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Consider the following statements:
S1: The number of relations which are both reflexive and asymmetric is 0
S2: The number of relations which are both symmetric and asymmetric is 0
Which option is correct :
- a)Only S1 is true
- b)Only S2 is true
- c)Both are true
- d)None is true
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider the following statements:S1: The number of relations which a...
S1: No relation is possible which is both reflexive and asymmetric because to be reflexive, all self-loops must be there but asymmetric doesn’t allow self-loops. Hence 0 relations.
S2: Only a single relation, i.e. empty relation is both symmetric and asymmetric.
So only S1 is correct.
Most Upvoted Answer
Consider the following statements:S1: The number of relations which a...
The given statements are:
S1: The number of relations which are both reflexive and asymmetric is 0.
S2: The number of relations which are both symmetric and asymmetric is 0.
Explanation:
To understand the given statements, let's first define what reflexive, symmetric, and asymmetric relations are:
- A relation R on a set A is said to be reflexive if every element of A is related to itself. In other words, (a, a) ∈ R for every a ∈ A.
- A relation R on a set A is said to be symmetric if for every pair (a, b) ∈ R, the pair (b, a) is also in R.
- A relation R on a set A is said to be asymmetric if for every pair (a, b) ∈ R, the pair (b, a) is not in R.
S1: The number of relations which are both reflexive and asymmetric is 0.
For a relation to be both reflexive and asymmetric, it should satisfy the following conditions:
- For every element a in the set A, (a, a) should be in the relation R (reflexive property).
- For every pair (a, b) in the relation R, (b, a) should not be in the relation R (asymmetric property).
However, these conditions cannot be simultaneously satisfied because if (a, a) is in the relation R (reflexive property), then (a, a) = (a, a) (symmetric property). Hence, S1 is true.
S2: The number of relations which are both symmetric and asymmetric is 0.
For a relation to be both symmetric and asymmetric, it should satisfy the following conditions:
- For every pair (a, b) in the relation R, (b, a) should also be in the relation R (symmetric property).
- For every pair (a, b) in the relation R, (b, a) should not be in the relation R (asymmetric property).
These conditions cannot be simultaneously satisfied because if (a, b) is in the relation R (symmetric property), then (b, a) = (a, b) (symmetric property). Hence, S2 is true.
Conclusion:
Based on the explanations above, it can be concluded that only statement S1 is true. The number of relations that are both reflexive and asymmetric is indeed 0.
S1: The number of relations which are both reflexive and asymmetric is 0.
S2: The number of relations which are both symmetric and asymmetric is 0.
Explanation:
To understand the given statements, let's first define what reflexive, symmetric, and asymmetric relations are:
- A relation R on a set A is said to be reflexive if every element of A is related to itself. In other words, (a, a) ∈ R for every a ∈ A.
- A relation R on a set A is said to be symmetric if for every pair (a, b) ∈ R, the pair (b, a) is also in R.
- A relation R on a set A is said to be asymmetric if for every pair (a, b) ∈ R, the pair (b, a) is not in R.
S1: The number of relations which are both reflexive and asymmetric is 0.
For a relation to be both reflexive and asymmetric, it should satisfy the following conditions:
- For every element a in the set A, (a, a) should be in the relation R (reflexive property).
- For every pair (a, b) in the relation R, (b, a) should not be in the relation R (asymmetric property).
However, these conditions cannot be simultaneously satisfied because if (a, a) is in the relation R (reflexive property), then (a, a) = (a, a) (symmetric property). Hence, S1 is true.
S2: The number of relations which are both symmetric and asymmetric is 0.
For a relation to be both symmetric and asymmetric, it should satisfy the following conditions:
- For every pair (a, b) in the relation R, (b, a) should also be in the relation R (symmetric property).
- For every pair (a, b) in the relation R, (b, a) should not be in the relation R (asymmetric property).
These conditions cannot be simultaneously satisfied because if (a, b) is in the relation R (symmetric property), then (b, a) = (a, b) (symmetric property). Hence, S2 is true.
Conclusion:
Based on the explanations above, it can be concluded that only statement S1 is true. The number of relations that are both reflexive and asymmetric is indeed 0.
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Consider the following statements:S1: The number of relations which are both reflexive and asymmetric is 0S2: The number of relations which are both symmetric and asymmetric is 0Which option is correct :a)Only S1 is trueb)Only S2 is truec)Both are trued)None is trueCorrect answer is option 'A'. Can you explain this answer?
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Solutions for Consider the following statements:S1: The number of relations which are both reflexive and asymmetric is 0S2: The number of relations which are both symmetric and asymmetric is 0Which option is correct :a)Only S1 is trueb)Only S2 is truec)Both are trued)None is trueCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
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