Which pair must belong to the smallest equivalence relation on A={1,2,3,4} containing R={(1,2),(2,3),(4,4)}?

Correct answer: (1,3)

The correct answer is A, (1,3). An equivalence relation must be reflexive, symmetric, and transitive. Since (1,2) and (2,3) are already present, transitivity forces (1,3). Symmetry also forces (2,1) and (3,2), and then the reverse of (1,3), namely (3,1). Reflexivity requires all diagonal pairs, including (1,1), (2,2), and (3,3); (4,4) is already present. The smallest closure therefore makes {1,2,3} one equivalence class and leaves {4} as a separate class. No pair joining 4 to 1, 2, or 3 is forced. Hence B, C, and D are not necessary. A useful method is to connect elements through chains: 1 is connected to 2, and 2 to 3, so 1 must connect to 3; element 4 has no such connection.