 In our previous lecture, we introduced the method which we call direct proof, which if you're trying to prove a statement which is conditional in form, so you're trying to prove the statement P implies Q, the template for the proof is the following. You assume the statement P is true, maybe unravel some definitions, bridge some logical gaps, and then you conclude that the statement Q is true as well, based upon the assumptions you made here. This is the method of direct proof. In this video, I want to provide some more examples of how do you prove something directly, unlike in the previous video where we actually sort of step-by-step wrote the proof together. In this video, I'll show you the proof already finished or de-polished, but try to motivate where this proof came from. And much like in the previous video, we're going to be focusing on conditional statements about integers and real numbers, trying to focus on arithmetic and algebraic properties that you may already be familiar with, as opposed to introducing sort of novel mathematical concepts. So consider this proposition here. If N is an even number, then N squared minus 6N plus 5 is an odd number. So this is an if-then statement. So if we wanted to prove this by direct proof, what we're going to do is we're going to assume N is an even number and then work to argue that N squared minus 6N plus 5 is an odd number. Now what is in, so that's exactly how we start this. Suppose that N is an even number. What does it mean for N to be an even number? What that means is there exists some integer k such that N equals 2k. So the very first thing I did was state the assumption. Assume that N is an even number. And then I decided to unravel the definition. What does it mean to be an even number? Well, there exists some integer k such that N equals 2k. Now if you were writing this as opposed to the finished version here, you would be like, okay, I'm going to end this proof by writing N squared minus 6N plus 5 is odd. Well, what does it mean to be odd? It means that there is some integer of, there's some way of writing my number as 2M plus 1. That's what we're looking for here. So we can see that. We have N equals 2k. We want to end up with some type of 2M plus 1. How would we accomplish that? Well, the idea is if I take, if I take the expression N squared minus 6N plus 5 and replace each of the Ns with 2k, maybe algebraic manipulation will allow me to then produce something of the form 2 times M plus 1. Okay, that's what we're going to do. So we start off with the value we're trying to consider for our conclusion. N squared minus 6N plus 5. Replace each of the Ns with a 2k. So we get 2k quantity squared minus 6, minus 6 times 2k plus 5. So let's do some algebraic calculations here. 2k quantity squared is the same thing as 4 times k squared. With the negative 6 times 2k, that's equal to negative 12k, you have a plus 5 right here. Now, notice my goal is I want to write this as 2, 2M plus 1, where M is any integer. I don't care what it is. It just needs to be an integer. So I need to find a 1 and I need to find a 2. So one way to try to do that is I'm just going to produce a 1, right? That is, there's a 5 here. 5 is just 4 plus 1. And so I'm just going to extract the 1, set it by itself, and then leave the plus 4. Because everything other than the plus 1 should be part of the 2M. And so that's exactly what we see right here. I have everything plus 1, okay? Now, this thing right here should be 2M. If this statement is true, that's got to be my 2M. Is this an even number? Sure enough, look at the coefficients. I don't know what k is, but 4, 12, 4, I can factor out a 2 from each of those coefficients. So I end up with 2 times 2k squared minus 6k plus 2. And then you add a 1 to that. And so you don't necessarily have to declare the symbol M. It doesn't need to be specified here, but it's like, okay, as the number 2k squared minus 6k plus 2 is an integer, the following form then tells us that n squared minus 6n plus 5 is in fact a non-number. And this improves our proposition by this method of direct proof. In summary, we started with our assumptions, we unravel definitions, we bridge the gap between the 2s, and then we finish with proving the statement we've set out to do, the conclusion of the conditional. Let's look at another example. This time we'll look at some real numbers. Let x and y be positive real numbers. So that, of course, means that x and y are going to be greater than 0. We then want to show, so this right here is basically just defining the set of objects we're going to consider. So the conditional statement actually follows here. If x is less than or equal to y, then the square root of x is less than or equal to the square root of y. This first statement was necessary so we know what we're describing. Are we describing integers? Are we describing real numbers? Are we describing ration numbers, complex numbers? In this case, the scope of our inquiry is only for positive real numbers. The conditional statement is right here. So you don't have to restate that x and y are positive numbers. That's already established. Instead, our conditional is right here, p implies q, so we start by assuming p. So the very first sentence will be, suppose that x is less than or equal to y. Our very last sentence will be that the square root of x is less