How are asymmetric, symmetric, and antisymmetric relations different?

"Antisymmetric is very different from asymmetric. An asymmetric relation is simply one that is not symmetric: in other words, there is some (x, y) in it without a matching (y, x). An antisymmetric relation, on the other hand, is one in which there are guaranteed to be no matching (y, x) pairs for any (x, y), except when x and y are the same. Most relations are neither symmetric nor antisymmetric. It is something of a coincidence for a relation to be symmetric: that would mean that for every single (x, y) it contains, it also contains a (y, x). Similarly, it is something of a coincidence for a relation to be antisymmetric: that would mean that for every single (x, y) it contains, it does not contain a (y, x). An ordinary relation can contain some (x, y) pairs that have matching (y, x) pairs and some that do not have matches; such relations are neither symmetric nor antisymmetric. It is possible for a relation to be both symmetric and antisymmetric, but not asymmetric. For instance, the empty relation, with no ordered pairs, is both symmetric and antisymmetric. It is symmetric because for every ordered pair (x, y) in it, of which there are zero, there is also the corresponding (y, x). Similarly, for every ordered pair (x, y), the corresponding (y, x) is not present. Another example is a relation with only doubles in it, such as {(3,3), (7,7), (Fred, Fred)}. This, too, is both symmetric and antisymmetric. The conditions are trivially satisfied because there are no distinct elements whose ordered pairs could conflict."