The binary relation S = ф (empty set) on set A = {1, 2, 3} is :
(A) Neither reflexive nor symmetric
(B) Symmetric and reflexive
(C) Transitive and reflexive
(D) Transitive and symmetric
Answer: (D)
Explanation:
- Reflexive : A relation is reflexive if every element of set is paired with itself. Here none of the element of A is paired with themselves, so S is not reflexive.
- Symmetric : This property says that if there is a pair (a, b) in S, then there must be a pair (b, a) in S. Since there is no pair here in S, this is trivially true, so S is symmetric.
-
Transitive : This says that if there are pairs (a, b) and (b, c) in S, then there must be pair (a,c) in S. Again, this condition is trivially true, so S is transitive.
Thus, option (D) is correct.
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