An asymmetric relation is a binary relation R on a set A such that for all a, b in A, if (a, b) in R, then (b, a) is not in R.
The complement of a relation R on a set A is the relation R^c on A such that for all a, b in A, (a, b) in R^c if and only if (a, b) is not in R.
So, the complement of an asymmetric relation R on a set A is a relation R^c on A such that for all a, b in A, if (a, b) in R^c, then (b, a) is not in R^c.
This does not mean that R^c is symmetric. For example, consider the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}. Then, R is an asymmetric relation. However, the complement of R is R^c = {(1, 3), (2, 1), (3, 2)}, which is not symmetric.