Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > Let A = {1, 2, 3} and consider the relation R...
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Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R 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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Most Upvoted Answer
Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3...
Understanding the Relation R
Let’s analyze the relation \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\} \) defined on the set \( A = \{1, 2, 3\} \).
Reflexivity
- A relation is reflexive if every element in the set is related to itself.
- In \( R \), we have:
- \( (1, 1) \)
- \( (2, 2) \)
- \( (3, 3) \)
- All elements of \( A \) are present in \( R \) as pairs related to themselves.
- Thus, R is reflexive.
Symmetry
- A relation is symmetric if for every \( (a, b) \) in \( R \), \( (b, a) \) is also in \( R \).
- Checking pairs:
- For \( (1, 2) \), \( (2, 1) \) is not in \( R \).
- For \( (2, 3) \), \( (3, 2) \) is not in \( R \).
- For \( (1, 3) \), \( (3, 1) \) is not in \( R \).
- Since not all pairs satisfy the condition, R is not symmetric.
Transitivity
- A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).
- Check:
- From \( (1, 2) \) and \( (2, 3) \), \( (1, 3) \) is in \( R \) (valid).
- However, there are no other combinations that violate transitivity.
- Thus, R is transitive.
Conclusion
- Hence, the correct classification for the relation \( R \) is that it is reflexive but not symmetric. Therefore, the correct answer is option 'A'.
Let’s analyze the relation \( R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\} \) defined on the set \( A = \{1, 2, 3\} \).
Reflexivity
- A relation is reflexive if every element in the set is related to itself.
- In \( R \), we have:
- \( (1, 1) \)
- \( (2, 2) \)
- \( (3, 3) \)
- All elements of \( A \) are present in \( R \) as pairs related to themselves.
- Thus, R is reflexive.
Symmetry
- A relation is symmetric if for every \( (a, b) \) in \( R \), \( (b, a) \) is also in \( R \).
- Checking pairs:
- For \( (1, 2) \), \( (2, 1) \) is not in \( R \).
- For \( (2, 3) \), \( (3, 2) \) is not in \( R \).
- For \( (1, 3) \), \( (3, 1) \) is not in \( R \).
- Since not all pairs satisfy the condition, R is not symmetric.
Transitivity
- A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).
- Check:
- From \( (1, 2) \) and \( (2, 3) \), \( (1, 3) \) is in \( R \) (valid).
- However, there are no other combinations that violate transitivity.
- Thus, R is transitive.
Conclusion
- Hence, the correct classification for the relation \( R \) is that it is reflexive but not symmetric. Therefore, the correct answer is option 'A'.
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Community Answer
Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3...
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:
Given that, A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}.
Now,
(1,1),(2,2),(3,3) ∈ R
⇒ R is reflexive.
(1,2),(2,3),(1,3) ∈ R but (2,1),(3,2),(3,1) ∉ R
⇒ R is not symmetric.
Also, (1,2) ∈ R and (2,3) ∈ R ⇒ (1,3) ∈ R
⇒ R is transitive.
∴ R is reflexive, and transitive but not symmetric.
Given that, A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}.
Now,
(1,1),(2,2),(3,3) ∈ R
⇒ R is reflexive.
(1,2),(2,3),(1,3) ∈ R but (2,1),(3,2),(3,1) ∉ R
⇒ R is not symmetric.
Also, (1,2) ∈ R and (2,3) ∈ R ⇒ (1,3) ∈ R
⇒ R is transitive.
∴ R is reflexive, and transitive but not symmetric.
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Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R 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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