1. Total number of relations from $\{x,y\}$ to $\{x,y\}$

A relation on $\{x,y\}$ is any subset of

There are $4$ possible ordered pairs, so total number of relations is

2. Condition for symmetry

A relation $R$ is symmetric if

For the set $\{x,y\}$:

• $(x,x)$ and $(y,y)$ are automatically symmetric with themselves.
• $(x,y)$ and $(y,x)$ must occur together or both be absent.

So a symmetric relation is determined by independent choices of:

• whether $(x,x)$ is included,
• whether $(y,y)$ is included,
• whether both $(x,y)$ and $(y,x)$ are included.

Hence there are

symmetric relations.

3. Now impose transitivity

Let us list all $8$ symmetric relations and test transitivity.

Write them as subsets of $\{(x,x),(x,y),(y,x),(y,y)\}$.

(i) $\varnothing$

This is transitive vacuously.

(ii) $\{(x,x)\}$

Since $(x,x)$ and $(x,x)$ imply $(x,x)$, which is present, it is transitive.

(iii) $\{(y,y)\}$

Similarly transitive.

(iv) $\{(x,x),(y,y)\}$

Both diagonal pairs are present, and compositions stay within the relation. So transitive.

(v) $\{(x,y),(y,x)\}$

Check transitivity:

but $(x,x)\notin R$. So this is not transitive.

Also,

which is also missing.

(vi) $\{(x,x),(x,y),(y,x)\}$

Check:

but $(y,y)\notin R$. So not transitive.

(vii) $\{(y,y),(x,y),(y,x)\}$

Check:

but $(x,x)\notin R$. So not transitive.

(viii) $\{(x,x),(y,y),(x,y),(y,x)\}$

This is the universal relation. It is transitive.

4. Count favorable relations

The symmetric and transitive relations are:

So number of favorable relations is

5. Probability

Therefore,

6. Compare with stored answer

Stored correct answer is A: $\frac{5}{16}$. This matches our derived result.