Understanding comparability in a partially ordered sets with Hasse diagrams I'm doing a section on directed graphs and Hasse diagrams right now on partially ordered sets, and I'm trying to understand what it means for two elements in a Hasse diagram to be comparable.
For example, on Wikipedia it states in the set of all subsets of a three element set, {x,y,z}, that the set {x} and {x,y,z} are comparable. And that for two elements to be comparable, it must satisfy the condition that $x \preceq y$ and $y \preceq x$
I'm having trouble understanding how the set {x} and the set {x,y,z} are anti-symmetric (because how does {x} = {x,y,z}?) first off, and how $\{x,y,z\} \preceq \{x\}$? Besides that how can I understand what is a partially ordered set of what in a Hasse diagram, or how to tell if elements are comparable within a Hasse diagram?
 A: A partially ordered set is any set $S$ with a relation $\preceq$ which is antisymmetric, transitive and reflexive.
Wikipedia correctly states that two elements are comparable if $x\preceq y$ or $y\preceq x$ so in your example everything is fine $\{x\}\preceq\{x,y,z\}$ but not the other way around.
As to how to understand a Hasse diagram this is fairly straightforward. The lines in a Hasse diagram are oriented (usually top down) and if there is a line from $y$ (above) to $x$ (below), this indicates that $x\preceq y$, $x\neq y$ and $\forall z (x\preceq z\preceq y)\implies (x=z)\vee(y=z)$. In words that just says a line from $y$ to $x$ when $x$ is below $y$, means that $x\prec y$ and there are no elements between them.
You can really think of a Hasse diagram of $\preceq$, as the diagram of the minimal relation $R$ such that it's transitive closure is $\preceq$.
When you are reading the relation off the diagram you can just follow the lines as pointed out in comments. If you can get from $x$ to $y$ along only upward sloping lines then $x\preceq y$.
