JEE 2022 — Mathematics PYQ

The set has $n=2$ elements. The number of elements in $A\times A$ is $2\times 2=4$.

Total relations = $2^4=16$.

2. Favorable Relations (Symmetric and Transitive):

A relation that is symmetric and transitive is an equivalence relation on some subset of $\{x,y\}$.

• Empty relation: $\varphi$ (Symmetric and Transitive) — 1 case.

• Relations on 1 element: $\{(x,x)\}$ or $\{(y,y)\}$ — 2 cases.

• Equivalence relations on 2 elements:

  • $\{(x,x),(y,y)\}$

  • $\{(x,x),(y,y),(x,y),(y,y)\}$ — 2 cases.

Wait, let's list them carefully:

1. $\varphi$

2. $\{(x,x)\}$

3. $\{(y,y)\}$

4. $\{(x,x),(y,y)\}$

5. $\{(x,x),(y,y),(x,y),(y,x)\}$

   Total favorable cases = $5$.

3. Probability:

$P=\frac{\text{Favorable}}{\text{Total}}=\frac{5}{16}$.

Correct Option: (A)