Solution
= Solution
A <symmetric relation> has $n$ independent choices on the diagonal. For each of the $\binom n2$ unordered pairs $\{a,b\}$ of distinct elements, either both ordered pairs are included or neither is. Thus there are $2^{n+\binom n2}$ <symmetric relations>, so the requested number is
$$
\boxed{2^{n^2}-2^{n+\binom n2}=2^{n^2}-2^{n(n+1)/2}.}
$$