Question:
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is
A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is
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Draw a three-by-three relation table. Check mirror symmetry across the diagonal, then look for missing diagonal loops and test the path $1\to2\to1$.
Updated On: Aug 14, 2026
- Reflexive only
- Reflexive and Transitive
- Symmetric and Transitive
- Symmetric only
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The Correct Option is D
Approach Solution - 1
Step 1: Check Reflexive Property.
A relation is reflexive if
\[ (a,a)\in R \] for every \(a\in A\).
Here \[ A=\{1,2,3\} \] For reflexive relation we must have \[ (1,1),(2,2),(3,3) \] But given relation contains only \[ (2,2) \] Thus the relation is not reflexive.
Step 2: Check Symmetric Property.
A relation is symmetric if \[ (a,b)\in R \Rightarrow (b,a)\in R \] Now observe the pairs:
\[ (1,2)\in R \] and \[ (2,1)\in R \] Thus symmetry condition is satisfied.
Step 3: Check Transitive Property.
A relation is transitive if \[ (a,b)\in R \text{ and } (b,c)\in R \] implies \[ (a,c)\in R \] Now \[ (1,2)\in R \] \[ (2,1)\in R \] Then transitivity requires \[ (1,1)\in R \] But \[ (1,1)\notin R \] Thus relation is not transitive.
Step 4: Conclusion.
The relation satisfies symmetry but does not satisfy reflexive or transitive properties.
Therefore the relation is symmetric only.
Final Answer: $\boxed{\text{Symmetric only}}$
A relation is reflexive if
\[ (a,a)\in R \] for every \(a\in A\).
Here \[ A=\{1,2,3\} \] For reflexive relation we must have \[ (1,1),(2,2),(3,3) \] But given relation contains only \[ (2,2) \] Thus the relation is not reflexive.
Step 2: Check Symmetric Property.
A relation is symmetric if \[ (a,b)\in R \Rightarrow (b,a)\in R \] Now observe the pairs:
\[ (1,2)\in R \] and \[ (2,1)\in R \] Thus symmetry condition is satisfied.
Step 3: Check Transitive Property.
A relation is transitive if \[ (a,b)\in R \text{ and } (b,c)\in R \] implies \[ (a,c)\in R \] Now \[ (1,2)\in R \] \[ (2,1)\in R \] Then transitivity requires \[ (1,1)\in R \] But \[ (1,1)\notin R \] Thus relation is not transitive.
Step 4: Conclusion.
The relation satisfies symmetry but does not satisfy reflexive or transitive properties.
Therefore the relation is symmetric only.
Final Answer: $\boxed{\text{Symmetric only}}$
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Approach Solution -2
Concept:
Step 1: Place the relation pairs in the table.
The marks are at $(1,2)$, $(2,1)$, and $(2,2)$.
The off-diagonal marks $(1,2)$ and $(2,1)$ mirror each other, and $(2,2)$ lies on the diagonal.
Therefore, the relation is symmetric.
Step 2: Inspect the diagonal for reflexivity.
A reflexive relation on $A=\{1,2,3\}$ needs $(1,1)$, $(2,2)$, and $(3,3)$.
The first and third loops are missing, so the relation is not reflexive.
Step 3: Follow a two-link path for transitivity.
The links $1\to2$ and $2\to1$ create a two-step path from $1$ back to $1$.
Transitivity would require $(1,1)$, but that pair is absent. Therefore, the relation is not transitive.
Final Answer: Symmetric only, option D.
- View the relation as marks in a $3\times3$ adjacency table.
- Symmetry appears as mirror-image marks across the main diagonal, while reflexivity requires every diagonal mark.
Step 1: Place the relation pairs in the table.
The marks are at $(1,2)$, $(2,1)$, and $(2,2)$.
The off-diagonal marks $(1,2)$ and $(2,1)$ mirror each other, and $(2,2)$ lies on the diagonal.
Therefore, the relation is symmetric.
Step 2: Inspect the diagonal for reflexivity.
A reflexive relation on $A=\{1,2,3\}$ needs $(1,1)$, $(2,2)$, and $(3,3)$.
The first and third loops are missing, so the relation is not reflexive.
Step 3: Follow a two-link path for transitivity.
The links $1\to2$ and $2\to1$ create a two-step path from $1$ back to $1$.
Transitivity would require $(1,1)$, but that pair is absent. Therefore, the relation is not transitive.
Final Answer: Symmetric only, option D.
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