The number of relations on the set \1,2,3,4\ that are reflexive and symmetric but not transitive is

A reflexive relation must contain the $4$ pairs $(i,i)$. Symmetry means each of the $\binom42=6$ unordered pairs $\{i,j\}$ is either included (both $(i,j),(j,i)$) or not, so there are $2^6=64$ reflexive symmetric relations. Of these, the transitive ones are exactly the equivalence relations, which correspond to partitions of a $4$-element set: $1+7+6+1=15$ (the Bell number $B_4$). Hence the required number is $64-15=49$. A brute-force count over all $2^{16}$ relations confirms this.