Why is the empty relation ∅ considered symmetric on any set A?
Answer and explanation
Correct answer: Because it has no pair that can violate the condition
A relation is symmetric if, for every pair (a,b) in the relation, the reverse pair (b,a) is also in it. The empty relation contains no ordered pairs at all. Consequently, there is no pair (a,b) that could serve as a counterexample with its reverse missing; the universal condition is therefore satisfied vacuously. Hence ∅ is symmetric on every set. It is not reflexive when A is non-empty, because reflexivity would require all pairs (a,a), and it is certainly not universal or a diagonal-only relation.
Frequently asked questions
What is the correct answer to this question?
Because it has no pair that can violate the condition
Why is this the correct answer?
A relation is symmetric if, for every pair (a,b) in the relation, the reverse pair (b,a) is also in it. The empty relation contains no ordered pairs at all. Consequently, there is no pair (a,b) that could serve as a counterexample with its reverse missing; the universal condition is therefore satisfied vacuously. Hence ∅ is symmetric on every set. It is not reflexive when A is non-empty, because reflexivity would require all pairs (a,a), and it is certainly not universal or a diagonal-only relation.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Symmetric relation.
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