 Welcome back to our lecture series, Math 42-20, abstract algebra one for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Missildine. This is the first video for lecture five, and you'll notice that we have now left section 1.2. If you're dyslexic, you might have not noticed that. I make fun only because I myself struggle with this a lot. 2.1 is about mathematical induction. And although many of us have seen this before, induction is a very important proof technique in abstract algebra. And so therefore, I do want to review the induction proof technique because I mean, we use it all the time. I remember as being a graduate student studying abstract algebra, I was actually studying from Joseph Rotman's algebra textbook. And I remember doing this like independent study, I felt like every other proof was induction. It felt like in his book. And it turns out it's a very important technique for many things in abstract algebra. So first of all, what is mathematical induction? This is actually a technique about proving statements about sequences of natural numbers and things related to that. And so we're gonna first begin with what's sometimes called the first principle of mathematical induction. I'll just call this the mathematical induction here. And I'll explain why I won't use any other terminology. Some people refer to this as weak induction. But imagine we have some statement about numbers, right? So let S in be a statement about integers, where in here is a natural number. Remember, natural numbers do include zero in our discussion right here. So we have some statement S in about integers. And suppose that S of n minus one is true for some integer as our n zero right there. It doesn't necessarily have to be zero. It doesn't have to be one. It's just some first number in a sequence here. If all integers k, where k is greater than or equal to n sub n. If S k, the statement implies S k and plus one is true, then S n is true for all integers greater than or equal to n sub zero. So this is the principle of mathematical induction. I like to think of it in the following way. An induction proof can be visualized as a string of dominoes, right? So if we're trying to set up these dominoes, we have all these dominoes lined up together and then we want to kind of push them over. As anyone who's seen a sequence of falling dominoes knows, there are two requirements for a good show. So like if you were to go on YouTube right now and look up some domino video, the first thing someone must push over the dominoes, right? Someone's gonna have to push the first domino over. If you just want me to see dominoes standing still for five minutes, that's not gonna be a very good video. At some point, someone has to push the dominoes over and then the dominoes have to be close enough together that when one falls over, the subsequent one falls over as well. Because if there is too large of a gap between the dominoes, when one falls, the next one won't fall, right? And therefore it's like, oh, that was a dud. Nothing, it would stop eventually when things knock over. So this domino effect actually describes how a typical induction proof goes in many, many ways. Because an induction, your typical induction proof has basically three parts to it. The first part is what we call the base case. And we'll do an example of this in just a moment. The base case, what you're gonna do is you're gonna show the statement S, the statement's true for N zero, that starting value. Now your base case is often the number one or the number zero, but there might be situations where the base case makes sense when you look at say seven. Maybe it's a statement about primes and seven is the first non-Fermont primer, you know, whatever. There's lots of reasons why things start where they are. It depends on the statement. So there's some base case. This is like someone's pushing the dominoes over. This shows us that at some point the dominoes start to fall. This is a very, the base case is typically trivial to prove, but it's very important to prove. The next part is what we call the inductive hypothesis. The inductive hypothesis, sometimes called IH for short. Did I spell hypothesis? Correct, I did not. Try that again, hypothesis. There we go. So the inductive hypothesis, what you're gonna do is you're gonna assume, you're gonna assume the statement is true for S sub K. So we're K, the only thing we know about K is that K is greater than or equal to N zero. So we're gonna assume that for some K, some integer K, this statement holds to be true. And then the next part, we'll refer to this as the inductive case, right? Cause we did the base case already. So the inductive application, there's a couple of different words you can call here. We're gonna call it this the inductive case. We're then going to prove the statement for S of K plus one. So we're gonna prove. So basically what you do is you assume it works for SK and then you prove it's true for SK plus one. You might be like, why do you assume this, right? Why are we assuming that? Well, think of the dominoes, right? What we're doing here, when you put the inductive hypothesis together with the inductive application, you're showing that SK and SK plus one as dominoes are sufficiently close to each other, that if SK falls, then SK plus one will fall as well. And since SK was chosen arbitrarily, this means that if SK plus one falls, then SK plus two will fall. And if SK plus two falls, then SK plus three will fall. And if SK plus three falls, then SK plus four will fall. You see where we're going here. This would continue on at an item here. Well, that tells us that the dominoes are sufficiently close together. They're not too far apart, but someone has to actually push them. At some point, the dominoes have to start falling. That's where the base case comes into play. And so when you put these principles together, like the dominoes, induction will then show that the statement is true for all numbers, all natural numbers greater than or equal to N sub zero. And so that's the idea of an induction proof. It's very versatile, but it's very powerful. And it turns out you can use it all the time. Another thing I should mention about induction is that we are not gonna prove the principle of mathematical induction. You'll notice that this is actually called principle 2.11, not theorem 2.11. One, for the most part, does not prove induction because mathematical induction is actually an axiom of the natural numbers. We don't prove axioms. Axioms are things we assume to be true. And some people might be like, well, what do you assume it to be true, right? It doesn't that lead to