Trouble constructing an ordered set that is not directed

Question

I want to try to construct an ordered set that is not directed to give myself a better understanding of what it means to be directed. Any good examples out there?

Answer 1

Trivial example: Let $S$ be any set with at least two elements, and deine a partial order on $S$ by saying $x\le y$ if and only if $x=y$.

More informative example: Consider the set of all $A\subset\Bbb Z$ such that $A$ has at most $3$ elements, ordered by inclusion.