than or equal to the square root of y. And then from there, we're going to apply some definitions when necessary and try to unravel these things. Now it is important to note here that y is the numbers being positive, important here. We'll get to that in a second, but let's start off with the thing we do know. We do know that x is less than or equal to y. If we start to manipulate this algebraic inequality, perhaps we can then produce the inequality we're looking for. After all, the statement has the form if an inequality is true, then another inequality that uses the same variables is also true. So maybe we can manipulate this one somehow or another. Notice that if you were to subtract y from both sides, you would turn this inequality to this one right here. But when we write a proof, we have to make sure we draw the picture with words, right? We can't necessarily use these little informal notations like we would if I was giving a lecture or in a classroom, much like this video is a recording of. So we describe it with words. Suppose x is less than or equal to y. Then we would say something like subtracting both sides by y, we have that x minus y is less than or equal to zero. Now there's a lot of liberty on how you can express such a statement. You have to always keep in mind who is your audience? Who are you writing to? Currently, this video is intended for students of a class similar to SUU's math 3120. This is a class for students transitioning into advanced mathematics and so this proof is written with that intention in mind. Now if you're writing to a very different audience, if you're writing to like mathematical scholars, you might not say this at all. You might say something like, suppose x is less than or equal to y, then x minus y is less than or equal to zero. You might not give any justification why the two statements are related, understanding that your audience would be able to feel the difference. Like, oh yeah, just distract y from both sides or if you move y to the left hand side, it becomes this inequality like here. So always keep your audience in mind. Writing for first grade math is different from writing 12th grade math, which is different for writing college math undergraduate, which is very different for how you might write for graduate students. Writing for like a general education math class like calculus or college algebra, it's very different how I present mathematics for my advanced math students, my math majors, much like we're doing in this video. I am providing some justification here, but I probably could get away with without it, but I got to write something, right? So subtracting both sides by y, we have y, x minus y is greater than or equal to zero or this is the same thing as x, the square root of x squared minus the square root of y squared is less than or equal to zero in that situation. Now look what I did. I explained exactly what I did to get from here to here in the first step, but in the second step, I didn't actually justify it. I just said that they're the same inequality. Understanding that my audience would understand that, oh yeah, we have positive numbers. The square root of x squared is equal to the absolute value of x, and since the numbers are positive, that's equal to x. Same thing for the square root of y squared. I expect my audience to be able to do that and it's not unfair to expect that of the audience I'm presenting right now. So I can go from here to here just by saying, oh yeah, these two things are the same. Now, when I use the word or here, I don't mean it in the logical sense. I mean that this is equivalent to this or it's equivalent to that. That is, you could pick either of those two forms, but they're actually both true, right? This is just a progression. We went from this inequality to this one to this one, okay? We're going to continue to do that until we get to this one down here. Now, the reason why this one is useful, I then explain to the reader here, factoring the left-hand side we have, because now you see this as a difference of squares. That was why we introduced the squares and the square roots. You can then factor this as a difference of squares. You have the square root of x minus the square of y times the square root of x plus the square root of y. That's less than or equal to 0, okay? And then again, I did preface this. We don't want our proof just to be a sequence of inequalities, much like we don't want our proof to be a sequence of equations. That's appropriate in a trigonometry class when you're trying to prove a trig identity. If you're trying to prove something like, oh, sine of 2 theta equals the usual identity 2 sine theta, cosine theta. If you were actually trying to prove this, then you might have some type of statement here, statement here. I should say expressions. You're just linking together a bunch of equations, equalities here. That's appropriate for like, sure, like a trigonometry class. But we have to write better proofs than that. This is not just a sequence of equations, or in this case, a sequence of equalities. We have to give some explanation. You don't want it just to be mathematical symbols. There should be some English in there as well to help our audience understand what's happening here. So we give some clarity that you're factoring this, and that's where this thing came