Which of the following relations is asymmetric?
Which of the following relations is asymmetric?
Answer: B. R = {(1, 2), (3, 4)} — Definition: An asymmetric relation means that for every pair (a, b) in the relation, the reverse pair (b, a) is not in the relation. This also implies that no…
- A.
R = {(1, 2), (2, 1), (3, 3)}
- B.
R = {(1, 2), (3, 4)}
- C.
R = {(1, 1), (2, 2), (3, 3)}
- D.
R = {(1, 2), (2, 1), (3, 2)}
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Correct answer: B
Definition: An asymmetric relation means that for every pair (a, b) in the relation, the reverse pair (b, a) is not in the relation. This also implies that no reflexive pair (a, a) can be present.
For R = {(1, 2), (2, 1), (3, 3)}: Not asymmetric because it contains (1, 2) and its reverse (2, 1), and it also contains a reflexive pair (3, 3).
For R = {(1, 2), (3, 4)}: Asymmetric because neither (1, 2) nor (3, 4) has its reverse present, and there are no reflexive pairs.
For R = {(1, 1), (2, 2), (3, 3)}: Not asymmetric because it contains reflexive pairs (a, a), which asymmetry forbids.
For R = {(1, 2), (2, 1), (3, 2)}: Not asymmetric because it contains the mutual pair (1, 2) and (2, 1).
Conclusion: The asymmetric relation is {(1, 2), (3, 4)} because it satisfies the definition while the others violate it.