Asymmetric relation: Given a set A and a relation R in A, R is asymmetric if it is never the case that for any ordered pair (x, y) in R, the pair (y, x) is in R.
Nonsymmetric relation: If for some (x, y) in R, the pair (y, x) is not in R then R is nonsymmetric.
Complement: The complement of a relation R contains all ordered pairs of the Cartesian product of A which are not members of R.
The book that I am reading says that the complement of an asymmetric relation is nonsymmetric.
Isn't this a counterexample: The set A is empty and relation R has no ordered pairs (i.e., R is empty).
R is asymmetric.
The complement of R is R, which is not nonsymmetric.
So, is the book incorrect?
Assume $(x,y) \in R$; then by asymmetry: $(y,x) \in R^C$.
Obviously $(x,y) \notin R^C$.
Thus, we have found a pair $(a,b) \in R^C$ such that the pair $(b,a) \notin R^C$, that means that $R^C$ is not symmetric.