The probability that a relation R from \ x, y\ to \ x, y\ is both...

Total number of elements in the Cartesian product will be $4$, so total possible relations will be ${2}^{4} = 16$

The relations which will be symmetric and transitive both will be \left\right, \left\{x, x\right\}, \left\{y, y\right\}, \left\{\left(x, x\right), \left(y, y\right)\right\}, \left\{\left(x, x\right), \left(x, y\right), \left(y, x\right)\left(y, y\right)\right\}$</p><p> i.e.$5$relations</p><p>Hence, required probability will be$\dfrac{5}{16}