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${\displaystyle S\ne \varnothing :}$ ${\displaystyle \begin{array}{r}minS\\ \text{exists}\end{array}}$ ${\displaystyle \begin{array}{r}a\vee b\\ \text{exists}\end{array}}$ ${\displaystyle \begin{array}{r}a\wedge b\\ \text{exists}\end{array}}$ ${\displaystyle aRa}$ ${\displaystyle \text{not }aRa}$ ${\displaystyle \begin{array}{r}aRb\Rightarrow \\ \text{not }bRa\end{array}}$
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 byY in the "Symmetric" column and✗ in the "Antisymmetric" column, respectively. All definitions tacitly require thehomogeneous relation${\displaystyle R}$ betransitive: 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.

Inmathematics, anasymmetric relation is abinary relation${\displaystyle R}$ on aset${\displaystyle X}$ where for all${\displaystyle a,b\in X,}$ if${\displaystyle a}$ is related to${\displaystyle b}$ then${\displaystyle b}$ isnot related to${\displaystyle a.}$^[1]

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 calledasymmetric 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${\displaystyle (b,a)\notin R.}$ This can be written in the notation offirst-order logic as$${\displaystyle \mathrm{\forall }a,b\in X:aRb⟹\mathrm{\neg }(bRa).}$$Alogically equivalent definition is:

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

which in first-order logic can be written as:$${\displaystyle \mathrm{\forall }a,b\in X:\mathrm{\neg }(aRb\wedge bRa).}$$A relation is asymmetric if and only if it is bothantisymmetric andirreflexive,^[2] so this may also be taken as a definition.

An example of an asymmetric relation is the "less than" relation betweenreal numbers: if 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, evenantitransitive relation is therock 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 andconverses 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${\displaystyle >}$ of is also asymmetric.

An asymmetric relation need not have theconnex property. For example, thestrict subset relation${\displaystyle ⊊}$ 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${\displaystyle \le }$. This is not asymmetric, because reversing for example,${\displaystyle x\le x}$ produces${\displaystyle x\le x}$ 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 "notsymmetric".

Theempty relation is the only relation that is (vacuously) both symmetric and asymmetric.

The following conditions are sufficient for a relation${\displaystyle R}$ to be asymmetric:^[3]

• ${\displaystyle R}$ is irreflexive and anti-symmetric (this is also necessary)
• ${\displaystyle R}$ is irreflexive and transitive. Atransitive relation is asymmetric if and only if it is irreflexive:^[4] if${\displaystyle aRb}$ and${\displaystyle bRa,}$ transitivity gives${\displaystyle aRa,}$ contradicting irreflexivity. Such a relation is astrict partial order.
• ${\displaystyle R}$ is irreflexive and satisfiessemiorder property 1 (there do not exist two mutually incomparable two-point linear orders)
• ${\displaystyle R}$ is antitransitive and anti-symmetric
• ${\displaystyle R}$ is antitransitive and transitive
• ${\displaystyle R}$ is antitransitive and satisfies semi-order property 1

• Tarski's axiomatization of the reals – part of this is the requirement that over the real numbers be asymmetric.

• Properties of binary relations
• Asymmetry

• Articles with short description
• Short description is different from Wikidata