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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.
Quote:
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