Asymmetrical relation
A two-digit relation on a set is called asymmetric if there is no pair for which the converse also applies.
Asymmetry is one of the prerequisites for an (irreflexive) strict order .
definition
Is a quantity and a two-digit relation to , it means asymmetric when
- applies.
Not symmetrical relation
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.
Examples
Are asymmetrical
- 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 .
properties
- Every asymmetrical relation is a non-symmetrical relation and also an antisymmetrical relation .
- The intersection of an asymmetrical relation and its converse relation is always empty, they are disjoint:
- Every subset of an asymmetric relation is again asymmetric.
Remarks
- ↑ see also: Ingmar Lehmann , Wolfgang Schulz: Sets - Relations - Functions. A clear introduction. 3rd, revised and expanded edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0162-3 , p. 64 f.