Asymmetric relation

Transitive binary relations v t e
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all ${\displaystyle a,b}$ and ${\displaystyle S\neq \varnothing :}$ ${\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}$ ${\displaystyle {\begin{aligned}aRb{\text{ and }&bRa\\\Rightarrow a={}&b\end{aligned}$ ${\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }&bRa\end{aligned}$ ${\displaystyle {\begin{aligned}\min S\\{\text{exists}\end{aligned}$ ${\displaystyle {\begin{aligned}a\vee b\\{\text{exists}\end{aligned}$ ${\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}\end{aligned}$ ${\displaystyle aRa}$ ${\displaystyle {\text{not }aRa}$ ${\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }bRa\end{aligned}$
Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation ${\displaystyle R}$ be transitive: for all ${\displaystyle a,b,c,}$ if ${\displaystyle aRb}$ and ${\displaystyle bRc}$ then ${\displaystyle aRc.}$ A term's definition may require additional properties that are not listed in this table.

In mathematics, an asymmetric relation is a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ where for all ${\displaystyle a,b\in X,}$ if ${\displaystyle a}$ is related to ${\displaystyle b}$ then ${\displaystyle b}$ is not related to ${\displaystyle a.}$^[1]

Formal definition

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.

Properties

• 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.

See also

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

References