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?

  • $\begingroup$ Sets aren't anti-symmetric (well some are, but ignore this), binary relations can be anti-symmetric. In a Hasse diagram two elements are comparable if you can go from one to another by going either straight up or straight down. $\endgroup$ – Git Gud Mar 4 '15 at 9:32
  • $\begingroup$ @GitGud So for the binary relation, where xRy is x is a subset of y, and anti-symmetric means xRy and yRx then x = y, I get how {x} is a subset of {x,y,z} but how is {x,y,z} a subset of {x}? and what defines the = then? I'm confused :( $\endgroup$ – Alex Mar 4 '15 at 9:36

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 $x\preceq y$.


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