Let $A = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$.
(1) Let $\rm R$ be the relation on the set $A$ defined as follows:
$∀a,b ∈ A, a\rm Rb$ $\iff$ every prime number that divides $a$ is a factor of $b$. Explain why the relation $\rm R$ is not a partial order relation.
(2) Let $\rm R$ be the relation on the set $A$ defined as follows: $∀a,b ∈ A, a\rm Rb$ $\iff$ every prime number that divides $a$ is a factor of $b$ and $a ≤ b$. The relation $\rm R$ is a partial order relation (you do not need to prove this). Draw the Hasse diagram for $\rm R$.
I found this question in my tutorial sheet, and I was unsure how to begin. From my understanding, a partial order relation is a relation that is Reflexive, Antisymmetric, and Transitive, and Im pretty sure I could do part(1). For part (2) im not quite sure how to go about drawing a Hasse diagram for the relation $\rm R$. Any assistance would be much appreciated. Thanks in advance :)
For #2: observe that $2$ is related to all the elements in the set $\{2,4,6,8,10,12\}$. Likewise $3$ is related to all the elements in the set $\{3,6,9,12\}$ and $9$ is related to $\{9,12\}$ because $3$ is the only prime that divides $9$ and $3 \mid 12$ and so on.
The Hasse diagram mainly exhibits the direct relationship between two elements (loops for reflexive and edges coming out of transitivity are dropped). So