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The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)},on the set A = {1 , 2 , 3, 4} is
- a)Reflexive, symmetric and transitive
- b)Neither reflexive, nor irreflexive but transitive
- c)Irreflexive, symmetric and transitive
- d)Irreflexive and antisymmetric
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1...
The relation R doesn’t contain (4, 4), so R is not reflexive relation.
Since relation R contains ( 1 ,1 ) , (2, 2) and (3, 3).
Therefore, relation R is also not irreflexive.
That R is transitive, can be checked by systematically checking for all (a, b) and (b, c) in R, whether (a, c) also exists in R.
Since relation R contains ( 1 ,1 ) , (2, 2) and (3, 3).
Therefore, relation R is also not irreflexive.
That R is transitive, can be checked by systematically checking for all (a, b) and (b, c) in R, whether (a, c) also exists in R.
Most Upvoted Answer
The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1...
Given binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)} on set A = {1, 2, 3, 4}.
Reflexive
A relation R on set A is reflexive if every element of A is related to itself.
For R to be reflexive, (1, 1), (2, 2), (3, 3), and (4, 4) must be present in R.
But (4, 4) is not present in R, hence it is not reflexive.
Irreflexive
A relation R on set A is irreflexive if no element of A is related to itself.
For R to be irreflexive, no pair of the form (a, a) should be present in R.
But (1, 1), (2, 2), and (3, 3) are present in R, hence it is not irreflexive.
Symmetric
A relation R on set A is symmetric if for every (a, b) in R, (b, a) is also in R.
For R to be symmetric, the following pairs must be present in R:
(1, 1), (2, 2), (3, 3), (1, 3), (1, 2), (2, 1), (3, 1), (2, 3), (4, 4).
But (1, 2) is present in R, but (2, 1) is not, hence it is not symmetric.
Antisymmetric
A relation R on set A is antisymmetric if for every (a, b) and (b, a) in R, a = b.
For R to be antisymmetric, there should be no pairs (a, b) and (b, a) in R where a ≠ b.
Since (2, 1) and (1, 2) are both present in R, and 2 ≠ 1, R is not antisymmetric.
Transitive
A relation R on set A is transitive if for every (a, b) and (b, c) in R, (a, c) is also in R.
For R to be transitive, the following pairs must be present in R:
(1, 1), (2, 2), (3, 3), (1, 3), (1, 2), (2, 1), (3, 1), (2, 3), (3, 4), (2, 4).
All possible pairs (a, b) and (b, c) are present in R, and their corresponding (a, c) pairs are also present in R. Hence, R is transitive.
Conclusion
From the above analysis, we can conclude that R is neither reflexive nor irreflexive, not symmetric nor antisymmetric, but transitive. Hence, the correct answer is option 'B'.
Reflexive
A relation R on set A is reflexive if every element of A is related to itself.
For R to be reflexive, (1, 1), (2, 2), (3, 3), and (4, 4) must be present in R.
But (4, 4) is not present in R, hence it is not reflexive.
Irreflexive
A relation R on set A is irreflexive if no element of A is related to itself.
For R to be irreflexive, no pair of the form (a, a) should be present in R.
But (1, 1), (2, 2), and (3, 3) are present in R, hence it is not irreflexive.
Symmetric
A relation R on set A is symmetric if for every (a, b) in R, (b, a) is also in R.
For R to be symmetric, the following pairs must be present in R:
(1, 1), (2, 2), (3, 3), (1, 3), (1, 2), (2, 1), (3, 1), (2, 3), (4, 4).
But (1, 2) is present in R, but (2, 1) is not, hence it is not symmetric.
Antisymmetric
A relation R on set A is antisymmetric if for every (a, b) and (b, a) in R, a = b.
For R to be antisymmetric, there should be no pairs (a, b) and (b, a) in R where a ≠ b.
Since (2, 1) and (1, 2) are both present in R, and 2 ≠ 1, R is not antisymmetric.
Transitive
A relation R on set A is transitive if for every (a, b) and (b, c) in R, (a, c) is also in R.
For R to be transitive, the following pairs must be present in R:
(1, 1), (2, 2), (3, 3), (1, 3), (1, 2), (2, 1), (3, 1), (2, 3), (3, 4), (2, 4).
All possible pairs (a, b) and (b, c) are present in R, and their corresponding (a, c) pairs are also present in R. Hence, R is transitive.
Conclusion
From the above analysis, we can conclude that R is neither reflexive nor irreflexive, not symmetric nor antisymmetric, but transitive. Hence, the correct answer is option 'B'.
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The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)},on the set A = {1 , 2 , 3, 4} isa)Reflexive, symmetric and transitiveb)Neither reflexive, nor irreflexive but transitivec)Irreflexive, symmetric and transitived)Irreflexive and antisymmetricCorrect answer is option 'B'. Can you explain this answer?
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The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)},on the set A = {1 , 2 , 3, 4} isa)Reflexive, symmetric and transitiveb)Neither reflexive, nor irreflexive but transitivec)Irreflexive, symmetric and transitived)Irreflexive and antisymmetricCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)},on the set A = {1 , 2 , 3, 4} isa)Reflexive, symmetric and transitiveb)Neither reflexive, nor irreflexive but transitivec)Irreflexive, symmetric and transitived)Irreflexive and antisymmetricCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)},on the set A = {1 , 2 , 3, 4} isa)Reflexive, symmetric and transitiveb)Neither reflexive, nor irreflexive but transitivec)Irreflexive, symmetric and transitived)Irreflexive and antisymmetricCorrect answer is option 'B'. Can you explain this answer?.
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