In
mathematics, a
binary relation on a
set is antisymmetric if there is no pair of ''distinct'' elements of
each of which is related by
to the other. More formally,
is antisymmetric precisely if for all
or equivalently,
The definition of antisymmetry says nothing about whether
actually holds or not for any
. An antisymmetric relation
on a set
may be
reflexive (that is,
for all
),
irreflexive (that is,
for no
), 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 ...
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
and
are distinct and
is a factor of
then
cannot be a factor of
For example, 12 is divisible by 4, but 4 is not divisible by 12.
The usual
order relation 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 ...
s is antisymmetric: if for two real numbers
and
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
* ...
and
hold, then
and
must be equal. Similarly, the
subset order on the subsets of any given set is antisymmetric: given two sets
and
if every
element in
also is in
and every element in
is also in
then
and
must contain all the same elements and therefore be equal:
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 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
irreflexive.
See also
*
*
References
*
*
nLab antisymmetric relation
{{DEFAULTSORT:Antisymmetric Relation
Binary relations