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![]() ![]() All definitions tacitly require the
homogeneous relation be
transitive: for all if and then |
In mathematics, an asymmetric relation is a binary relation on a set where for all if is related to then is not related to [1]
A binary relation on is any subset of Given write if and only if which means that is shorthand for The expression is read as " is related to by "
The binary relation is called asymmetric if for all if is true then is false; that is, if then This can be written in the notation of first-order logic as
which in first-order logic can be written as:
An example of an asymmetric relation is the " less than" relation between real numbers: if then necessarily is not less than 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 then does not beat and if beats and beats then does not beat
Restrictions and converses of asymmetric relations are also asymmetric. For example, the restriction of from the reals to the integers is still asymmetric, and the converse or dual of is also asymmetric.
An asymmetric relation need not have the connex property. For example, the strict subset relation is asymmetric, and neither of the sets and 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, produces 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.
The following conditions are sufficient for a relation to be asymmetric: [3]
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![]() ![]() All definitions tacitly require the
homogeneous relation be
transitive: for all if and then |
In mathematics, an asymmetric relation is a binary relation on a set where for all if is related to then is not related to [1]
A binary relation on is any subset of Given write if and only if which means that is shorthand for The expression is read as " is related to by "
The binary relation is called asymmetric if for all if is true then is false; that is, if then This can be written in the notation of first-order logic as
which in first-order logic can be written as:
An example of an asymmetric relation is the " less than" relation between real numbers: if then necessarily is not less than 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 then does not beat and if beats and beats then does not beat
Restrictions and converses of asymmetric relations are also asymmetric. For example, the restriction of from the reals to the integers is still asymmetric, and the converse or dual of is also asymmetric.
An asymmetric relation need not have the connex property. For example, the strict subset relation is asymmetric, and neither of the sets and 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, produces 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.
The following conditions are sufficient for a relation to be asymmetric: [3]