[Discrete Mathematics] Indexed Sets and Well Ordering Principle

Today we discuss indexed sets and the well ordering principle.

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Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

[Trevtutor]

It comes with video lectures, text lectures, practice problems, solutions, and a practice final exam!

[RoszyPeterz]

I'm grateful for a comment section because I got confused at 5:40.
I know Tutor Trev once mentioned that positive integers are a subset of Natural numbers meaning, 0 is not the least element of the positive integer.

[Nabilliban]

For the positive integers, wouldn't 1 be the smallest element, not 0?

[yoitslemonboy]

correct me if i am wrong but i think there is a mistake on 1:55 that summation starts with a1 when the index starts with 0 so it show really be
...= a0 + a1 +a2... an

[adamisforgiants]

Thanks for the video! Why is negative infinity considered to contradict the idea of a least number? Although the sign is different, wouldn't the magnitude indicate that the least number is still zero?

[bradalex]

Tysm for what you post here. It'd be so helpful if you can get your site working. Maybe I can find something that would help me with the mid term exams that are scheduled from Oct 18.

[jeremiahmort]

I have read my text over this section several times, watched your videos pertaining to the subject (very helpful by the way) and I am still confused on how I am supposed to answer here. After working on this problem for an hour and still being uncertain I would like to ask for some help. Here is the problem:

Let Ai = {…, -2, -1, 0, 1, …, i}. Find
n
a) ∪Ai = I think this is all integers but I don't know.
i=1

n
b) ∩Ai = And I have no idea here.
i=1

I want the answer but more than that I want to understand what this is even asking. Thank you again for your videos! They have been very helpful.

[nickblick]

I'm probably being dumb here as I'm using these clips to learn but wouldn't a "non empty subset of N" include a set with a single number in it e.g { 5 } is a subset of N, and for it to have a "least element" does it need another number in the set for it to relate to as < is a binary operation? Or is it just that it is the smallest element by virtue of it being the only one in the set?

[MagnusTheUltramarine]

in the beginning what is the meaning of A1?
I mean, A1, A2, A3.. are all subsets of a bigger A?
like A={1, 2, 3, 4, 5, 6}, so A1={1, 2, 3} A2={4} A3={5, 6}
or they are all different sets, like A1={1, 2, 3} A2={6, 7, 1, -5}?

[jumpercrumper]

I dont understand, wouldnt there still be a least element for the set of all integers? Even if we tend to negative infinity for the universe, couldnt you just say the least element is the number i that satisfies i < j for all j != i in the set?

[rebekahshtayfman]

Hi Trev,

I was wondering if maybe you could answer a quick question for me...Is there an error in The Book of Proof, or am I not seeing something? On page 26, example 1.11, it says, "Let I = [0, 2] = {x ε R : 0 <_ x <_ 2}..." and then it gives more information with different ranges for x (between 1 and 2). How is it denoted that x is between 0 and 2? I thought that would be alpha, not x.

[XXgamemaster]

Why can't the Well-Ordering Principle be extended to the integers? Suppose we have a non-empty subset of the integers, then wouldn't it also be the case that there is a smallest element?

Let's say I have the set {1, 2, 3} which is a non-empty subset of the integers. The smallest element here is clearly 1. If the set consisted of negative integers, say {-1, -2, -3} then the smallest integer would be -3. I don't see why WOP doesn't hold for the integers...

[DiegoCastro-oome]

I'm pretty sure you can prove the well ordering principle using induction

[DeathToAmericaAllahuAkubar]

The well ordering principle is not an axiom. It is a theorem and it should be proved. The video is great though.

[kevinjones]

"Yeah let me just start talking out of my ass. That should make a good video"