sort of like bad assumptions or problems with logic and things? Well, without getting into a deep, deep, deep logical discussion about the necessity for axioms, which are unproven truths. Again, if you're really into that, you should check out a video I have from my geometry class, where I actually do get into deep videos like that. But when it comes to axioms, we don't prove them essentially because it's the definition. We don't prove that, well, how do I best say this? Like we don't prove that cats meow because meowing is by definition the sound a cat makes. That it's like the definition. So what does it mean to be a natural number, right? What's the definition of this set in here? Well, it's the set of natural numbers. What's a natural number? If we actually were to define it, you might give me examples like zero, one, two, three, four. That's not the definition of a natural number. Those are just examples of it. If we actually go into the definition of natural numbers, probably the best approach would be the piano axioms for the natural numbers, in which case one of the axioms of the natural numbers is the induction axiom. Thing, something is a natural number only if it has this inductive principle. If you don't have the inductive principle, then you're not really a natural number. So honestly, anything that has the inductive principle is sort of a natural number-like object. And again, I digress too much here. What I wanna do for the end of this video here is actually give you an example of a typical induction argument. And let's do one, just some basic number theory type of thing here. So let's say if we take the sum of one plus two plus three plus four all the way up to N, this is equal to the formula N times N plus one over two. I guess I should call this a combinatorial argument here. And we can prove this by induction. And it's gonna come in three phases here, right? We wanna prove these things are equal to each other. And so when you start an induction proof, you probably should start off by saying something like, we will prove this by induction. So we proceed by induction. And so since this is an induction proof, we're gonna start off with the base case, in which case you might list it as such by the base case. So when is this statement going to be true? You could, one could actually prove this for N equals zero, but I'm just gonna start with N equals one here. So let N equal one, right? In this situation. So consider the left-hand side, namely you're just gonna get a single one by itself, right? Cause you add one plus two plus three up to N, but if N is itself one, you just terminate there already. On the other hand, if we consider the right-hand side of this equation, you're gonna end up with one times one plus one over two, which equals one times two over two, the two's cancel and you're left with just one. And so you can see that equality holds in this situation, and thus the base case holds. Like I said, this base case is often typical, but it is necessary. If you don't prove the base case, we don't actually know the dominoes are gonna be falling down. So then for our inductive hypothesis, what we're gonna do here is we're gonna assume, and I didn't mention this earlier, but I should mention that this right here is our statement. This is our S of N right here. We're trying to prove this statement. So our statement is an equation. We wanna prove this equation holds in general. So for the inductive hypothesis, we're gonna assume that K, how do I wanna say this? We're going to assume that one plus two plus three all the way up to K equals K times K plus one over two, four K greater than or equal to one. That's what we're gonna assume here. So the inductive hypothesis is just an assumption we're assuming it's true. And so then when we come to the inductive case here, I see we didn't need to prove that the statement S of K plus one is true. So we're gonna start with the left-hand side of S K plus one. So consider the sum one plus two plus three all the way up to K plus K plus one. So this is the left-hand side of the K plus one statement. But you'll notice that one plus two plus three all the way up to K here. This is the left-hand side for the S K statement. So now using my inductive hypothesis, I see that one plus two plus three up to K, this is equal to K times K plus one over two plus K plus one. So this right here, this first equality is where I use my inductive hypothesis. You don't have, you probably should. I mean, when you write a proof, you should make it very clear where you're using your deductive hypothesis. This doesn't just make it a good proof, but it also helps you as the learner better understand how induction works. So be very explicit about your induction hypothesis. I used it right there. The rest of this exercise is just gonna need to try to add these together, right? I should have a common fraction, right? So you're gonna get K plus one. I wanna have a common denominator of two. So I'll take two over two here. And so if we add these together, this is just principles of addition at this moment. We have K times K plus one. We're gonna have two times K plus one, like so. I noticed that we have a K plus one and a K plus one. I'm gonna factor this thing out. And this gives us K plus one times K plus two over two. And notice that K plus two, of course, is just K plus one plus one, which then proves the case. And so then we'll make a comment about this, right? So this proves the identity for K plus one. All right, so we've now finished the inductive case, this inductive hypothesis. So then we'll say something like the following. Therefore, the result follows by induction. And then to finish your little proof, you're gonna draw your little tombstone right there. This QED symbol actually, it's supposed to represent a tombstone. Cause when you actually look at the Latin phrase QED, it's an acronym for Latin phrase, which I won't say it at the moment. You can Google it sometime if you want to. But basically, it's a Latin phrase that means the question is dead. Our doubt about the truthfulness of this statement is now dead. And so you put a little tombstone to commemorate that our doubt is now dead. How morbid and sweet that is, isn't it? But this gives us the typical proof by induction, a base case, an inductive hypothesis and an inductive application. If you do all three steps, then the statement will then follow by induction. In the next video, I'll do some more examples of proofs by inductions, some other type of number theoretic, combinatorial statements. Take a look at that if you wanna see some more examples how to prove things by induction.