L = {all binary relations except reflexive relation set S}
=> L has no xRx pair
=> So no diagonal element present.
=> Total size of L = 2^(n^{2} - n)
In symmetric relations, xRy and yRx should both come. For each x and y (x!=y), either (x,y) or (y,x) can be present.
Number of symmetric relations in L = (No diagonal element selected)*(One of the non-diagonal pairs)
No of non-diagonal pairs = (n^{2} - n)/2
=> Number of symmetric relations = 2^((n^{2} - n)/2)