Example of Partial Order that’s not a Total Order and why?

I’m looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two.

An explanation of why the example is a partial order but not a total order would also be greatly appreciated.

Answer

Think about the subsets of {0,1}. They are: ,{0},{1}, and {0,1}. Now, we can make these subsets into a partial order with . For instance, {0} and {1}{0,1}. You can show this satisfies the axioms for a partial order:

(AA and AB, and BC)AC AB,BAA=B

But a total order < drops the first axiom above and replaces it with the following:

x<y or y<x for all x,y

And we see that our example of subsets of {0,1} does not satisfy this. For instance, neither {0}{1} nor {1}{0} are true. In a total order, we want to be able to compare any two elements. In a partial order, we don't.

Attribution
Source : Link , Question Author : eZanmoto , Answer Author : amWhy

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