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Wikipedia / Asymmetric relation

In mathematics, an asymmetric relation is a binary relation on a set is not

Formal definition

Preliminaries

"

Definition

is called This can be written in the notation of first-order logic A logically equivalent definition is:

  1. is ,

A relation is asymmetric if and only if it is both antisymmetric and irreflexive, so this may also be taken as a definition.

Examples

An example of an asymmetric relation is the "less than between real numbers More generally, any strict partial order is an asymmetric relation. Not all asymmetric relations are strict partial orders. An example of an asymmetric non-transitive, even antitransitive relation is the

Restrictions and converses is also asymmetric.

An asymmetric relation need not have the connex property. For example, the strict subset is a strict subset of the other. A relation is connex if and only if its complement is asymmetric.

and both are true. The less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric, showing that asymmetry is not the same thing as "not symmetric".

The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.

Properties

to be asymmetric:

  • is irreflexive and anti-symmetric (this is also necessary)

  • is irreflexive and transitive. A transitive relation is asymmetric if and only if it is irreflexive: contradicting irreflexivity. Such a relation is a strict partial order.

  • is irreflexive and satisfies semiorder property 1 (there do not exist two mutually incomparable two-point linear orders)

  • is antitransitive and anti-symmetric

  • is antitransitive and transitive

  • is antitransitive and satisfies semi-order property 1

See also

References