Let \(A = \{1, 2, 3, 4\}\) and let R be the relation on A given by \(R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (2, 4), (4, 2)\}\). Then R is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → an equivalence relation
R is reflexive, since (1, 1), (2, 2), (3, 3) and (4, 4) are all in R.
R is symmetric, since the pairs that are not of the form (a, a) come in reversed pairs:
(1, 3) and (3, 1)
(2, 4) and (4, 2)
R is transitive. Each chain (a, b), (b, c) in R gives a pair (a, c) that is again in R:
(1, 3), (3, 1) give (1, 1)
(3, 1), (1, 3) give (3, 3)
(2, 4), (4, 2) give (2, 2)
(4, 2), (2, 4) give (4, 4)
Chains that use a pair (a, a) just give back the other pair. So R is reflexive, symmetric and transitive, that is, an equivalence relation. Its classes are {1, 3} and {2, 4}.