JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Sets Relations and Functions

A relation from $\{x,y\}$ to $\{x,y\}$ is a subset of $\{x,y\}\times \{x,y\}=\{(x,x),(x,y),(y,x),(y,y)\}$.

Total number of relations = $2^4=16$.

For a relation to be both symmetric and transitive, we enumerate the favorable cases:

1. $\varphi$ (empty relation) — symmetric and transitive
2. $\{(x,x)\}$ — symmetric and transitive
3. $\{(y,y)\}$ — symmetric and transitive
4. $\{(x,x),(y,y)\}$ — symmetric and transitive
5. $\{(x,x),(y,y),(x,y),(y,x)\}$ — symmetric and transitive

Any other combination either breaks symmetry or transitivity. For example, $\{(x,y)\}$ is not symmetric.

Favorable relations = 5.

Probability = $\frac{5}{16}$.