Question

A relation $R$ on set $A=\{1,2,3\}$ defined as $R=\{(1,2),(2,1),(2,2)\}$ is

• A. Reflexive only
• B. Reflexive and Transitive
• C. Symmetric and Transitive
• D. Symmetric only

Hint:

To test symmetry check pair reversal. If $(a,b)$ exists then $(b,a)$ must also exist.

Answer 1

The Correct Option is D

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: $\boxed{\text{Symmetric only}}$