Mathematics Exam > Mathematics Questions > If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2,...
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If, A = {1,2,3,4} and
R = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4) is a relation on A x A, then which one of the following is correct?
R = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4) is a relation on A x A, then which one of the following is correct?
- a)R is reflexive
- b)R is symmetric and transitive
- c)R is reflexive and symmetric
- d)R is neither reflexive nor transitive
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3...
Correct Answer :- c
Explanation : Let A = {1,2,3,4} and R be a relation on A given by R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(3,1),(1,3)}.
Now for all 1,2,3,4 ∈ A, (1,1),(2,2),(3,3),(4,4) ∈ R, this gives the relation R is reflexive.
Again for (1,2) ∈ R
⇒(2,1)∈R and (1,3)∈R
⇒(3,1)∈R for 1,2∈A.
This gives the relation R is symmetric.
But the relation is not transitive as (2,1),(1,3)∈R but (2,3)does not ∈ R.
Most Upvoted Answer
If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3...
R is transitive, but not reflexive.
Explanation:
Reflexive property:
A relation R on a set A is said to be reflexive if every element of A is related to itself. In other words, for every element a in A, (a, a) must be in R.
In the given relation R = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4)}, we can see that (2, 2) is missing. Since the element 2 is not related to itself, R is not reflexive.
Transitive property:
A relation R on a set A is said to be transitive if for every (a, b) and (b, c) in R, (a, c) is also in R.
In the given relation R = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4)}, we can observe that for (1, 3) and (3, 1), we have (1, 1) and (3, 1) in R. But (1, 1) and (3, 1) imply (1, 1). Therefore, (1, 3) and (3, 1) imply (1, 1) which is in R. Hence, R is transitive.
Conclusion:
Based on the above analysis, we can conclude that the given relation R is transitive, but not reflexive. Therefore, the correct option is C) R is transitive, but not reflexive.
Explanation:
Reflexive property:
A relation R on a set A is said to be reflexive if every element of A is related to itself. In other words, for every element a in A, (a, a) must be in R.
In the given relation R = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4)}, we can see that (2, 2) is missing. Since the element 2 is not related to itself, R is not reflexive.
Transitive property:
A relation R on a set A is said to be transitive if for every (a, b) and (b, c) in R, (a, c) is also in R.
In the given relation R = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4)}, we can observe that for (1, 3) and (3, 1), we have (1, 1) and (3, 1) in R. But (1, 1) and (3, 1) imply (1, 1). Therefore, (1, 3) and (3, 1) imply (1, 1) which is in R. Hence, R is transitive.
Conclusion:
Based on the above analysis, we can conclude that the given relation R is transitive, but not reflexive. Therefore, the correct option is C) R is transitive, but not reflexive.
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If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3...
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If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4) is a relation on A x A, then which one of the following is correct?a)R is reflexiveb)R is symmetric and transitivec)R is reflexive andsymmetricd)R is neither reflexive nor transitiveCorrect answer is option 'C'. Can you explain this answer?
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If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4) is a relation on A x A, then which one of the following is correct?a)R is reflexiveb)R is symmetric and transitivec)R is reflexive andsymmetricd)R is neither reflexive nor transitiveCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4) is a relation on A x A, then which one of the following is correct?a)R is reflexiveb)R is symmetric and transitivec)R is reflexive andsymmetricd)R is neither reflexive nor transitiveCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If, A = {1,2,3,4} andR = {(1, 1), (1, 3), (2, 2), (3, 1) (3, 4), (4, 3) (4, 4) is a relation on A x A, then which one of the following is correct?a)R is reflexiveb)R is symmetric and transitivec)R is reflexive andsymmetricd)R is neither reflexive nor transitiveCorrect answer is option 'C'. Can you explain this answer?.
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