If relation is both symmetric and anti-symmetric, what can be
If a relation R is both symmetric and anti-symmetric, what can be said about R?
- A.
R must contain no elements
- B.
R must contain all possible elements
- C.
R must contain only elements of the form (a, a)
- D.
R must contain only elements of the form (a, b) where a ≠ b
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Correct answer: C
Answer: The relation must consist only of pairs of the form (a, a) (i.e., it is a subset of the diagonal).
Reasoning:
Symmetry: If (a, b) is in R, then (b, a) must also be in R.
Anti-symmetry: If (a, b) and (b, a) are both in R, then a = b.
Combining these: if (a, b) were present with a ≠ b, symmetry would force (b, a) too, and anti-symmetry would then force a = b, a contradiction. Therefore no pair with a ≠ b can be in R.
Conclusion: R can only contain pairs (a, a). This includes the empty relation or any subset of the diagonal {(a,a) : a is in the set}.