College Algebra: Prove Formulas Using Induction

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Now a lot of places in this course we actually prove that certain things hold true, like the formula for distance and the slope and so forth. We actually sat down and really saw why those things were true. In fact, in mathematics that really is what's at the heart of this business. Namely, why things are true, not just a formula for something, but the proof or the verification that that formula really holds in general. Although we didn't do a lot of proving of results and proving of formulas or proving of theorem, there's the technique that's actually really powerful and sort of fun to look at to actually prove a whole bunch things true all at once.
I want to tell you the idea first and then take a look at some examples, because in fact, they dovetail nicely in our discussions about sequencing and numbers and so forth. Now, let's just think about some of these from a philosophical point of view. Suppose that I had a whole bunch of things, a whole bunch of statements or a whole bunch of whatevers. I need at the very first statement were to be true. So, that first statement, that's true. Suppose that I tell you the following was a fact. If you knew that a particular statement in this list of statements was to be true, then that would actually force the next statement to be true. Let me say it again what I'm talking about.
Suppose I have a list of statements and I tell you that the first statement is true. Just the first statement is true. Then I tell you a basic sort of fact about life. That if you know a particular statement in this list is true, that forces the next statement to be true. I claim that just those two principles allow me to say that every single thing is true and here's why. Since I know the first thing is true, then with the second principle, which tells me if I know something is true the next thing is true, that means that if the first thing is true, the second thing must be true. For the second principle is that if something is true the statement that follows it must be true.
But now if I know the second statement is true, that forces the third statement to be true. If I know the third statement is true, that forces the fourth statement to be true and so on. So, in fact, I get all the statements true, for free, just my knowing that the first statement is true and the basic principle that if any statement is true that forces the next statement to be true. In fact, there's a great way to think about this. It really is sort of thinking about dominos lined up. Take a look here what I made. Here I have a whole bunch of dominos. Think of these as actually statements. For the statement to be true, that means the domino falls.
Now think about it for a second. Just sort of from a domino point of view. It's definitely the case that if this domino were to fall that would force this domino to fall. Watch. You see? So, if one domino falls then we know the next domino will fall. So, if I know that the first domino falls, in particular if I push the first domino, that should force all the dominos to fall. Thus proving that every single statement is true. So, look what I'm using? The fact that I know the first domino falls. I know the first statement is true and I know the general fact that if any domino falls then its neighbor domino will fall as well.
That tells me that all the dominos must fall. That's induction. Let's see if induction works. Are you ready? I'm nervous. Okay, so this is partial induction. This is induction where these fall, but then nothing else falls. Not as powerful as it could have been, but let's just see what happens here. Okay, so that's a two step induction, which I just made up, but I think you get a sense of what's going on here. The idea is something falls, that causes the next thing to fall, causes the next thing to fall and so on.
So, in fact, that really is at the heart of induction. That allows us to actually prove things inductively. Where here's the recipe. You prove that the very first case is true. You check that. Just check the first case is true. Then you prove the following is true. That if an arbitrary statement in this list is true, that forces the very next statement to be true as well. If you can do that, if you can show that the first thing is true and that if any one is true that forces the next one to be true, then you've really have got a dominosque kind of thing going on here. Because what you really see is that you prove that well, the first one's true and they all sort of line up and you get them all to fall. Up next I'll show you some real specific math examples that go beyond dominos and actually take a look at actually proving certain formulas are true no matter what values you plug in. We'll take a look at those formulas and induction in action up next. I'll see you there.
Further Topics in Algebra
Induction
Using Formulas Using Mathematical Induction Page [1 of 1]