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

In summary, the problem is trying to figure out the greatest, least, maximal, and minimal elements for the "divides" relation on a set of numbers.
  • #1
mr_coffee
1,629
1
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!
 
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  • #2
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.
 
  • #3
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.
 

1. What is the "divides" relation in mathematics?

The "divides" relation is a mathematical concept that describes the relationship between two numbers, where one number can be evenly divided by the other without resulting in a remainder. For example, 5 divides 15 because 15 ÷ 5 = 3 with no remainder.

2. How do you find the greatest element for the "divides" relation?

The greatest element for the "divides" relation is the largest number that can divide both numbers in the relationship without resulting in a remainder. This is also known as the greatest common divisor (GCD) or highest common factor (HCF). To find the greatest element, you can use methods such as prime factorization or the Euclidean algorithm.

3. What is the difference between the maximal and greatest elements for the "divides" relation?

The greatest element is the largest number that divides two numbers without a remainder, while the maximal element is the largest number that divides at least one of the two numbers without a remainder. In other words, the maximal element may not necessarily divide both numbers, but it is still the largest number that is a factor of at least one of the numbers.

4. How do you find the least element for the "divides" relation?

The least element for the "divides" relation is the smallest number that can divide both numbers in the relationship without resulting in a remainder. This is also known as the least common multiple (LCM). To find the least element, you can use methods such as prime factorization or the Euclidean algorithm.

5. Can there be multiple greatest, least, maximal, or minimal elements for the "divides" relation?

Yes, there can be multiple greatest, least, maximal, and minimal elements for the "divides" relation. For example, for the numbers 6 and 12, the greatest element is 6, the least element is 12, and the maximal and minimal elements are both 3. This is because all of these numbers can divide both 6 and 12 without resulting in a remainder.

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