalence relation?

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

The relation already contains all diagonal pairs, so it is reflexive. It is also symmetric because (1,2) and (2,1), as well as (2,3) and (3,2), are both present. The missing requirement is transitivity. Since (1,2) and (2,3) belong to R, transitivity requires (1,3). Also, because (3,2) and (2,1) are present, it requires (3,1). Equivalently, symmetry would require the reverse of (1,3) once that pair is added. After inserting both pairs, every pair among {1,2,3} is present, so the resulting universal relation is certainly an equivalence relation. Adding only one pair leaves either transitivity or symmetry violated. Thus option A is necessary and sufficient.