# 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.

Think about the subsets of $${0,1}\{0,1\}$$. They are: $$∅,{0},{1}\emptyset, \{0\}, \{1\}$$, and $${0,1}\{0,1\}$$. Now, we can make these subsets into a partial order with $$⊂\subset$$. For instance, $$∅⊂{0}\emptyset \subset \{0\}$$ and $${1}⊂{0,1}\{1\} \subset \{0,1\}$$. You can show this satisfies the axioms for a partial order:
$$(A⊂A and A⊂B, and B⊂C)⇒A⊂C(A \subset A \text{ and } A \subset B, \text{ and } B \subset C) \Rightarrow A \subset C \\ \\$$ $$A⊂B,B⊂A⇒A=BA \subset B, B \subset A \Rightarrow A = B$$
But a total order $$<<$$ drops the first axiom above and replaces it with the following:
$$x or $$y for all $$x,yx,y$$
And we see that our example of subsets of $${0,1}\{0,1\}$$ does not satisfy this. For instance, neither $${0}⊂{1}\{0\} \subset \{1\}$$ nor $${1}⊂{0}\{1\} \subset \{0\}$$ are true. In a total order, we want to be able to compare any two elements. In a partial order, we don't.