All definitions tacitly require the homogeneous relation
be transitive: for all
A term's definition may require additional properties that are not listed in this table.
In mathematics, an asymmetric relation is a binary relation
[1] A binary relation on
" The binary relation
is called asymmetric if for all
This can be written in the notation of first-order logic as
A logically equivalent definition is: which in first-order logic can be written as:
A relation is asymmetric if and only if it is both antisymmetric and irreflexive,[2] so this may also be taken as a definition.
An example of an asymmetric relation is the "less than" relation
between real numbers: if
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 rock paper scissors relation: if
beats
does not beat
beats
Restrictions and converses of asymmetric relations are also asymmetric.
from the reals to the integers is still asymmetric, and the converse or dual
An asymmetric relation need not have the connex property.
For example, the strict subset relation
is asymmetric, and neither of the sets
is a strict subset of the other.
A relation is connex if and only if its complement is asymmetric.
A non-example is the "less than or equal" relation
This is not asymmetric, because reversing for example,
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.
The following conditions are sufficient for a relation