Finding Greatest, Least, Maximal & Minimal Elements for "Divides" Relation

[mr_coffee]

confused on the "divides" relation, find all greatest, least, maximal, minal elememen

Hello everyone I'm not sure how I'm suppose to do this...

The problem is the following:
Find all greatest, least, maximal, minimal elemnts for the relations.

It says to find that on exercise 16b. I looked at 16b. and it says:

Consider the "divides" relation on each of the following sets A. Draw the Hasse diagram for each relation.

b. A = {2,3,4, 6, 8, 9, 12, 18}

Well i looked up divides relation and its defined as the following:

a|b if and only if b = ka for some integer k.

Im not sure how figure anything out with that definition but maybe I'm missing somthing.

in part a of this problem, htey found the following hasse diagram if this helps any:
A = {1,2,4,5,10,15,20}
20
4 10 15
2 5
1

20 connects to 4 and 10
4 connects to 2, 2 connects to 1
10 connects to 5 and 5 connects to 15 and 1

i have no idea how they got this either, any explanation would be great!

 

[HallsofIvy]

I would think that anyone who can do arithmetic would know about "divides"! "a divides b" if and only if b/a is an integer.

It was "Hasse diagram" you should have defined! I had to look that up. A Hasse diagram of a set of numbers has the largest number at the top, smaller numbers below with lines going downward from the large numbers to those smaller numbers that divide it.

The numbers are {2,3,4, 6, 8, 9, 12, 18}. 18= (2)(9)= 3(6) so 2, 3, 6, and 9 all divide 18 and there should be lines from them to 18. 12= (4)(3)= 2(6) so there should be lines from 2, 3, 4, and 6 to 12. 9= (3)(3) so there should be a line from 3 to 9. 8= (2)(4) so there should be lines from 2 and 4 to 8. 6= (2)(3) so there should be lines from 2 and 3 to 6. 4= (2)(2) so there should be a line from 2 to 4.

 

[mr_coffee]

Thanks again Ivy, I got the hang of the diagram after awhile. The transitivity property kicks in so you don;'t have to draw all the lines to 12 if there is already a "node" connecting to it.