 We have seen previously that if one wants to write a mathematical proof, that's really the same thing as just writing a logical argument. All right, and what is a logical argument? You have some list of premises that you list right here. You have some conclusion and we say that the argument is valid if the premises being true then imply that the conclusion is likewise true. Now, we've also discussed that every logical argument can be rewritten as a single conditional statement where you take as a very large conjunction, the conjunction where i ranges from 1 to n of all of the statements pi, that then implies the statement q. And now this argument is valid if and only if this resulting conditional statement is always true. And as such, providing valid arguments, which are the only out arguments that we will ever want to provide, is really logically equivalent to proving that this statement over here is true. So the most important and the most fundamental of any proof patterns is being able to prove that a conditional statement is true because in the end every proof is proving a conditional and when you look at in terms of logical equivalent. So how does one prove that a conditional is true? Well, let's remember what we've discussed with truth tables, right? So let's say we have two statements p and q and these could be very compounded statements. Don't worry about that. We have some conditional here. Well, the possibilities are you have true, true, true, false, false, true and false, false. Okay. And when your premise and conclusion are true, the conditional is true. When your premise is true and conclusion is false, that's false. When the premise is false and the conclusion is true, that is true. And when the premise is false and the conclusion is false, that's likewise false. Okay. Now these last two cases that are on this table here, when the premise is false, then the conditional is considered true. It's what we call vacuously true. And likewise, when the conclusion is true, regardless of what happens with the premise, the conditional is likewise true, what we call trivially true. And so the only case that really deserves attention is this one right here. We have to basically argue that this possibility never happens. What is this possibility? This is the possibility where the premise is true, but the conclusion is false. And so to prove that a conditional is a true statement, all you have to do is you can assume that the premise is true. And if there's more than one premise, then you list all of them. You assume that all of the premises are true. Then you provide some more statements, more arguments as necessary, and then you conclude that the conclusion is likewise true. This is the method known commonly as direct proof. And the method of direct proof, which is a technique used to prove a conditional statement, you assume the hypothesis, you assume the premise, and then you use that to show that the conclusion is likewise true. As such, because of the method of direct proof, the hypothesis or premise of a conditional is sometimes called the assumption, because we constantly are assuming it to be true. And then we argue that the conclusion follows by the assumptions that we then hold. And so what I want to do in this video then is demonstrate by example how one can write a proof of a conditional statement using this method of direct proof. The formula for writing the proof is always the same. You will start with the assumptions, and then you end with the conclusion. What happens in the middle depends on the argument itself. So consider the following conditional statement. If n is an odd integer, then n squared is likewise odd. Now, the basic structure of this proof, which we will do via direct proof here is you will start with your assumptions. So the first part says if n is an odd integer, so we're going to change that to say something like assume n is an odd integer or suppose n is an odd integer or let n be an odd integer, something like that. That is then the very first sentence of the proof. That's what you write there. Then the last sentence of the proof should be your conclusion. If we have an if, then there should be a then. And what's the then n squared is likewise odd. Now you can change some of the verbiage, right? If you want to drop the word likewise, that's fine. But a direct proof will always start with something like assume the hypothesis, therefore the conclusion. Now, some other things that we typically write when we write a proof is we typically will start the proof with a word like proof, right? We indicate to the reader that we're now beginning the proof of the statement that then preceded it, which is probably marked as a proposition or theorem or something like that. And then at the very end, there's typically going to be a little square of some kind. This is referred to as the QED symbol that this is an acronym for a Latin phrase that basically says that the question has now been demonstrated. That is, we've proven, we've proven the thing that we went out to go improve. And the little square that indicates we're at the end. It's sometimes called a tombstone symbol or the QED symbol. Tombstones a little bit more morbid, it's like our doubt of the truth of the statement is now dead. You can say that if you want to. But those are also just things you're going to see all the time, the QED symbol and proof indicates where you start and stop to prove everything in the middle is then the proof. For the direct proof, we start off with the hypothesis, which we then assumed to be true. Let N be an integer. And then we write the conclusion. What are we going to do next? Well, there's