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

Solution 1:

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

$$(A \subset A \text{ and } A \subset B, \text{ and } B \subset C) \Rightarrow A \subset C \\ \\$$ $$A \subset B, B \subset A \Rightarrow A = B$$

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 $\{0\} \subset \{1\}$ nor $\{1\} \subset \{0\}$ are true. In a total order, we want to be able to compare any two elements. In a partial order, we don't.

Solution 2:

Take your favourite set, which is $\,X:=\{a,b\}\,$ and then its power set

$$P(X):=\left\{\emptyset\,,\,X\,,\,\{a\}\,,\,\{b\}\right\}$$

Partial order $\,P(X)\,$ by set inclusion: $\,A\le B\iff A\subset B\;,\;\;A,B\in P(X)\,$

Check the above is a partial not total order.

Solution 3:

There are some small differences in the way people define order (partial or total). Roughly speaking, they correspond to the difference between $\lt$ and $\le$. We opt for the $\le $ version. You can undoubtedly adapt the example below to the other version, if that's the one being used in your course.

Let our set be $\{1,2,3,6\}$. If $x$ and $y$ are elements of this set, we will say that $x\le y$ if $x$ divides $y$. So for example $2\le 6$, and $3\le 3$.

Note that it is not true that $2\le 3$, since $2$ does not divide $3$. Also, it is not true that $3\le 2$. The two objects $2$ and $3$ are incomparable with respect to the order just defined.

In a total order $\le$, any two objects $x$ and $y$ are comparable. Either $x\le y$ or $y\le x$ or both. ("Both" happens when $x=y$.)

For a non-mathematical example, let $A$ be the set of all people. If $x$ and $y$ are people, write $x\le y$ if $x=y$ or $x$ is an ancestor of $y$. This is a partial order. However, it is not total, since for example Obama is not an ancestor of Putin, and Putin is not an ancestor of Obama.