Asymmetric relation

A binary relation on ${\displaystyle X}$ is any subset ${\displaystyle R}$ of ${\displaystyle X\times X.}$ Given ${\displaystyle a,b\in X,}$ write ${\displaystyle aRb}$ if and only if ${\displaystyle (a,b)\in R,}$ which means that ${\displaystyle aRb}$ is shorthand for ${\displaystyle (a,b)\in R.}$ The expression ${\displaystyle aRb}$ is read as "${\displaystyle a}$ is related to ${\displaystyle b}$ by ${\displaystyle R.}$" The binary relation ${\displaystyle R}$ is called asymmetric if for all ${\displaystyle a,b\in X,}$ if ${\displaystyle aRb}$ is true then ${\displaystyle bRa}$ is false; that is, if ${\displaystyle (a,b)\in R}$ then ${\displaystyle (b,a)\not \in R.}$ This can be written in the notation of first-order logic as

$${\displaystyle \forall a,b\in X:aRb\implies \lnot (bRa).}$$

A logically equivalent definition is:

for all ${\displaystyle a,b\in X,}$ at least one of ${\displaystyle aRb}$ and ${\displaystyle bRa}$ is false,

which in first-order logic can be written as:

$${\displaystyle \forall a,b\in X:\lnot (aRb\wedge bRa).}$$

An example of an asymmetric relation is the "less than" relation ${\displaystyle \,<\,}$ between real numbers: if ${\displaystyle x<y}$ then necessarily ${\displaystyle y}$ is not less than ${\displaystyle x.}$ The "less than or equal" relation ${\displaystyle \,\leq ,}$ on the other hand, is not asymmetric, because reversing for example, ${\displaystyle x\leq x}$ produces ${\displaystyle x\leq x}$ and both are true. Asymmetry is not the same thing as "not symmetric": the less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric. The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.

• A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]
• Restrictions and converses of asymmetric relations are also asymmetric. For example, the restriction of ${\displaystyle \,<\,}$ from the reals to the integers is still asymmetric, and the inverse ${\displaystyle \,>\,}$ of ${\displaystyle \,<\,}$ is also asymmetric.
• A transitive relation is asymmetric if and only if it is irreflexive:[3] if ${\displaystyle aRb}$ and ${\displaystyle bRa,}$ transitivity gives ${\displaystyle aRa,}$ contradicting irreflexivity.
• As a consequence, a relation is transitive and asymmetric if and only if it is a strict partial order.
• 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 ${\displaystyle X}$ beats ${\displaystyle Y,}$ then ${\displaystyle Y}$ does not beat ${\displaystyle X;}$ and if ${\displaystyle X}$ beats ${\displaystyle Y}$ and ${\displaystyle Y}$ beats ${\displaystyle Z,}$ then ${\displaystyle X}$ does not beat ${\displaystyle Z.}$
• An asymmetric relation need not have the connex property. For example, the strict subset relation ${\displaystyle \,\subsetneq \,}$ is asymmetric, and neither of the sets ${\displaystyle \{1,2\}}$ and ${\displaystyle \{3,4\}}$ is a strict subset of the other. A relation is connex if and only if its complement is asymmetric.

• Tarski's axiomatization of the reals – part of this is the requirement that ${\displaystyle \,<\,}$ over the real numbers be asymmetric.