In $A=\{2, 3, 6, 12, 36, 72, 108\}$ we define the relation $R$ by $aRb$ if $b=a$, or $b=2a$ or $b=3a$.
Q1: Draw the graph of $R$ and list which properties $R$ has.
A1:
The properties are: reflexive, anti-symmetric, and non-transitive.
Q2: Determine the transitive closure $R^{+}$ and draw it in the graph.
A2: (reflexivity is implied in the graph since it would get too messy)
Q3: Show that $R^{+}$ is a partial ordering and draw the Hasse diagram of $(A, R^{+})$. What are the minimal and maximal elements?
A3: For it to be a partial ordering, it should have the following properties: reflexive, anti-symmetric, and transitive. Looking at the graph of the previous answer A2, it shows that it satisfies all of these properties.
The Hasse diagram:
The minimal elements are 2 and 3, and the maximal elements are 72 and 108.
I would like to know if my answers are correct, and as well if every two elements has a supremum and an infimum. Wouldn't that be... 30 comparisons in all? Also, it is said that $R^{+}$is a very familiar relation, but I'm not seeing it. Any clues?
I think the undirected edges need to be directed. E.g. $$36\ R\ 72$$ since $72=2 \times 36$, but $72$ is not related to $36$ since $36 \neq 72$, $36 \neq 2 \times 72$, and $36 \neq 3 \times 72$.
It seems you also omitted determining whether or not the relation is reflexive, symmetric, transitive and antitransitive, i.e., "which properties $R$ has".
There's some edges missing, e.g. from $6$ to $108$.
"Looking at the graph of the previous answer A2, it shows that it satisfies all of these properties." Yes, it is obvious from the picture, but perhaps it's worthwhile being a bit more specific. E.g. $R^+$ is, by definition, transitive; it's reflexive since $a\ R a$ for all $a \in A$; it's antisymmetric since if $a\ R\ b$ and $b\ R\ a$ then $a$ divides $b$ and $b$ divides $a$ and hence $a=b$.
Hasse Diagram looks fine to me. Hint for your final question: What is the supremum of $72$ and $108$? (Same argument works for infimum.)