Question:

Key Concept: A symmetric relation is determined by: (i) for each of the $n$ diagonal pairs $(a, a)$: include or not ($2^n$ choices); (ii) for each of the $\binom{n}{2} = n(n - 1)/2$ unordered off-diagonal pairs $\{a, b\}$: include both or neither ($2^{n(n-1)/2}$ choices). Total: $2^n \cdot 2^{n(n-1)/2} = 2^{n(n+1)/2}$.