Answer these questions for the poset ({3,5,9,15, 24,45}, |) a) Find the maximal elements. b) Fin

step 1 So they want us to answer these questions about this post set here. So we're using division or d

Answer these questions for the poset ({3,5,9,15, 24,45}, |)...

Key Concepts

Partially Ordered Set (Poset)

A poset is a set equipped with a binary relation that is reflexive, antisymmetric, and transitive. This structure allows elements to be compared in a way that does not necessarily yield a comparison for every pair, distinguishing posets from total orders.

Divisibility Relation

Divisibility is a classic example of a partial order when considering integers. One number divides another if there is an integer multiplier such that the product equals the latter. This relation naturally satisfies the conditions of reflexivity, antisymmetry, and transitivity required for a poset.

Maximal and Minimal Elements

In a poset, a maximal element is one for which no other distinct element is strictly greater, meaning there is no element in the set that it is related to as 'smaller.' Conversely, a minimal element has no distinct element that is strictly smaller. These definitions help identify elements that cannot be exceeded or undermined within the ordered structure.

Greatest and Least Elements

A greatest element (or maximum) in a poset is an element that is greater than or equal to every other element, while a least element (or minimum) is one that is less than or equal to all other elements. Their existence implies a unique envelope that encompasses all elements in the poset, which is a stronger condition than being merely maximal or minimal.

Upper and Lower Bounds, Least Upper Bound and Greatest Lower Bound

An upper bound of a subset within a poset is an element that is greater than or equal to every element of that subset, whereas a lower bound is an element that is less than or equal to every element of the subset. The least upper bound (or supremum) is the smallest among all upper bounds, and the greatest lower bound (or infimum) is the largest among all lower bounds, provided these exist within the poset.

Transcript

00:01 So they want us to answer these questions about this post set here.

00:04 So we're using division or divisibility of on 35, 9, 15, 24, and 45.

00:12 So something that will help is if we make the hos diagram for this.

00:15 So let's go ahead and do that first.

00:17 So i think it'll be easier if we start on the right, since we're doing divided by.

00:22 So we'd have 45.

00:26 Then 45 doesn't divide anything in this set.

00:28 24 doesn't divide anything in this set either so these would both be on top here 15 divides 45 so we'd have 15 here 9 divides 45 and then 5 divides 45 but it also divides 15 so we have that there and now 3 divides we'll divide 9, it would divide 15, it would divide 24, and then it also divides 45.

01:12 All right, so remember all these should be directed in this direction here, but when we do the haas diagram, we assume it's directed upwards.

01:22 But the other thing about the osd diagram is if we have one path going from one number to another, we don't need to draw any extra paths to it.

01:32 So for example, we remember we're assuming transivity in this.

01:38 So for five, five divides 15, but 15 divides 45.

01:42 And then we also have five divides 45.

01:44 Well, we don't need that one.

01:50 Well, three divides nine, which divides 45, but we also have three dividing 45.

01:55 So we can go ahead and get rid of that one there.

02:00 And so then this here would be our diagram.

02:02 And we just would erase the directions on all these.

02:08 All right, but i'll just go ahead and leave the arrows because it really won't matter all that much.

02:13 So we have this, though.

02:14 So this is the hoss diagram.

02:19 So maximal elements are going to be elements where there is nothing that is above it in the hoss diagram.

02:27 You can think about it that way.

02:28 Or in our case, we want elements that do not divide anything else.

02:34 So in that case, that would be 24 and 45.

02:39 And then we can just say since these do not divide anything in our set s.

02:57 All right.

03:00 Now for minimal elements, so instead of...

03:08 Actually, let me go ahead and scoot this over a little, because i'll need to move this down as we go.

03:13 So for the minimal elements now, well, the maximal meant nothing.

03:19 Could divide it.

03:21 So a minimal means that we have it on the opposite side.

03:30 So it can divide things, but nothing can divide into it.

03:34 So that would be like the bottom of our chart here, which is going to be five and, so three and five.

03:43 Since five divides 15, three divides nine.

03:50 So we just need to show that they divide one, but nothing.

03:57 Divides 3 or 5 in s.

04:03 Okay.

04:05 Now, we want to see if we have a greatest element...

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