The number of a relation(s) which is/are symmetric and asymmetric on a set of…
The number of a relation(s) which is/are symmetric and asymmetric on a set of cardinality n is,
Answer: B. 1 — Answer: 1 Explanation: Symmetric: If a pair (x,y) is in the relation, then (y,x) must also be in the relation for all x,y. Asymmetric: If a pair (x,y) is in…
- A.
0
- B.
1
- C.
2n^2-n/2
- D.
Data insufficient
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Correct answer: B
Answer: 1
Explanation:
Symmetric: If a pair (x,y) is in the relation, then (y,x) must also be in the relation for all x,y.
Asymmetric: If a pair (x,y) is in the relation, then (y,x) must not be in the relation for all x,y. In particular, no reflexive pair (x,x) can be present.
Combining these properties shows that no off-diagonal pair (x,y) with x ≠ y can be present: symmetry would require its reverse, but asymmetry forbids that. Also no diagonal pair (x,x) can be present because asymmetry forbids reflexive pairs. Therefore every possible ordered pair must be absent, so the relation must be the empty relation.
Hence there is exactly one relation that is both symmetric and asymmetric: the empty relation.