Question

If a relation $R$ in the set $\{1,2,3\}$ be defined by $R=\{(1,1),(2,2)\}$, then $R$ is

• A. Symmetric but not transitive
• B. Transitive but not symmetric
• C. Symmetric and transitive
• D. Neither symmetric nor transitive

Hint:

Pairs of the form $(a,a)$ always satisfy symmetry and transitivity.

Answer 1

The Correct Option is C

Step 1: Write the given relation.
The relation is defined on the set \[ A=\{1,2,3\} \] and \[ R=\{(1,1),(2,2)\} \] Thus the relation contains only ordered pairs where each element is related to itself.

Step 2: Check the symmetric property.
A relation is symmetric if \[ (a,b)\in R \Rightarrow (b,a)\in R \] Now check the elements:
\[ (1,1)\in R \] Its reverse is also \[ (1,1) \] Similarly \[ (2,2)\in R \] Its reverse is also \[ (2,2) \] Thus the relation satisfies the symmetric condition.

Step 3: Check the transitive property.
A relation is transitive if \[ (a,b)\in R \text{ and } (b,c)\in R \Rightarrow (a,c)\in R \] Here \[ (1,1)\in R \] and \[ (1,1)\in R \] Therefore \[ (1,1)\in R \] which satisfies transitivity. Similarly for \[ (2,2) \] Thus the relation is transitive.

Step 4: Conclusion.
The relation satisfies both symmetric and transitive properties.
Final Answer: $\boxed{\text{Symmetric and Transitive}}$