Let \(A = \{1, 2, 3\}\). Consider the following statements:
(A) The empty relation on A is symmetric.
(B) The empty relation on A is reflexive.
(C) The universal relation \(A \times A\) is an equivalence relation.
(D) The empty relation on A is transitive.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
The empty relation contains no pair at all.
Reflexive needs (1, 1), (2, 2), (3, 3) to be present, and they are not.
Symmetric and transitive are rules of the form “if a pair (or a chain of pairs) is in the relation, then another pair must be in it”. With no pairs present, there is nothing to break these rules.
The universal relation \(A \times A\) contains every pair, so every pair any rule asks for is present.
So the empty relation is symmetric and transitive but not reflexive, and \(A \times A\) is reflexive, symmetric and transitive.
Checking the statements
(A) The empty relation on A is symmetric. — Correct. It has no pair (a, b) that needs (b, a).
(B) The empty relation on A is reflexive. — Incorrect. (1, 1) is not in it.
(C) The universal relation \(A \times A\) is an equivalence relation. — Correct. It contains all 9 pairs, so all three properties hold.
(D) The empty relation on A is transitive. — Correct. It has no chain (a, b), (b, c) to check.