I was asked on a test to find the number of relations on a set with 10 elements which is both symmetric and asymmetric.
Now since any element of the type
$$ (a,b) \quad where \quad b \neq a $$ cannot exist in such a relation (I think).
This problem essentially reduces to the statement whether a relation which only has elements of the type $ (a,a)$ (elements found on the main diagonal of the matrix form of R) is a relation which is both symmetric and asymmetric?
No, for any such $R$, you can't have any $(a,a) \in R$ since that would go against asymmetry (If $(a,a) \in R$, then by asymmetry, $(a,a) \not \in R$ ... so we can't have $(a,a)$!)
The only relation that works is the empty relation, i.e. $R=\{ \}$, since if any $(a,b) \in R$, we must have (by symmetry) $(b,a) \in R$, but we also must have (by asymmetry) $(b,a) \in R$. Since we can't have both, we can't have any $(a,b) \in R$.