A binary relation which never occurs in both directions
In mathematics, an asymmetric relation is a binary relation formula_1 on a set formula_2 where for all formula_3 if formula_4 is related to formula_5 then formula_5 is "not" related to formula_7
Formal definition.
Preliminaries.
A binary relation on formula_2 is any subset formula_1 of formula_10 Given formula_3 write formula_12 if and only if formula_13 which means that formula_12 is shorthand for formula_15 The expression formula_12 is read as "formula_4 is related to formula_5 by formula_19"
Definition.
The binary relation formula_1 is called asymmetric if for all formula_3 if formula_12 is true then formula_23 is false; that is, if formula_24 then formula_25 
This can be written in the notation of first-order logic as
formula_26
A logically equivalent definition is:
for all formula_3 at least one of formula_12 and formula_23 is false,
which in first-order logic can be written as:
formula_30
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" relation formula_31 between real numbers: if formula_32 then necessarily formula_33 is not less than formula_34 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 formula_2 beats formula_36 then formula_37 does not beat formula_38 and if formula_2 beats formula_37 and formula_37 beats formula_42 then formula_2 does not beat formula_44
Restrictions and converses of asymmetric relations are also asymmetric. For example, the restriction of formula_31 from the reals to the integers is still asymmetric, and the converse or dual formula_46 of formula_31 is also asymmetric.
An asymmetric relation need not have the connex property. For example, the strict subset relation formula_48 is asymmetric, and neither of the sets formula_49 and formula_50 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 formula_51. This is not asymmetric, because reversing for example, formula_52 produces formula_52 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.
The following conditions are sufficient for a relation formula_1 to be asymmetric:
References.
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