Question

Consider the set $A = \{1,2,3\}$ along with the relation $R = \{(1,1),(2,2),(1,2),(2,1),(3,3)\}$. Which of the following statements is true?

• A. The relation is symmetric but not transitive
• B. The relation is transitive but not symmetric
• C. The relation is neither symmetric nor transitive
• D. The relation is both symmetric and transitive
• E. The relation is a function

Hint:

Always check all possible pair combinations for transitivity carefully.

Answer 1

The Correct Option is D

Step 1: Check symmetry.
A relation is symmetric if: \[ (a,b)\in R \Rightarrow (b,a)\in R \] Check pairs: \[ (1,2)\in R \Rightarrow (2,1)\in R \; \checkmark \] \[ (2,1)\in R \Rightarrow (1,2)\in R \; \checkmark \] Diagonal elements: \[ (1,1),(2,2),(3,3) \text{ are symmetric by nature} \] Thus, relation is symmetric.

Step 2: Check transitivity.
If $(a,b)$ and $(b,c)$ are in $R$, then $(a,c)$ must be in $R$. Check: \[ (1,2),(2,1) \Rightarrow (1,1)\in R \; \checkmark \] \[ (2,1),(1,2) \Rightarrow (2,2)\in R \; \checkmark \] \[ (1,1),(1,2) \Rightarrow (1,2)\in R \; \checkmark \] \[ (2,2),(2,1) \Rightarrow (2,1)\in R \; \checkmark \] All conditions satisfied.
Final Answer: \[ \boxed{\text{both symmetric and transitive}} \]