Define a relation $D_n$ on $S$ by $xD_ny$ if and only if $x\mid y$. Determine if it's a poset.

Question

Here is the question I am currently working on (screenshot):

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I'd appreciate some suggestions/guidance for part (a), proving that $D_n$ is a partial order.

Reflexive: Let $x \in \mathbb{Z}$ such that $xD_nx$.

Let $n$ be a positive integer such that $x \mid x$ can be restated as $x=xn$. Thus, $xD_nx$ and $D_n$ is reflexive.

Antisymmetric: Let $x,y \in \mathbb{Z}$ such that $xD_ny$ and $yD_nx$.

Thus we have that $x\mid y$ and $y \mid x$. Let $k,j \in \mathbb{Z}$ such that $xk=y$ and $yj=x$. By substitution, we have that $x=xjk$ and $y=yjk$. This implies that $x=y$ and that the relation is antisymmetric.

^I definitely feel like there are some holes here, and this is probably not correct.

Transitive: Let $x,y,z \in \mathbb{Z}$ such that $xD_ny$ and $yD_nz$.

Let $m,n \in \mathbb{Z}$ such that $xm=y$ and $yn=z$.

^ This is where I am stuck for transitivity. Generally I would do some type of addition to get to reach the proper conclusion, but I don't think that would work in this case.

Thanks for the guidance! As for part (b), I am comfortable drawing Hasse diagrams, with the exception that I'm not sure how to plug in these numbers for $S$ into the relation.

I'm terrible at these relation questions, and working through a review for my final next week. Up until this point, I have been mostly okay with the intro/proofs class. If anyone has any good suggestions for sites/resources, I'd appreciate it. I haven't found anything too great online.

Answer 1

You are done though! $$xm=y, yn=z \implies xmn= z \implies x|z \implies x D_n z$$

Substitution, you were doing so well.

Answer 2

Reflexive: Does $xD_nx \forall x\in S$? Yes, because $x|x \implies x|x$.

Anti-symmetric: Does $xD_ny \land yD_nx \implies x=y$? Your proof here is correct, you might want to add $jk=1 \implies j=1,k=1$.

Transitive: Does $xD_ny \land yD_nz \implies xD_nz$? We have $jx=y$ and $ky=z$ so $z=ky=kjx \implies jkx=z \implies xD_nz$.

For the Hasse diagrams, I would use horizontal levels $\{1\}, \{2,3\}, \{4,6\}, \{8,12\}, \{24\}$ for $n=24$. $D_{16}$ is ordered.