The relation R on the set A = {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 3), (2, 3), (1, 2)} is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → reflexive and transitive
Reflexive: (1, 1), (2, 2) and (3, 3) are all in R, so R is reflexive.
Symmetric: (1, 3) ∈ R but (3, 1) ∉ R, so R is not symmetric.
Transitive: the only chain through two different elements is (1, 2) and (2, 3), and (1, 3) ∈ R. Chains that use a pair (a, a) give back a pair already in R. So R is transitive.
Hence R is reflexive and transitive, but not symmetric.
Why the other options are incorrect
Option (1) reflexive but neither symmetric nor transitive — Incorrect. R is transitive, since the only chain (1, 2), (2, 3) gives (1, 3), which is in R.
Option (3) reflexive and symmetric — Incorrect. R is not symmetric: (1, 3) is in R but (3, 1) is not.
Option (4) an equivalence relation — Incorrect. An equivalence relation must be symmetric, and R is not.
Tip: R here is the relation “≤” on {1, 2, 3}: every pair (a, b) with a ≤ b. The relation ≤ is always reflexive and transitive but not symmetric.