mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

R

on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

X

is antisymmetric if there is no pair of ''distinct'' elements of

X

each of which is related by

R

to the other. More formally,

R

is antisymmetric precisely if for all

a, b \in X,

\text \,aRb\, \text \,a \neq b\, \text \,bRa\, \text,

or equivalently,

\text \,aRb\, \text \,bRa\, \text \,a = b.

The definition of antisymmetry says nothing about whether

aRa

actually holds or not for any

a

. An antisymmetric relation

R

on a set

X

may be reflexive (that is,

aRa

for all

a \in X

irreflexive In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...

(that is,

aRa

for no

a \in X

), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.

Examples

The

divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

relation on the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s is an important example of an antisymmetric relation. In this context, antisymmetry means that the only way each of two numbers can be divisible by the other is if the two are, in fact, the same number; equivalently, if

n

and

m

are distinct and

n

is a factor of

m,

then

m

cannot be a factor of

n.

For example, 12 is divisible by 4, but 4 is not divisible by 12. The usual

order relation Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...

\,\leq\,

on the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s is antisymmetric: if for two real numbers

x

and

y

both

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

x \leq y

and

y \leq x

hold, then

x

and

y

must be equal. Similarly, the subset order

\,\subseteq\,

on the subsets of any given set is antisymmetric: given two sets

A

and

B,

if every element in

A

also is in

B

and every element in

B

is also in

A,

then

A

and

B

must contain all the same elements and therefore be equal:

A \subseteq B \text B \subseteq A \text A = B

A real-life example of a relation that is typically antisymmetric is "paid the restaurant bill of" (understood as restricted to a given occasion). Typically, some people pay their own bills, while others pay for their spouses or friends. As long as no two people pay each other's bills, the relation is antisymmetric.

Properties

Partial Partial may refer to: Mathematics * Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...

and

total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflex ...

s are antisymmetric by definition. A relation can be both

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

and antisymmetric (in this case, it must be coreflexive), and there are relations which are neither symmetric nor antisymmetric (for example, the "preys on" relation on biological

species In biology, a species is the basic unit of classification and a taxonomic rank of an organism, as well as a unit of biodiversity. A species is often defined as the largest group of organisms in which any two individuals of the appropriate s ...

). Antisymmetry is different from asymmetry: a relation is asymmetric if and only if it is antisymmetric and

References

* *
nLab antisymmetric relation
{{DEFAULTSORT:Antisymmetric Relation Binary relations

Examples

Properties

See also

References