Editing Antisymmetric relation (section)

== Examples ==

The [[divisibility]] relation on the [[natural number]]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 <math>n</math> and <math>m</math> are distinct and <math>n</math> is a factor of <math>m,</math> then <math>m</math> cannot be a factor of <math>n.</math>  For example, 12 is divisible by 4, but 4 is not divisible by 12.

The usual [[order relation]] <math>\,\leq\,</math> on the [[real number]]s is antisymmetric: if for two real numbers <math>x</math> and <math>y</math> both [[Inequality (mathematics)|inequalities]] <math>x \leq y</math> and <math>y \leq x</math> hold, then <math>x</math> and <math>y</math> must be equal. Similarly, the [[subset order]] <math>\,\subseteq\,</math> on the subsets of any given set is antisymmetric: given two sets <math>A</math> and <math>B,</math> if every [[Element (mathematics)|element]] in <math>A</math> also is in <math>B</math> and every element in <math>B</math> is also in <math>A,</math> then <math>A</math> and <math>B</math> must contain all the same elements and therefore be equal:
<math display=block>A \subseteq B \text{ and } B \subseteq A \text{ implies } A = B</math>
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.