Find a compatible total order for the divisibility relation on the set {1,2,3,6,8,12,24,36}

Name: Problem 62, Chapter 9: Discrete Mathematics and its Applications
Uploaded: 2019-09-17T19:02:35-08:00
Duration: 5 min
Channel: Prathan Jarupoonphol
Description: Find a compatible total order for the divisibility relation on the set {1,2,3,6,8,12,24,36}

Find a compatible total order for the divisibility relation...

Key Concepts

Total Order

A total order, or linear order, is an ordering where every pair of elements is comparable. This means that for any two elements in the set, one must precede the other according to the ordering rule. A total order that is consistent with the divisibility relation must arrange the elements so that if one element divides another, it appears before it in the order.

Linear Extension

A linear extension is a total (or linear) ordering of the elements of a partially ordered set that respects the original partial order. In this context, finding a compatible total order means constructing a linear extension of the divisibility relation on the set, ensuring that if an element divides another, it comes earlier in the extended order.

Partial Order

A partial order is a binary relation that is reflexive, antisymmetric, and transitive. In the context of divisibility, some elements may be incomparable since one might neither divide the other nor be divided by the other, which leads to a partially ordered set (poset).

Divisibility Relation

The divisibility relation is a way of ordering numbers where one number is considered less than or equal to another if it divides the other without a remainder. It naturally forms a partial order because not every two numbers are comparably related via divisibility.

Transcript

00:01 In this question we are asked to find a compatible total ordering for this following set with divisibility relation.

00:11 So when we have a post set is and some some ordering, when we say compatible total ordering, it means we want a total ordering such that whenever we have a real order, in our partial ordering, it will also appear in our total ordering.

00:43 So can we just use the partial ordering as it as is? well, it depends.

00:52 In this case, our divisibility is not total order.

01:01 By that, by that it means that not everything is compatible to each other, but we want to find.

01:10 Some total ordering that still preserve the order of things inside this divisibility ordering.

01:20 And one way to help solving this is to first draw a healthy diagram because it will contain all the information we need and also simplify the ordering enough so that we don't have to look to every single like pair of of elements okay so i have drawn the hacd diagram out this is divisibility of the set and here we can create total order by just going from starting from the bottom make our way up and on each on each layer the elements on the same layer we can make up their precedence or their relation as we want so let me show what i mean so we start with one right next floor has two and three anything can come first so we can have three first and then two later it doesn't matter what matter is this two both of them must be below the next floor which is six, and next floor is 12.

02:54 And now again, last floor is 24 and 36.

03:01 I can have 24 first, 36 later.

03:06 This inequality side does not mean divisibility anymore.

03:16 It would mean just some ordering that we arrange up...

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