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Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.
- a)R = {(1, 2), (2, 1)}
- b)R = {(1, 2), (1, 3), (1, 4)}
- c)R = {(1, 1), (2, 2), (3, 3)}
- d)R = {(1, 1), (1, 2), (2, 3)}
Correct answer is option 'A'. Can you explain this answer?
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Which of the following relations is symmetric but neither reflexive no...
Symmetric, Reflexive and Transitive Relations
Symmetric Relation:
A relation R on a set A is said to be symmetric if (a, b) ∈ R implies (b, a) ∈ R for every (a, b) ∈ R.
Reflexive Relation:
A relation R on a set A is said to be reflexive if (a, a) ∈ R for every a ∈ A.
Transitive Relation:
A relation R on a set A is said to be transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R for every a, b, c ∈ A.
Analysis of the given options:
a) R = {(1, 2), (2, 1)}:
- Symmetric: (1, 2) ∈ R implies (2, 1) ∈ R, so it is symmetric.
- Reflexive: (1, 1) and (2, 2) are not in R, so it is not reflexive.
- Transitive: (1, 2) ∈ R and (2, 1) ∈ R, but (1, 1) is not in R, so it is not transitive.
b) R = {(1, 2), (1, 3), (1, 4)}:
- Symmetric: (1, 2) ∈ R implies (2, 1) is not in R, so it is not symmetric.
- Reflexive: (1, 1) is not in R, so it is not reflexive.
- Transitive: There are no two pairs (a, b) and (b, c) in R, so it is transitive vacuously.
c) R = {(1, 1), (2, 2), (3, 3)}:
- Symmetric: All pairs are of the form (a, a), so it is symmetric vacuously.
- Reflexive: All pairs are of the form (a, a), so it is reflexive vacuously.
- Transitive: All pairs are of the form (a, a), so it is transitive vacuously.
d) R = {(1, 1), (1, 2), (2, 3)}:
- Symmetric: (1, 2) ∈ R implies (2, 1) is not in R, so it is not symmetric.
- Reflexive: (2, 2) and (3, 3) are not in R, so it is not reflexive.
- Transitive: (1, 2) ∈ R and (2, 3) ∈ R, but (1, 3) is not in R, so it is not transitive.
Therefore, option 'A' is the only relation that is symmetric but neither reflexive nor transitive for the given set A = {1, 2, 3}.
Symmetric Relation:
A relation R on a set A is said to be symmetric if (a, b) ∈ R implies (b, a) ∈ R for every (a, b) ∈ R.
Reflexive Relation:
A relation R on a set A is said to be reflexive if (a, a) ∈ R for every a ∈ A.
Transitive Relation:
A relation R on a set A is said to be transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R for every a, b, c ∈ A.
Analysis of the given options:
a) R = {(1, 2), (2, 1)}:
- Symmetric: (1, 2) ∈ R implies (2, 1) ∈ R, so it is symmetric.
- Reflexive: (1, 1) and (2, 2) are not in R, so it is not reflexive.
- Transitive: (1, 2) ∈ R and (2, 1) ∈ R, but (1, 1) is not in R, so it is not transitive.
b) R = {(1, 2), (1, 3), (1, 4)}:
- Symmetric: (1, 2) ∈ R implies (2, 1) is not in R, so it is not symmetric.
- Reflexive: (1, 1) is not in R, so it is not reflexive.
- Transitive: There are no two pairs (a, b) and (b, c) in R, so it is transitive vacuously.
c) R = {(1, 1), (2, 2), (3, 3)}:
- Symmetric: All pairs are of the form (a, a), so it is symmetric vacuously.
- Reflexive: All pairs are of the form (a, a), so it is reflexive vacuously.
- Transitive: All pairs are of the form (a, a), so it is transitive vacuously.
d) R = {(1, 1), (1, 2), (2, 3)}:
- Symmetric: (1, 2) ∈ R implies (2, 1) is not in R, so it is not symmetric.
- Reflexive: (2, 2) and (3, 3) are not in R, so it is not reflexive.
- Transitive: (1, 2) ∈ R and (2, 3) ∈ R, but (1, 3) is not in R, so it is not transitive.
Therefore, option 'A' is the only relation that is symmetric but neither reflexive nor transitive for the given set A = {1, 2, 3}.
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Community Answer
Which of the following relations is symmetric but neither reflexive no...
A relation in a set A is said to be symmetric if (a1, a2)∈R implies that (a1, a2)∈R,for every a1, a2∈R.
Hence, for the given set A={1, 2, 3}, R={(1, 2), (2, 1)} is symmetric. It is not reflexive since every element is not related to itself and neither transitive as it does not satisfy the condition that for a given relation R in a set A if (a1, a2)∈R and (a2, a3)∈R implies that (a1, a3)∈ R for every a1, a2, a3∈R.
Hence, for the given set A={1, 2, 3}, R={(1, 2), (2, 1)} is symmetric. It is not reflexive since every element is not related to itself and neither transitive as it does not satisfy the condition that for a given relation R in a set A if (a1, a2)∈R and (a2, a3)∈R implies that (a1, a3)∈ R for every a1, a2, a3∈R.
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Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer?
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Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer?.
Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer?.
Solutions for Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.a)R = {(1, 2), (2, 1)}b)R = {(1, 2), (1, 3), (1, 4)}c)R = {(1, 1), (2, 2), (3, 3)}d)R = {(1, 1), (1, 2), (2, 3)}Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 12.
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