Is it possible for a relation to be transitive and symmetric but not reflexive with only one element? E.g. On the set $A = \{1,2,3,4,5,6\}$, is the relation set $R = \{(1,1)\}$ a transitive and symmetric relation but not reflexive?

Yes. The relation $R$ that you have given is transitive and symmetric, but not reflexive. To see this, note that $(2,2)\not\in R$, but by definition, $R$ is reflexive if $(a,a)\in R$ for all $a\in A$. It's easy to check that $R$ is symmetric and transitive. For example, note that for all $(a,b)\in R$, we also have $(b,a)\in R$, since $R=\{(1,1)\}$.
