The number of relations on the set A=\1,2,3,4\ that are both reflexive and symmetric is

A reflexive relation must contain all four pairs $(a,a)$. A symmetric relation is fixed by deciding, independently, whether each unordered pair $\{a,b\}$ with $a\neq b$ is included. There are $\binom{4}{2}=6$ such pairs, each included or not, so the count is $2^6=64$. The values $1024$ (symmetric only, $2^{10}$) and $4096$ (reflexive only, $2^{12}$) count relations with just one of the two properties.