 So in this final video for lecture five, it turns out that there is another version of the induction axiom, referred to as the well ordering principle. This one is crafted a little bit differently, so let me first define what a well-ordered set is. So first of all, let S be a partially ordered set. We will properly define this later on. Basically just means that if we have elements X and Y, we could decide if X is less than or equal to Y. We'll define what that is more properly later on, but we've probably seen something like this before. We say that a set is well-ordered if every non-empty subset contains a least element. So if you take any subset of the set, there's always a smallest element inside that set. The natural numbers are a well-ordered set. And so this is called the well-ordering principle. The set of natural numbers is well-ordered. If you take any subset of natural numbers that's non-empty, there always will be a smallest natural number inside of that set. It turns out that the well-ordering principle is equivalent to the induction axiom. The natural numbers, in fact, is the canonical example of a well-ordered set. Well-ordered sets are trying to generalize this principle for the natural numbers. And the well-ordering principle is equivalent to the induction hypothesis on the natural numbers. Now, when I talk about the well-ordering principle for the natural numbers, that is not the same thing as the well-ordering principle for the real numbers. The well-ordering principle in that context is actually for the real numbers is equivalent to the axiom of choice, which is a much deeper mathematical topic that I'm not talking about right here. The well-ordering principle of the natural number is actually just the induction hypothesis. I'm sorry, the induction principle. And the equivalence of the well-ordering principle with the induction principle is really just by showing a statement is true by induction if and only if it's true by looking at the smallest counter-example argument. So what do we mean by the smallest counter-example? Smallest counter-example. We'll do a proof of this in just a second. So this is a proof by contradiction, in which case you're like, if a statement is true for all natural numbers, let's prove this by contradiction. Let's assume the statement's not true for all natural numbers. Well, by the well-ordering principle, if the statement is not true for some natural numbers, there's a smallest natural number where that statement is not true. So the smallest counter-example, okay? The well-ordering principle gives us that. And then you argue that taking your smallest counter-example, that its predecessor as a natural number is also a counter-example, thus violating that we have the smallest counter-example present. And so you can argue logically speaking that this well-ordering principle is equivalent to the induction argument, the induction proofs we did earlier by showing that a smallest counter-example proof is essentially the same argument just in a slightly different direction. So if you're ever asked to prove something by induction, you can try to prove it using the smallest counter-example. And as an example of this, let n be a positive number. Let's prove that the sum of the first in odd integers is equal to n squared. So the sum of one plus three plus five all the way up to two n minus one, that equals n squared. The sum of odd numbers adds up to be a perfect square. So we will prove this, we will prove this using, we'll prove this by smallest counter-example. Let me phrase it that way. By smallest counter-example. Great, how does one do that? So then we talk about that. So like by proof of, by way of contradiction, assume the statement is false. The statement is false, okay? So then by the well-ordering principle, our WAP right here, right? By the well-ordering principle, there exists a smallest counter-example. Let's call it K, a smallest number K. Such that one plus three plus five, going all the way up to two K minus one, this doesn't equal K squared, all right? So we can then kind of check, so we have this smallest counter-example, right? So we can notice like note that K doesn't equal one since, in that situation you get one equals one squared. So the sum of odd integers from one to one is just one, and then one, that's equal to one squared. So K is not one, because that would be sort of like the, that doesn't work here, right? And so then let's consider the number smaller than K, right? So note also that K minus one is less than K, right? And so because of that, since it's smaller than it, that implies that one plus three plus five, all the way up to two times K minus one, plus one, sorry, minus one there, this is equal to K minus one squared. So because K minus one is strictly smaller than K, that means that the statement has to be true for that one because K was the smallest situation where that actually works. And so when you actually start putting these things together then, if you were to take this statement right here, you can see that this thing is gonna be equal to K minus one, K minus one squared, plus two K minus one, because basically if you take the first, the sum of the first few terms here, which is this friend right here, that's gonna equal K minus one squared. And then if we start working through this equation, notice if you foil out the K minus one squared, you get K squared minus two K plus one, plus two K minus one, you're gonna see some cancellation right there, right? Two K, cancels with the two K, the ones cancel, and you're left with just K squared. And so this right here then shows that the K minus one, basically you would see that it's like, oh, it violates that, right? So we get a contradiction at this moment because we assume that K was a counter-example. So this statement didn't happen, right? Equality did not hold right here, but it did, we show that it did, right? And so we got a proof by contradiction. And so therefore the statement holds for all, for all natural numbers, statement holds for all n. And so you can see in this proof, how similar this proof by smallest counter-example feels like an induction proof, right? Because we kind of have an assumption, kind of like the induction hypothesis that follows because of our counter-example. It's not exactly what happens, but the fact that K is the smallest counter-example means its predecessor has the property. So we kind of get our induction hypothesis. Then we show that because this statement holds, it holds for K, which gives us a counter-example because K was the smallest counter-example. Why did we show, why did we show that K couldn't equal one? Well, the reason we show that K didn't equal one is because in our original assumption, K was supposed to be a positive number. So is K minus one positive or negative? Well, if K is bigger than one, then K minus one will be positive. So we had to kind of do this initial case to make sure that it didn't happen there, kind of like a base case. This proof by short, smallest counter-example is very much the same proof as an induction proof. And so although this doesn't show the two techniques are logically equivalent, you can see here the similarities. And I think you can kind of imagine how the rest of it would work. So this finishes our lecture five about induction. This is hopefully a good review. This probably is not your first exposure to induction, but we can always benefit from some more practice. We'll use induction throughout this semester in this course, right? And so feel free to use it on assignments, homework and proofs that yourself are doing and we'll use it in proofs we see all the time in abstract algebra. If you liked what you saw or you learned something, feel free to hit the like button. Feel free to just subscribe if you wanna learn some more about abstract algebra or other mathematical topics. And if any questions come up along the way while you're watching these videos, feel free to post those in the comments below. Hopefully I'll see you next time for lecture six. Have a great day, everyone. Bye.