Show that if A1, A2, …, An are sets where n ≥2, and for all pairs of integers i and j with 1 ≤i<

step 1 Hi, in this video we are asked to prove the following statement. So if we have a set A1, A2 to A

Show that if A1, A2, …, An are sets where n ≥2, and for all...

Key Concepts

Partial Order

A partial order is a binary relation over a set that is reflexive, antisymmetric, and transitive. In the context of sets, the subset relation (?) is a classic example of a partial order because every set is a subset of itself (reflexivity), if one set is a subset of another and vice versa then they are equal (antisymmetry), and if one set is a subset of a second and that second is a subset of a third, then the first is a subset of the third (transitivity). This concept provides the framework for comparing sets in terms of their elements.

Chain in a Partially Ordered Set

A chain is a subset of a partially ordered set in which every pair of elements is comparable; that is, given any two sets in the chain, one is a subset of the other. This total comparability implies that the sets can be arranged in a linear order by inclusion, which is crucial for arguments about minimal or maximal elements within the collection.

Least Element

A least element in a partially ordered set is an element that is a subset of (or related to) every other element in the set. In the context of chains of sets, proving the existence of a least element means showing that there is one set that is contained in every other set in the chain. This concept is often used to demonstrate minimality or a form of 'base case' in ordered structures.

Subset Relation

The subset relation is a fundamental concept in set theory where one set is entirely contained within another set. Understanding this relation is key in problems involving collections of sets, as it allows for the ordering of sets by inclusion and facilitates reasoning about chains and minimal elements.

Transcript

00:01 Hi, in this video we are asked to prove the following statement.

00:05 So if we have a set a1, a2 to an, they are sets such that any pair of a .i .a .j.

00:18 That is not the same set.

00:21 We have either ai is a subset of a .j.

00:26 Or a .j is a subset of ai.

00:28 This properties means that all ai are compatible or comparable in a sense that we can know, we pick two of them, we can know which one is smaller or which one is a subset of another.

00:49 We can do this for any pair we picked.

00:53 If you are familiar with the concept of partial order or total order, you will.

01:00 Recognize right away that this is a total order on on this collection of sets and then there is next statement then there is i0 here i will just name it i zero there is an integer between one and such that a of i0 so there is this spatial a that acts as the smallest subset among all of them so this a i0 will be a subset of all other a j in this in this collection so we don't know what the number is but we know they exist or we want to show that they exist okay and there are many ways to do this.

02:00 I will show one of them.

02:02 I will use the proof by suppose the contrary and find contradiction.

02:12 Okay, so to suppose the contrary, the statement here said that there exists this special i -0 that acts as the smallest, like ai0 act as the smallest subset.

02:27 So i will suppose that it doesn't exist.

02:32 When there is no such integer i -0, what happened? then it means any set in the collection that we choose cannot be the smallest element or smallest subset, right? otherwise it will be that i0 that we assume not exist so any a j can be smallest which mean there must be another another set smaller than it so there will be some other a i that is a subset of a j that we choose always and you probably can guess what this will lead to but i will just write it out anyway.

03:28 So when this follows from our assumption, what will happen? well, first consider a1...

Please give Ace some feedback