from, okay? Might it be reasonable for the audience to figure that? Yeah, but that doesn't mean you never say anything that your audience could do on their own. Because honestly, if your audience is math 31-20 students, which mine is right now, it's reasonable that I could expect them to write this proof. In which case, I might write the proof as, in the following here, the proof is trivial. Duh. It's like you guys can figure it on your own, or textbooks do this all the time. The proof is delegated to a homework assignment. It's like you get to prove it true. Yeah, sure, you can do that. But that's not going to be very instructive for students if I just say this is a trivial proof. You wouldn't learn why it's a trivial proof. And so you should give something. You've got to give something to the audience. And so it is a give and take. You don't want to give too much because then your proofs get long and verbose and difficult to read because it's too long. It's like, I get it, move on. But you also don't want to do too little so that the audience is expected to do everything on their own. Finding the right balance is important here. This is all about writing composition. How do you compose it to make a well-written beautiful proof? It's give and take. An experience is how we can fill in the gaps there. So returning to the, returning to the proof here, we have the square root of x minus square root of y times square root of x plus the square root of y is less than or equal to zero. Now we can say that well, because like, let's note that the square root of x plus the square of y is greater than zero, right? Since x and y are positive numbers, the square root of x is always positive. The square root of y is always positive. The sum of two positive numbers is positive. Regardless, I know that the square root of x plus square root of y is a positive quantity. Why is that necessary? Well, if we divide both sides by the square root of x plus the square root of y, that means we divide both sides here, zero divide by anything of course is going to be zero again. I can then simplify the inequality to be this one right here. The square root of x minus the square root of y is less than or equal to zero. Remember, when you work with inequalities here, if you divide by a positive, you preserve the direction, but if you divide by a negative, it actually flips it around. That's why I need to specify this was a positive number. Be aware that writing a proof is not just about composition. It should be well composed so that it reads well, okay? We also have to worry about things like clarity. Is it clear to the reader what's going on here? If you skip too many things, then it might not be very clear. But also, if you put too many things in there, the audience might get stuck in the weeds and not understand what's going on either. That's what we were talking about earlier. We also have to worry about completeness. We have to make sure we include everything that we need to say. Now, if you start dividing inequalities by willy nilly real numbers, we should be cautious. Did you just divide by zero? Because that might actually make the statement mathematically incorrect. Is it a correct statement? But then also, in this case, since we're divided by a positive quantity, it doesn't switch the inequality around. But if you divide by a negative, it would switch the inequality around. So to be correct, to have the correct ordering right here, we have to divide by a positive. And if we don't state that, then our proof is somewhat incomplete. If there's two large gaps, then that makes the proof incomplete. Now, admittedly, the gaps we leave could be because we expect our audience to do it and it's clear how the audience is able to do it. But as students writing this, we have to be very careful. Is there a gap there because the audience can answer it and you're just choosing not to? Or is that you don't know how to fill in the gap and you're hoping no one will notice if you claim that the emperor is wearing some new clothes or something like that. We'd be cautious of such things as we write. Okay, so for completeness sake, we would specify this as a positive quantity. Therefore, this inequality implies this inequality, we get that the square root of X minus square root of Y is less than or equal to zero, for which then I say finally, that implies the square root of X is less than or equal to square root of Y. I didn't specify this time that we're actually adding the square root of Y to both sides. I mentioned it before. But really, it's like I expect by now the audience to see what happens. You move this to the other side, and then we get the inequality we wanted right there. And so that then provides the proof of this one. And we went through it very slowly explaining why things were written. Every word in this proof was mentioned for a reason. None of it was intentional to be fluff. It was every word was put in there for the following rubric, the so-called foresees. I did it for the sake of composition or for the sake of clarity, the sake of completeness, or for the sake of correctness. If any word was modified, it's possible that one of these four rubrics would have been weakened inside of this proof. Let's look at another example of a direct proof. Let X and Y again be positive numbers, then the quantity two times the square root