Let \(A = \{1, 2, 3\}\) and \(R = \{(1, 1), (2, 2), (3, 3), (1, 3)\}\). Which ordered pair should be added to R to make it an equivalence relation on A?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (3, 1)
R is already reflexive, since (1, 1), (2, 2) and (3, 3) are in R.
R is not symmetric: (1, 3) is in R, but (3, 1) is not. So (3, 1) has to be added.
The new relation is
\(\{(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)\}\)
It is symmetric, and it is transitive: the chains (1, 3), (3, 1) and (3, 1), (1, 3) give (1, 1) and (3, 3), which are in it. So it is an equivalence relation, with classes {1, 3} and {2}.