going to be some type of gap right here. Now, if you were writing this as scratch paper, you might want to leave a lot of space for you to write some sentences here, much like other types of writing, proof writing comes in phases. There's going to be some type of prewriting phase where you will outline or think about how is this proof going to go? There might be some organizational phase where we have to change how things are revision is going to take place there as well. We don't start writing a final draft, the final draft is written and revised as we go through it. Now, in this example, you will see it finalize pretty quickly, because I'm practicing the direct proof method and not actually demonstrating all the steps of writing. So because we have an odd integer, and we have to end with an odd integer, it's often a good practice to unravel the definitions. So after you make an assumption, the assumption basically is something of the following form, we have an object that belongs to a set. And so we then remind the reader, what does it mean to belong to that set? And we might also establish notation. So if n is an odd integer, that means there exists some integer k, such that n is equal to 2k plus one. So let's rewrite that first sentence. Let n be an odd integer. If you want to end the sentence right now with a period that's okay, you can then say something like there exists an integer k such that n is odd, right? I actually chose to just make this into one sentence. I think it reads okay in that manner. Let n be an odd integer. That is, this is going to suggest that we're going to clarify what the previous thing just said. Let n be an odd integer. Therefore, that is, there exists some integer k such that n equals 2k plus one. Be aware that in mathematical proofs, it is okay to use symbols. It's also okay to use words. So you could say there exists an integer k and if you spell it out, you're probably not going to put the symbols in there, but you can do the symbols too if that's how it reads. Now you want to be careful not to use too many symbols, but the appropriate use of symbols in a proof can help simplify the language here. Let n be an odd integer. That is, there exists some integer k such that n equals 2k plus one. This is what it means by n being odd. I don't know what n is, but it is odd. So there is some integer k so that this happens. I don't know what k is, but whatever k is, it's an integer. Two times k will give you an even number plus one to an even give you an odd number. This is what it means to be an odd number. So we've unraveled the definition. Maybe that'll help us somehow get from the assumptions to the conclusion. But what does it mean for the conclusion? Sometimes we unravel backwards. We can start with the assumptions of the problem. We can also work with the conclusion and try to come together, right? At the beginning of writing a proof, the logical gaps between them might be so large, it's hard to see how to connect them. But if we work closer and closer to the logical gap in the middle, maybe we can reach a point where we can bridge that gap. And for these easier exercises, that's exactly what we're going to do. So the next thing I might do is I might unravel the definition. What does it mean for n squared to be likewise odd, right? Well, since n is odd, that's what the likewise means. Okay, it just means two things are odd. n squared would be odd if there exists some integer m such that n squared is equal to two m plus one. Now, unlike this one, I didn't necessarily mention that m is an integer. That's probably because I would have discovered that previously. I don't know what m is yet, but there's, if n squared is odd, that means I have to be able to find some integer m so that two times m plus one gives you the n squared. And I bet you somehow or another the numbers m and the numbers k are somehow related to each other, because after all, n and n squared are related to each other. So this is what we're looking for. We have some k, we have to inform the m. That's how we're going to bridge the gap. And we're going to use the relationship between n and n squared here, because after all, n squared is just n times n. But we also have this assumption that n is two k plus one. And so if I replace the n with two k plus one, maybe I could foil this thing out to then get you something that looks like two m plus one. And that's how I'm going to try to bridge the gap here. So let's take n squared, which is the same thing as two k plus one squared by the previous assumption here. So then if you foil it out, I'm not going to go through all the details of the foil for this audience, those type of mechanical algebraic things are things I can expect the reader to be able to do. So if I were to provide the details, we take two k plus one, we take two k plus one, we foil that out, we get two k times two k, which is four k squared, we get two k times one, we get two k plus one, and then we have one times one, right? We could then combine the like terms, we're going to end up with something like this. But like I said, it's reasonable to expect of this audience that they could foil this out and arrive upon this discovery as well. So two k plus one quantity squared is the same thing as four k squared plus four k plus one. Now, I'm looking for something of the form two m plus one, I have the plus one. So if we separate the plus one, you're left with the four k squared plus four k, both of those coefficients are divisible by two, I could factor out the two and get two times two k squared plus two k plus one. This right here feels like my m that I'm looking for. So I'm then going to