Thread: Hasse Diagram

Draw the poset diagram (Hasse) diagram for the poset ({a | a is a positive integer divisor of 20}, <=), where <= is a denotes the divisibility relation.

I have only worked on two other Hasse diagrams and they were much easier. I not sure if you can construct the diagram on the forum, so an explanation might have to do. Thanks for your help.

Originally Posted by Plato

Thanks Plato! So basically 1,2,4,5,10, and 20 divide into 20 evenly and are positive integers. I see that. How do you know which ones to connect each one to another? Thanks!

Generally the idea is to picture a poset in a minimal way. If a<=b and there is no c such that a<=c<=b then there is an edge between a and b in the diagram. We arrange the graph so as to give the relation is top-down.

In the diagram 2 divides 4, 10, & 20. There are edges between 2 and 10, 2 and 4, but not between 2 and 20 because they are strictly adjacent.

Would I be right if I said ((1,2,4,5,10,20} <=)? I am still working on understanding the arrangement of elements. Thanks.

Are you sure that understand just what a relation is?
A relation is a set of ordered pairs.
If A is a set a relation is a subset AxA. In the case at hand, the positive divisors of 20, x is related to y, (x,y) is in the relation, if and only if x divides y. I will list the complete relation: {(1,1),(2,2),(4,4),(5,5),(10,10),(20,20),(1,2),(1, 4),(1,5),(1,10),(1,20),(2,4),(2,10),(2,20),(4,20), (5,10),(5,20),(10,20)}. That is what is shown in the Hasse diagram.

Ahhhh, okay I see. Thank you Plato for your help on this problem!

I think the following page has the hasse diagram you need

hasse diagram of {2,4,5,10,20}under the relation aRb iff a divides b