Question:
Assertion (A) : A relation R on the set {1, 2, 3\ defined as R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} is an equivalence relation.
Reason (R) : A relation that is reflexive, symmetric and transitive is an equivalence relation.}
Assertion (A) : A relation R on the set {1, 2, 3\ defined as R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} is an equivalence relation.
Reason (R) : A relation that is reflexive, symmetric and transitive is an equivalence relation.}
Reason (R) : A relation that is reflexive, symmetric and transitive is an equivalence relation.}
Show Hint
To verify an equivalence relation on a small set, check the diagonal elements \((a,a)\) first for reflexivity.
In Assertion-Reason questions, ask "Is A true?", "Is R true?", then "Is A true BECAUSE of R?".
In Assertion-Reason questions, ask "Is A true?", "Is R true?", then "Is A true BECAUSE of R?".
Updated On: Sep 10, 2026
- Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
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The Correct Option is A
Solution and Explanation
Concept:
• Reflexive: \( (a, a) \in R \) for every \( a \in A \).
• Symmetric: \( (a, b) \in R \Rightarrow (b, a) \in R \).
• Transitive: \( (a, b) \in R \) and \( (b, c) \in R \Rightarrow (a, c) \in R \).
• Equivalence: A relation is equivalence if it satisfies all three properties.
Step 1: Evaluate the Assertion (A) for reflexivity
Set \( A = \{1, 2, 3\} \). For \( R \) to be reflexive, it must contain \( (1, 1), (2, 2), (3, 3) \). Check \( R \): \( (1, 1) \in R, (2, 2) \in R, (3, 3) \in R \). Thus, \( R \) is reflexive.
Step 2: Evaluate for symmetry and transitivity
Symmetry: \( (1, 2) \in R \) and its flip \( (2, 1) \in R \). All other pairs are self-symmetric. So, \( R \) is symmetric.
Transitivity: We have \( (1, 2) \) and \( (2, 1) \). Their combination \( (1, 1) \) is in \( R \). Also \( (2, 1) \) and \( (1, 2) \) gives \( (2, 2) \), which is in \( R \). No other non-trivial chains exist. So, \( R \) is transitive.
Since it satisfies all three, Assertion (A) is true.
Step 3: Evaluate the Reason (R) and its relationship to (A)
Reason (R) states the standard definition of an equivalence relation. This is a true statement. Furthermore, the assertion is categorized as an equivalence relation precisely because it fulfills the conditions of being reflexive, symmetric, and transitive. Thus, Reason (R) correctly explains Assertion (A).
• Reflexive: \( (a, a) \in R \) for every \( a \in A \).
• Symmetric: \( (a, b) \in R \Rightarrow (b, a) \in R \).
• Transitive: \( (a, b) \in R \) and \( (b, c) \in R \Rightarrow (a, c) \in R \).
• Equivalence: A relation is equivalence if it satisfies all three properties.
Step 1: Evaluate the Assertion (A) for reflexivity
Set \( A = \{1, 2, 3\} \). For \( R \) to be reflexive, it must contain \( (1, 1), (2, 2), (3, 3) \). Check \( R \): \( (1, 1) \in R, (2, 2) \in R, (3, 3) \in R \). Thus, \( R \) is reflexive.
Step 2: Evaluate for symmetry and transitivity
Symmetry: \( (1, 2) \in R \) and its flip \( (2, 1) \in R \). All other pairs are self-symmetric. So, \( R \) is symmetric.
Transitivity: We have \( (1, 2) \) and \( (2, 1) \). Their combination \( (1, 1) \) is in \( R \). Also \( (2, 1) \) and \( (1, 2) \) gives \( (2, 2) \), which is in \( R \). No other non-trivial chains exist. So, \( R \) is transitive.
Since it satisfies all three, Assertion (A) is true.
Step 3: Evaluate the Reason (R) and its relationship to (A)
Reason (R) states the standard definition of an equivalence relation. This is a true statement. Furthermore, the assertion is categorized as an equivalence relation precisely because it fulfills the conditions of being reflexive, symmetric, and transitive. Thus, Reason (R) correctly explains Assertion (A).
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