currently learning about hasse diagrams and equivalences.
given the following relations $ R_1,R_2,R_3$ how should I generate the Hasse diagram (could someone show me how it would look like?).
I understand that only one of these relations is able to generate the Hasse (my guess is $R_3$ but correct me if I am wrong)
The set is made up of:
{a, b, c, d, e, f}
I would assume I need the hasse diagram to find the minimal elements,lowest bound and upper bound elements.
Any help would be appreciated 🙂
Best Answer
You are correct that $R_3$ is the only among these relations which is a partial ordering.
While they're all reflexive, $R_1$ is clearly not anti-symmetric; indeed, it's the equivalence relation with classes $\{a,c,f\}$, $\{b\}$ and $\{d,e\}$.
Relation $R_2$ is also not anti-symmetric: for example $cR_2d$ and $dR_2c$.
On the other hand, it isn't symmetric either: $cR_2b$ but not the converse.
Hence $R_2$ is neither an equivalence relation, nor a partial order.
Finally, as you suspected, $R_3$ is a partial order.
Here's its Hasse diagram: