Partial Relations: draw Hasse Diagram for partial ordering

[benderzfloozy]

Hi. I have the problem where given the set {2, 3, 5, 7, 21, 42, 105, 210}, I have to draw the Hasse Diagram for the partial ordering of x divides y. I also have to mention the minimal, maximal, greatest, and least elements.

I am confused here. The book doesn't really explain it in great detail. I was wondering if anyone could really break it down for me so I can gain a better understanding. Thanks!

 

[pka]

The above is the Hasse Diagram for divisibly in that set.
Note that it is a minimal graph.
That is, although 3 divides 21, 42, and 210 because of transitivity it is sufficient to have only the edge (3,21).

 

[benderzfloozy]

Hi. Thanks for your help, but I am still confused. I have no idea on what you mean when you say the diagram is a minimal graph. By looking at the diagram, the least element is an element below all others, and the minimal element is one that have no elements below it right?

So, would this mean that 2, 3, 7, 5 are all minimal elements and that 21, 42, and 10 are least elements?

Would the greatest element be one that have elements above it on the diagram, and the maximal element is the one that have no elements above it?

 

[pka]

Hi. Thanks for your help, but I am still confused. I have no idea on what you mean when you say the diagram is a minimal graph.

Do you have any idea about the exact definition of Hasse Diagrams?
One cannot work a problem if one does not bother to learn the definitions.