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The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set, A = {1, 2, 3} is
- a)reflexive but not symmetric
- b)reflexive but not transitive
- c)symmetric and transitive
- d)neither symmetric nor transitive
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set, A...
(a) Since, (1 ,1 ); (2 ,2 ); (3 ,3 ) ∈ R
therefore R is reflexive.
(1, 2) ∈ R but (2, 1) ∉ R, therefore R is not symmetric. It can be easily seen that R is transitive.
(c) Here,R{(l, 3), (2,2); (3,2)}, &= {(2,1); (3,2); (3,2); (2 ,3 )}. Then RoS = {(2,3), (3,2); (2,2)}
therefore R is reflexive.
(1, 2) ∈ R but (2, 1) ∉ R, therefore R is not symmetric. It can be easily seen that R is transitive.
(c) Here,R{(l, 3), (2,2); (3,2)}, &= {(2,1); (3,2); (3,2); (2 ,3 )}. Then RoS = {(2,3), (3,2); (2,2)}
Most Upvoted Answer
The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set, A...
Explanation:
The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set A = {1, 2, 3} is defined by the given set of ordered pairs.
Reflexive:
A relation R on a set A is 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.
Let's check if the relation R is reflexive:
- (1, 1) is in R
- (2, 2) is in R
- (3, 3) is in R
So, for every element 'a' in A, (a, a) is in R. Hence, the relation R is reflexive.
Symmetric:
A relation R on a set A is symmetric if for every pair (a, b) in R, (b, a) must also be in R.
Let's check if the relation R is symmetric:
- (1, 2) is in R, but (2, 1) is not in R
- (2, 3) is in R, but (3, 2) is not in R
- (1, 3) is in R, but (3, 1) is not in R
Since there exist pairs (a, b) in R such that (b, a) is not in R, the relation R is not symmetric.
Transitive:
A relation R on a set A is transitive if for every pair (a, b) and (b, c) in R, (a, c) must also be in R.
Let's check if the relation R is transitive:
- (1, 2) and (2, 3) are in R, but (1, 3) is also in R
Since for every pair (a, b) and (b, c) in R, (a, c) is also in R, the relation R is transitive.
Conclusion:
The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set A = {1, 2, 3} is reflexive but not symmetric. It is also transitive. Therefore, the correct answer is option 'A' (reflexive but not symmetric).
The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set A = {1, 2, 3} is defined by the given set of ordered pairs.
Reflexive:
A relation R on a set A is 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.
Let's check if the relation R is reflexive:
- (1, 1) is in R
- (2, 2) is in R
- (3, 3) is in R
So, for every element 'a' in A, (a, a) is in R. Hence, the relation R is reflexive.
Symmetric:
A relation R on a set A is symmetric if for every pair (a, b) in R, (b, a) must also be in R.
Let's check if the relation R is symmetric:
- (1, 2) is in R, but (2, 1) is not in R
- (2, 3) is in R, but (3, 2) is not in R
- (1, 3) is in R, but (3, 1) is not in R
Since there exist pairs (a, b) in R such that (b, a) is not in R, the relation R is not symmetric.
Transitive:
A relation R on a set A is transitive if for every pair (a, b) and (b, c) in R, (a, c) must also be in R.
Let's check if the relation R is transitive:
- (1, 2) and (2, 3) are in R, but (1, 3) is also in R
Since for every pair (a, b) and (b, c) in R, (a, c) is also in R, the relation R is transitive.
Conclusion:
The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set A = {1, 2, 3} is reflexive but not symmetric. It is also transitive. Therefore, the correct answer is option 'A' (reflexive but not symmetric).
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The relation R = {(1, 1), (2,2), (3,3), (1,2), (2,3), (1,3)} on set, A= {1, 2, 3} isa)reflexive but not symmetricb)reflexive but not transitivec)symmetric and transitived)neither symmetric nor transitiveCorrect answer is option 'A'. Can you explain this answer?
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