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 for 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 ${\displaystyle a.}$

Formal definition

Preliminaries

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 ${\displaystyle R.}$"

Definition

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: $${\displaystyle \forall a,b\in X:\lnot (aRb\wedge bRa).}$$ 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 ${\displaystyle \,<\,}$ between real numbers: if ${\displaystyle x<y}$ then necessarily ${\displaystyle y}$ is not less than ${\displaystyle x.}$ 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 ${\displaystyle X}$ 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 ${\displaystyle Z.}$

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 converse or dual ${\displaystyle \,>\,}$ of ${\displaystyle \,<\,}$ is also asymmetric.

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.

A non-example is the "less than or equal" relation ${\displaystyle \leq }$. This is not asymmetric, because reversing for example, ${\displaystyle x\leq x}$ produces ${\displaystyle x\leq x}$ 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 ${\displaystyle R}$ to be asymmetric:

• ${\displaystyle R}$ is irreflexive and anti-symmetric (this is also necessary)
• ${\displaystyle R}$ is irreflexive and transitive. A transitive relation is asymmetric if and only if it is irreflexive: if ${\displaystyle aRb}$ and ${\displaystyle bRa,}$ transitivity gives ${\displaystyle aRa,}$ contradicting irreflexivity. Such a relation is a strict partial order.
• ${\displaystyle R}$ is irreflexive and satisfies semiorder property 1 (there do not exist two mutually incomparable two-point linear orders)
• ${\displaystyle R}$ is anti-transitive and anti-symmetric
• ${\displaystyle R}$ is anti-transitive and transitive
• ${\displaystyle R}$ is anti-transitive and satisfies semi-order property 1

See also

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

References

1. Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 273.
2. Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.
3. Burghardt, Jochen (2018). "Simple Laws about Nonprominent Properties of Binary Relations". arXiv:1806.05036 [math.LO].
4. Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics - Physics Charles University. p. 1. Archived from the original (PDF) on 2013-11-02. Retrieved 2013-08-20. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".