If there is a relation that is not symmetric , then there is at least one pair for which the inverse relation does not hold; so applies
A non-empty asymmetric relation is never symmetric. An asymmetrical relation is also always irreflexive . The concept of antisymmetry , which also allows reflexivity, must be distinguished from asymmetry . An asymmetrical relation is therefore a special case of an antisymmetrical relation.
There are relations that are neither symmetrical nor antisymmetrical, and certainly not asymmetrical. An example is provided by the definition on the natural numbers.
the relation “is (real) smaller than” on the real numbers , which is also a strict total order . The same applies to the relation “is (real) greater than”.
the relation “is a real subset of” and also the relation “is a real superset of” as relationships between sets. In a system of sets or of subsets of a given set, they are also a strict partial order .