Let \(A = \{1, 2, 3\}\) and \(R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)\}\) be a relation on A. Consider the following statements:
(A) R is reflexive.
(B) R is symmetric.
(C) R is transitive.
(D) R together with the pairs (1, 3) and (3, 1) is an equivalence relation on A.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
All three pairs (1, 1), (2, 2), (3, 3) are in R, and the other pairs come in reversed pairs: (1, 2), (2, 1) and (2, 3), (3, 2).
For transitivity, look at the chain through 2:
\((1, 2) \in R\) and \((2, 3) \in R\), but \((1, 3) \notin R\)
After adding (1, 3) and (3, 1), the relation contains all 9 pairs of \(A \times A\), which is reflexive, symmetric and transitive.
So R is reflexive and symmetric, not transitive, and R with (1, 3), (3, 1) added is \(A \times A\).
Checking the statements
(A) R is reflexive. — Correct. (1, 1), (2, 2), (3, 3) are all in R.
(B) R is symmetric. — Correct. (1, 2), (2, 1) and (2, 3), (3, 2) are both in R.
(C) R is transitive. — Incorrect. (1, 2) and (2, 3) are in R, but (1, 3) is not.
(D) R together with the pairs (1, 3) and (3, 1) is an equivalence relation on A. — Correct. The result is \(A \times A\), which is an equivalence relation.