example of antisymmetric

The axioms of a partial ordering demonstrate that every partial ordering is antisymmetric. That is: the relationMathworldPlanetmath on a set S forces

ab and ba implies a=b

for every a,bS.

For a concrete example consider the natural numbersMathworldPlanetmath ={0,1,2,} (as defined by the Peano postulates (http://planetmath.org/PeanoArithmetic)). Take the relation set to be:


Then we denote ab if (a,b)R. That is, 57 because (5,7)=(5,5+2) and both 5,2.

We can prove this relation is antisymmetric as follows: Suppose ab and ba for some a,b. Then there exist n,m such that a+n=b and b+m=a. Therefore


so by the cancellation property of the natural numbers, 0=m+n. But by the first piano postulateMathworldPlanetmath, 0 has no predecessor, meaning 0m+n unless m=n=0.

Title example of antisymmetric
Canonical name ExampleOfAntisymmetric
Date of creation 2013-03-22 16:00:36
Last modified on 2013-03-22 16:00:36
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 8
Author Algeboy (12884)
Entry type Example
Classification msc 03E20