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[parent] example of antisymmetric (Example)

The axioms of a partial ordering demonstrate that every partial ordering is antisymmetric. That is: the relation $ \leq $ on a set $ S$ forces

$ a\leq b$ and $ b\leq a$ implies $ a=b$
for every $ a,b\in S$.

For a concrete example consider the natural numbers $ \mathbb{N}=\{0,1,2,\dots\}$ (as defined by the Peano postulates). Take the relation set to be:

$\displaystyle R=\{(a,a+n):a,n\in \mathbb{N}\}\subset \mathbb{N}\times \mathbb{N}.$
Then we denote $ a\leq b$ if $ (a,b)\in R$. That is, $ 5\leq 7$ because $ (5,7)=(5,5+2)$ and both $ 5,2\in\mathbb{N}$.

We can prove this relation is antisymmetric as follows: Suppose $ a\leq b$ and $ b\leq a$ for some $ a,b\in\mathbb{N}$. Then there exist $ n,m\in\mathbb{N}$ such that $ a+n=b$ and $ b+m=a$. Therefore

$\displaystyle b=a+n=b+m+n$
so by the cancellation property of the natural numbers, $ 0=m+n$. But by the first piano postulate, 0 has no predecessor, meaning $ 0\neq m+n$ unless $ m=n=0$.



"example of antisymmetric" is owned by Algeboy.
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Cross-references: postulate, property, relation set, natural numbers, implies, forces, relation, antisymmetric, partial ordering, axioms
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This is version 5 of example of antisymmetric, born on 2006-06-15, modified 2006-06-15.
Object id is 8044, canonical name is ExampleOfAntisymmetric.
Accessed 1493 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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