Answer: Symmetry and transitivity hold vacuously with no pairs to check, but reflexivity actively demands that (a, a) be present.
- A The empty relation is not a valid relation on any set by definition
- B Reflexivity holds vacuously as well, so the relation is in fact an equivalence relation
- C Reflexivity requires the set itself to be empty in all cases
- D Symmetry and transitivity hold vacuously with no pairs to check, but reflexivity actively demands that (a, a) be present
Correct answer: D. Symmetry and transitivity hold vacuously with no pairs to check, but reflexivity actively demands that (a, a) be present
Explanation: Symmetry and transitivity are conditional statements satisfied when the hypothesis never occurs; reflexivity is an existence requirement that fails.
Concept context
Types of relations, equivalence classes, one-one and onto functions, composition, and invertible functions