Asymmetric relation

Asymmetry is, a binary relation on a set, if there is no pair with, for the converse also holds.

The asymmetry is one of the prerequisites for a ( irreflexive ) Strict order.

Definition

Is a set and a binary relation on, then that means asymmetric when

Non- symmetric relation

Is a ratio which is not symmetrical, then there is at least one pair, of which the reverse ratio is not the case; so true

A nonempty asymmetric relation is thus not symmetrical. An asymmetric relation is also always irreflexive. To be distinguished from the asymmetry is thus the concept of anti-symmetry, which also allows for reflexivity. An asymmetric relation is thus a special case of an antisymmetric relation.

Examples

Are asymmetric

• The relation " is (real ) is less than " the real numbers that a strict total ordering is beyond. The same applies to the relation " is (real ) greater than."
• The relation " is a proper subset of" and also the relation " is a proper subset of" as relationships between quantities. They are a strict partial order in a system of quantities or of subsets of a given set beyond.

Properties

Each asymmetric relation is a non- symmetric relation and also an anti-symmetric relation.

• For the asymmetric relation and its converse relation of the section is empty, they are disjoint:
• Every subset of an asymmetric relation is asymmetric again.

Comments

• Order theory
• Set theory