declare it. Let m equal two k squared plus one, sorry, two k squared plus two k, which is an integer, then n squared is equal to two m plus one, which means it's an odd number. And so look at this together now, our finalized proof. Let n be an odd integer, that is, there exists some integer k such that n equals two k plus one, then n squared is equal to two k plus one squared, which is equal to four k squared plus four k plus one, which is equal to two times in parenthesis, two k squared plus two k close parenthesis plus one. Well, if we let m equal two k squared plus two k, which itself is an integer, then that means n squared is equal to two m plus one, which shows that n squared is an odd integer. So this has all of the ingredients of a direct proof. We assumed the hypothesis, we proved the conclusion, and we provided details in the middle. The basic steps were we made an assumption, we unraveled, we unraveled definitions, we bridged the gap, and we then end with our conclusion. Those are the ingredients of every direct proof, which I confess some of these steps can get more complicated depending on the complexity of the statement we're trying to prove. Let's look at another example of this. This time let's do one with some integer divisibility. Let A, B, C be integers. If A divides B and B divides C, then we want to conclude that A divides C like so. Now we're going to prove this by direct proof. Now this right here, this first sentence, you could take that as an assumption, but really what we're saying is we're trying to describe what's the universe in play. We have three numbers. What kind of numbers are they? They are integers. So those really aren't the assumptions of the problem. The if then statement is right here. This is just adding context to the following statement. So we understand what A, B, and C are. The conditional that we're trying to prove is this one right here. If A divides B and B divides C, then A divides C. So our hypotheses show up right here and they're connected by an AND. It's a compounded statement. So what I'm going to do for my direct proof is I'm going to assume both of the hypotheses. Suppose that A divides B and B divides C. Then the very last sentence of my proof is going to be the conclusion. Therefore, A divides C. So that's how we begin. I might want to unravel some definitions. What does it mean for A to divide B? What does it mean for B to divide C? What does it mean for A to divide C? If we talk about these terms, we can unravel them and get the following. Suppose A divides B and B divides C. Hence, there exist integers M and N such that B is equal to AM and C is equal to BM. The fact that A divides B means there is an integer M such that AM is equal to B. So this is what it means for A to divide B. And likewise, since B divides C, there has to exist some integer such that B times that integer is equal to C. So this statement right here is unraveling the definition of B divides C. Now, I can also do that with the definition of the conclusion. I want to end up with A divides C. So what does it mean for A to divide C? That means there exists some integer K such that AK is equal to C. Now, notice how I phrase this. When you're working on the back, you actually work backwards. The last thing I want is the conclusion A divides C, which means if I unravel the definition, the definition of divisibility will probably be the thing I see immediately before it. So I get something like therefore C equals AK, which oh wait, if C equals AK, that means A divides C. So we've unraveled the definitions. Now we want to try to bridge the gap here. How can we connect these statements of divisibility with this statement of divisibility? Well, divisibility comes down to equations of integers. Can we connect these equations together to get this one? Well, this last equation involves A and C. This one involves A and B. And this one involves C and B. Could I somehow remove the middle man of B and go forward from there? My hope is exactly yes. And I could, because B, I don't need B in the final statement, A divides C has nothing to do with B. So maybe I could substitute this equation into that one, thus removing B from consideration. And maybe I can find my integer K in that process. Let's spell exactly that out. So what I'm going to do is I'm going to take the equation involving C, it's equal to BN. So that was already established by this assumption. Then by the other assumption, B equals AM, I'm going to substitute this in for B. Thus we get that C equals AM times N. Now as this is an equation of integers, I can use all of the algebraic properties I know about integers. For example, I can use the associative property. AM times N is the same thing as A times AM. And so notice here what we have here, C is equal to A times something, that something is an integer. If K is equal to MN, which is an integer, product of integers or integers, that then gives us the final statement that we're looking for. So now let's read the completed proof here. Suppose that A divides B and B divides C. Hence there exist integers M and N such that B is equal to AM and C is equal to BN. Thus C, which is equal to BN, is also equal to AM times N, which is the same thing as A times MN. Now if K is equal to MN, which is an integer, we then have that C equals AK, and that of course implies that A divides C, thus completing the proof. We'll do some more examples of direct proof in lecture 14, because I don't want this video to be too long, but this gives us some good examples of how does one prove a conditional statement. Direct proof is going to be your go-to move for many exercises like this in the future.