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
For equivalence relation questions, always verify the identity pairs \( (a, a) \) first to confirm reflexivity. If any identity pair is missing, it cannot be an equivalence relation.
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.
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
• A relation is reflexive if \( (a, a) \in R \) for all \( a \) in the set.
• A relation is symmetric if \( (a, b) \in R \implies (b, a) \in R \).
• A relation is transitive if \( (a, b) \in R \) and \( (b, c) \in R \implies (a, c) \in R \).
• An equivalence relation satisfies all three properties.
Step 1: Analyze the Assertion (A)
Set \( S = \{1, 2, 3\} \). Given relation \( R = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)\} \).
1. Reflexive: Since \( (1,1), (2,2), (3,3) \in R \), it is reflexive.
2. Symmetric: Here \( (1,2) \in R \) and its reverse \( (2,1) \) is also in \( R \). All others are identity pairs which are symmetric. So, it is symmetric.
3. Transitive: Since \( (1,2) \in R \) and \( (2,1) \in R \), we check if \( (1,1) \in R \). Yes. Similarly for other pairs. It is transitive.
Since all three hold, Assertion (A) is true.
Step 2: Analyze the Reason (R)
The Reason (R) states the standard definition of an equivalence relation.
Since this definition is universally true in set theory, Reason (R) is true.
Step 3: Determine if Reason explains Assertion
The Assertion states that a specific relation is an equivalence relation. The Reason provides the criteria for a relation to be classified as such.
Because the Assertion was verified using the exact criteria mentioned in the Reason, Reason (R) is the correct explanation.
• A relation is reflexive if \( (a, a) \in R \) for all \( a \) in the set.
• A relation is symmetric if \( (a, b) \in R \implies (b, a) \in R \).
• A relation is transitive if \( (a, b) \in R \) and \( (b, c) \in R \implies (a, c) \in R \).
• An equivalence relation satisfies all three properties.
Step 1: Analyze the Assertion (A)
Set \( S = \{1, 2, 3\} \). Given relation \( R = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)\} \).
1. Reflexive: Since \( (1,1), (2,2), (3,3) \in R \), it is reflexive.
2. Symmetric: Here \( (1,2) \in R \) and its reverse \( (2,1) \) is also in \( R \). All others are identity pairs which are symmetric. So, it is symmetric.
3. Transitive: Since \( (1,2) \in R \) and \( (2,1) \in R \), we check if \( (1,1) \in R \). Yes. Similarly for other pairs. It is transitive.
Since all three hold, Assertion (A) is true.
Step 2: Analyze the Reason (R)
The Reason (R) states the standard definition of an equivalence relation.
Since this definition is universally true in set theory, Reason (R) is true.
Step 3: Determine if Reason explains Assertion
The Assertion states that a specific relation is an equivalence relation. The Reason provides the criteria for a relation to be classified as such.
Because the Assertion was verified using the exact criteria mentioned in the Reason, Reason (R) is the correct explanation.
Was this answer helpful?
0
0
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
View More Questions
Top CBSE CLASS XII Relations and its properties Questions
- A relation $R$ on the set $A = \{1, 2, 3\}$ defined as $R = \{(1, 2), (2, 1), (2, 2)\}$ is:
- A relation \(R\) on \[ A=\{1,2,3\} \] is defined as \[ R=\{(1,1),(3,3),(1,2)\}. \] Is \(R\) a symmetric relation? Justify your answer.
Write the smallest relation set \(R_1\) such that \[ R\cup R_1 \] becomes an equivalence relation on the set \[ \{1,2,3\}. \] - Let $n$ be a fixed positive integer. A relation $R$ is defined in the set of integers $\mathbb{Z}$ such that $R = \{(x, y) : (x - y) \text{ is divisible by } n, x, y \in \mathbb{Z}\}$. Determine if $R$ is an equivalence relation.
- A relation \(R\) is defined on \(Z\), the set of integers, as \[ R = \{(x,y) : |x-y| \text{ is divisible by a prime number } 'p', x,y \in Z\}. \] Check whether \(R\) is an equivalence relation or not.
- A relation \( R \) on set \( A = \{1, 2, 3\} \) is defined as \( R = \{(1, 3), (3, 3), (1, 1), (2, 2), (3, 1)\ \) is}
View More Questions
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
View More Questions