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I am still having a little difficultly with the covering relation, specifically that when y covers x, $x \prec y$ there is no element in between them, $ x \prec z \prec y$, where x,y, and z are elements of the arbitrary set in question.

For instance, I am looking at the problem

Draw the Hasse diagram for divisibility on the set

a) $\{1,2,3,4,5,6,7,8\}$

From what I have read, in order for one to be able to construct a Hasse diagram, you have to use the covering relation.

But in this example, it seems that the covering relation is contradicted. For example, $1|3$, and nothing comes between them; and $1|6$, but in this circumstance, 3 is between 1 and 6. Could someone please help me?

Thank you!!

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Here’s a rough sketch of the Hasse diagram:

enter image description here

The covering relation is therefore

$$\Big\{\langle 1,2\rangle,\langle 1,3\rangle,\langle 1,5\rangle,\langle 1,7\rangle, \langle 2,4\rangle,\langle 2,6\rangle,\langle 3,6\rangle,\langle 4,8\rangle\Big\}\;.$$

I’m not sure, but I think that you were confusing membership in the covering relation with membership in the set $\{1,2,3,4,5,6,7,8\}$, which is the underlying set for both the divisibility relation and its associated covering relation.

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