What is the covering relation of the partial ordering {(A, B) | A ⊆B} on the power set of S, whe

What is the covering relation of the partial ordering {(A,...

Key Concepts

Power Set

The power set of a set S is the set of all possible subsets of S, including the empty set and the set S itself. It is a key concept in set theory and is used to explore various structures and relationships between subsets, particularly in problems involving inclusion relations.

Covering Relation

In a partially ordered set, the covering relation captures the idea of an immediate successor or a 'next step' in the order. Specifically, an element B is said to cover an element A if A is strictly less than B with respect to the order, and there is no intermediate element C such that A is less than C and C is less than B. This concept is crucial for constructing simplified visual representations of posets, like Hasse diagrams.

Partial Order

A partial order is a binary relation defined on a set that is reflexive, antisymmetric, and transitive, meaning that not all elements need be comparable. It is a fundamental concept in order theory that allows us to structure elements in a way that some are considered 'less than' others in the order, even if the relation does not apply to all pairs.

Transcript

00:01 So they want us to come up with the covering relation for this partial ordering where a -b and a is a subset of b on the power set of s and s is just a -b -c.

00:14 Well, if you're unsure of what this definition is for a covering set, they explain it to us on page 6 -6, or not 6 -6 -55 that they talk about this.

00:31 Essentially what they say it is is if we make the host diagram the edges between those will be our covering set so let's go ahead and start so with the power set well we have just the set s itself so up top something that's going to be a subset of all these or contain every element which is a b c so we have this up top here now the next kind of sets we can have are sets of element size 2.

01:06 So we could have a, b here, and then we could have a, c, and then we could have b, c.

01:20 So remember, i'm making the hoss diagram here, so we don't need to draw the self -relations or the directed for this, because it's all implied going up.

01:31 So those would be the sets of element size 2.

01:34 Now, what about the sets of element size 1? well, that would be a, b, and c.

01:42 And, well, how do these relate? well, a is a subset of ab, and a is a subset of ac.

01:48 B is a subset of a -b, and is a subset of b -c, and then c is a subset of ac, and is a subset of b -c.

01:57 And then, remember, we have a set of size 0, just being the empty set.

02:02 And so this would be a subset of all those.

02:04 So normally, remember, we would draw like the empty set going to here, to here, to here, and to there.

02:11 And likewise, a would also want to go to here, b would go to here, c would go to here.

02:19 But remember, since we're doing the hoss diagram, it's implied the transitivity properties hold.

02:25 So we don't need to draw all those other lines.

02:27 And doing it this way, it's not as messy either.

02:29 So we have this now.

02:31 Well, all we care about is the ones that have these edges attached.

02:38 So let's just go from the bottom and we can work our way to the top.

02:42 So first we have, so v and a, so, or empty set, and a.

02:50 So this is going to be one part of our covering set, our covering relation.

02:57 So let's go ahead and erase that so we don't try to use it again...

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