of XY, the product, is less than or equal to X plus Y. So again, how do we start this thing? We can mention our assumptions here. So let X and Y be positive numbers. Now, you'll look at this one. It's like this one doesn't exactly look like the previous one. The previous one was like X and Y are positive numbers. Let that happen. Then you have this inequality. If X is less than or equal to Y, then the square root of X is less than or equal to square root of Y. This time, I'm starting off by actually stating that the X and Y are both real numbers. I didn't know the previous one because it was built into it. I didn't even mention it, but unlike the other ones, like I don't have anything else to say, I got to start with somewhere. So I'm just going to mention, okay, X and Y are positive real numbers. Then since they're positive real numbers, it is true that zero is greater than or equal to X minus Y squared. I want to be aware that this statement holds not because the numbers are positive, but because they're real numbers. If you take any real number in existence and you square it, it's always going to be greater than or equal to zero. If you square zero, you get back zero. If you square anything else, a positive or negative, you always get something positive. And so regardless of what the number X minus Y is, it could be positive, it could be negative, it could be zero. I know it's square will be non-negative. This is a universal statement about real numbers. The positivity is not being used there. Then because this is true, I can algebraically foil this out and get X squared minus 2XY plus Y squared. Just these two quantities are equal. And so since this is greater than or equal to zero, this will be greater than or equal to zero. Okay, I got that. Taking this inequality, I could then add 4X to both sides of the inequality. This would then give something the following. Well, zero plus 4XY would be 4XY. And then if you add 4XY here, you end up with a positive 2XY. And then on the right hand side, if we then refactor that, we end up with X plus Y squared. Okay, so now we have the inequality that 4XY is less than or equal to X plus Y squared. Then I claim by proposition 434, which be aware, the proposition 434 was the one we just proved on the previous page. If X is less than or equal to Y, then the square root of X is greater than or equal to square root of Y. Now in this situation, what meant X before is now replaced with 4XY. And what meant Y before is now replaced with X plus Y squared. And so by proposition 434, we get that the square root of 4XY, which is the same thing as two times the square root of XY. So notice those things are equal. That'll be less than or equal to the square root of X plus Y quantity squared, which will simplify to be X plus Y, like so. Thus giving us the inequality we are looking for provided the proof. Now you look at this one, it's like, wait a second, I thought this video was about direct proofs. How is this a direct proof? Well, it turns out the direct proof is a lot more subtle in this situation. By all means, sure, we started with our assumption, we ended with our conclusion, but we're not actually proving a conditional, so to speak. I mentioned beforehand in a previous video that essentially every proof you ever write is essentially a conditional, because there's always some hypotheses and some conclusion. So you can be like, oh, if X and Y are positive integers, then this inequality happens. So in that regard, it's the case, that's the case as well. But I also mentioned, where was positivity ever used? When did it ever use the fact that the numbers were positive real numbers? You didn't need it here, because that inequality would be true for any real numbers. You didn't use it here, that's just an algebraic expansion. You didn't use that here, because again, you just algebraic could manipulate the inequality, no real number was needed there. Where was it used? Aha, it's actually used quite secretly right here. The fact that the integers were positive was used in Proposition 434. If you recall, Proposition 434 says, let X and Y be positive real numbers, and then this inequality or this conditional holds in that situation. X and Y being positive real numbers were part of the assumptions of Proposition 434. If you do not have them, then you can't use the proposition. And so as your proof, as your proof writing gets more complicated and more detailed, sometimes the assumptions of the conditionals you use will not be used directly, they'll be used indirectly as hypotheses of previously proven theorems like in this case. So I want to include this example because we didn't use our hypothesis directly in this proof. We used it indirectly in another proof that we had just done previously. This is common mathematical writing. And so this is an example of the direct proof method, but the uses of the assumptions were actually inside of a proposition. If you don't use your assumptions, then it begs the question, why did you use the assumptions at all? The assumptions that are there for a reason. If you don't need them, remove the assumption and that gives you a stronger statement. We use them not directly, but indirectly as the assumptions of a different proposition. And that's something we're going to see all the time as we proof things by this method of direct proof.