R = {(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)} on the set A = {1,2,3} is:

Option a: reflexive but not symmetric. Solution: We are given set A = {1,2,3} and relation R = {(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)}. Let's check the properties of R: 1. Reflexive: For R to be reflexive, it must contain (1,1), (2,2), (3,3) since A = {1, 2, 3}. Since these pairs are in R, the relation is reflexive. 2. Symmetric: For R to be symmetric, (a, b) in R implies (b, a) in R. Here, (1, 2) in R but (2, 1) notin R. Thus, the relation is not symmetric. 3. Transitive: For R to be transitive, (a, b) in R text{ and } (b, c) in R implies (a, c) in R. Here, (1, 2) in R and (2, 3) in R implies (1, 3) in R. Thus, the relation is transitive. Therefore, the relation R is reflexive but not symmetric. Hence, Option A is the correct answer.

$R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$ on the set $A = \{1,2,3\}$ is:

The correct answer is Option a: reflexive but not symmetric. We are given set $A = \{1,2,3\}$ and relation $R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$. Let's check the properties of $R$: 1. **Reflexive**: For $R$ to be reflexive, it must contain $(1,1), (2,2), (3,3)$ since $A = \{1, 2, 3\}$. Since these pairs are in $R$, the relation is reflexive. 2. **Symmetric**: For $R$ to be symmetric, $(a, b) \in R \implies (b, a) \in R$. Here, $(1, 2) \in R$ but $(2, 1) \notin R$. Thus, the relation is not symmetric. 3. **Transitive**: For $R$ to be transitive, $(a, b) \in R \text{ and } (b, c) \in R \implies (a, c) \in R$. Here, $(1, 2) \in R$ and $(2, 3) \in R \implies (1, 3) \in R$. Thus, the relation is transitive. Therefore, the relation $R$ is reflexive but not symmetric. Hence, **Option A** is the correct answer.

Set, Relations and FunctionsMTP Dec 22 Series IQuestion 1956 of 217

All Questions

$\displaystyle R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$ on the set $\displaystyle A = \{1,2,3\}$ is:

Options

Areflexive but not symmetric

Bsymmetric but not transitive

Csymmetric and transitive

Dneither symmetric nor transitive

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Correct Answer

✅ Option a — reflexive but not symmetric

All Options:

Areflexive but not symmetric
Bsymmetric but not transitive
Csymmetric and transitive
Dneither symmetric nor transitive

Detailed Solution & Explanation

We are given set

\displaystyle A = \{1,2,3\}

and relation

\displaystyle R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}

.
Let's check the properties of

\displaystyle R

:
1. **Reflexive**: For

\displaystyle R

to be reflexive, it must contain

\displaystyle (1,1), (2,2), (3,3)

since

\displaystyle A = \{1, 2, 3\}

.
Since these pairs are in

\displaystyle R

, the relation is reflexive.
2. **Symmetric**: For

\displaystyle R

to be symmetric,

\displaystyle (a, b) \in R \implies (b, a) \in R

.
Here,

\displaystyle (1, 2) \in R

but

\displaystyle (2, 1) \notin R

. Thus, the relation is not symmetric.
3. **Transitive**: For

\displaystyle R

to be transitive,

\displaystyle (a, b) \in R \text{ and } (b, c) \in R \implies (a, c) \in R

.
Here,

\displaystyle (1, 2) \in R

and

\displaystyle (2, 3) \in R \implies (1, 3) \in R

. Thus, the relation is transitive.
Therefore, the relation

\displaystyle R

is reflexive but not symmetric.
Hence, **Option A** is the correct answer.

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R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}\displaystyle R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)} on the set A={1,2,3}\displaystyle A = \{1,2,3\}A={1,2,3} is: