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:

(A⊂A and A⊂B, and B⊂C)⇒A⊂C A⊂B,B⊂A⇒A=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*