Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > If a relation R on the set {1, 2, 3} be defin...
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If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is
- a)reflexive
- b)transitive
- c)symmetric
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then ...
Concept:
Reflexive relation: Relation is reflexive If (a, a) ∈ R ∀ a ∈ A.
Symmetric relation: Relation is symmetric, If (a, b) ∈ R, then (b, a) ∈ R.
Transitive relation: Relation is transitive, If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R,
If the relation is reflexive, symmetric, and transitive, it is known as an equivalence relation.
Reflexive relation: Relation is reflexive If (a, a) ∈ R ∀ a ∈ A.
Symmetric relation: Relation is symmetric, If (a, b) ∈ R, then (b, a) ∈ R.
Transitive relation: Relation is transitive, If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R,
If the relation is reflexive, symmetric, and transitive, it is known as an equivalence relation.
Explanation:
Let A = {1, 2, 3}
The relation R is defined by R = {(1, 2)}
Since, (1, 1) ∉ R
∴ It is not reflexive.
Since, (1, 2) ∈ R but (2, 1) ∉ R
∴ It is not symmetric.
But there is no counter-example to disapprove of transitive condition.
∴ It is transitive.
Let A = {1, 2, 3}
The relation R is defined by R = {(1, 2)}
Since, (1, 1) ∉ R
∴ It is not reflexive.
Since, (1, 2) ∈ R but (2, 1) ∉ R
∴ It is not symmetric.
But there is no counter-example to disapprove of transitive condition.
∴ It is transitive.
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Community Answer
If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then ...
Explanation:
To determine if the relation R on the set {1, 2, 3} defined by R = {(1, 2)} is transitive, we need to check if for every (a, b) and (b, c) in R, the pair (a, c) is also in R.
In this case, R = {(1, 2)}. So, we need to check if (1, 2) and (2, 3) are both in R.
Since (1, 2) is in R, we check if (2, 3) is in R. However, there is no pair in R with 2 as the first element. Therefore, (1, 2) and (2, 3) are not both in R, and the relation R is not transitive.
Hence, the correct answer is option 'B' - R is not transitive.
To determine if the relation R on the set {1, 2, 3} defined by R = {(1, 2)} is transitive, we need to check if for every (a, b) and (b, c) in R, the pair (a, c) is also in R.
In this case, R = {(1, 2)}. So, we need to check if (1, 2) and (2, 3) are both in R.
Since (1, 2) is in R, we check if (2, 3) is in R. However, there is no pair in R with 2 as the first element. Therefore, (1, 2) and (2, 3) are not both in R, and the relation R is not transitive.
Hence, the correct answer is option 'B' - R is not transitive.
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