The
divisibility relation on the
natural numbers 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 \,\leq\, on the
real numbers is antisymmetric: if for two real numbers x and y both
inequalities 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{ and } B \subseteq A \text{ implies } 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 ==