 In the previous video, we introduced the method of proof by cases, which is a really useful proof technique when, well, multiple different cases appear, where you have to make different considerations for the same problem here. Now, there are some language considerations that have to be addressed when one talks about proof by cases. And so I actually wanted to treat that in this video right here. Typically, it has to do with the idea of treating similar cases as the title of the section seems to suggest right there. Because many times one proves a statement by cases, one has two, maybe three or more cases that are so similar to each other that the proof of one case is virtually identical to the proof of the second one. It's the exact same argument. And it sort of begs the question, should I include the second proof when it's really just the same proof as the first one? And I'll show you an example of what that means in just a second. Like the author of the proof has to sort of make a judgment call. Like, do I include the second case when the argument is identical and it's reasonable that the reader could read one proof and then automatically know the proof for the second case because their similarities are so much the same. They're like, they're so similar. It almost is a disservice to the reader and the writer to include the second case, okay? And because after all, I mean, we want arguments to be valid. We have to consider all the cases, but if the two cases are so similar, the second consideration might actually be unnecessary. And so in mathematical writing, it is customary and acceptable to first prove the first case, whatever that turns out to be. And then the second case or the similar case you can then exclude by giving some statement like the following, the second case is handled similarly. That is the writer acknowledges to the reader that there is another case that needs to be considered, but because the proof is so similar, we're not gonna write it, but you and I will both know what that proof is, okay? This is a valid and acceptable proof technique, but for the simplicity of reading and understanding, we can write a shorter proof because the second case is so similar. So you might say something like this, the second case is handled similar or the second case is similar, that you use this word similarity of some kind to describe it. If you wanna throw out fun Latin jargon, you can use the phrase mutatis, mutandis, which if you translate the Latin into English, it basically would translate as change what should be changed or change the necessary parts. That the two cases are different, but the differences are so insignificant that the same argument, the same logical argument can be used to prove it. And so let me provide you a proof of to provide an example of that. So consider the proposition you see on the screen right now. Two integers, if two integers have opposite parity, that means one of them's even, one of them's odd, then their sum is automatically an odd number. And so okay, let A and B be two integers with opposite parity. Well, there's two possibilities. A is even and B is odd. That's the first possibility. The second possibility is that A is odd instead and B is even, okay? Now we're adding together numbers, the sum. So we have two possibilities, A and B, which in case one it would be this is even, this is odd. Then there's the other possibility, A and B where A is odd and B is even. In all reality though, since addition is commutative, if the odd comes first, you could always switch the order of the sum so that the first number is the even one and the second one. And so in this case, there's two possibilities but the two cases are so similar, it maybe is justifiable to not consider the second case. Now for the first case where N is even and B is odd, well, if A is even, that may exist some integer K such that A equals two K. And if B is odd, then there exists some integer L such that B equals two L plus one. That makes it an odd number, that makes this an even number. And so when we add together A and B, A is equal to two K, B is equal to two L plus one. And so when you add two K with two L plus one, since two K and two L are both divisible by two, you can factor out the common divisor of two. And so this gives us two plus K plus L plus one. Since K and L are integers, their sum is an integer, two times an integer would give you an even integer, adding one makes it an odd integer. So we've demonstrated that A plus B is an odd number. That was when A was even and B was odd. But the other situation where A is odd and B is even, this would be like B and A right here. So you swap the order of these two, it'll still become like two times L plus K plus one, that's still gonna be an odd number. The second possibility where A is odd and B is even is so similar to the first one that it's reasonable that the reader would know that and not need a second argument because like, oh yeah, you could just swap the roles of A and B and now it would be the argument for the second one. And so this exactly falls into this situation where we're justified in saying the second case is handled similarly. And there's only the two cases since they have odd parity, opposite parity, excuse me, one is even, one is odd. So depending on who's even, who's odd, there's the two possibilities. Either situation leads to A plus B being an odd number, okay? Now this way of writing a similar case argument, you could argue is like the retroactive way of doing it. Excuse me, retroactive, the retro way of doing. As in like, you've acknowledged there's two cases you deal with the first case and then after the first case is done, you're like, oh yeah, the two cases are similar so I don't have to do the second one. It turns out though that a lot of authors prefer a more proactive way of dealing with these similar cases. That is the author is like, hey, I know the cases are similar and I'm gonna let the reader know that beforehand. And in that case, if you want this proactive way, people typically use the phrase without the loss of generality. What this means is that we are going to consider multiple similar cases at the same time. In this case, there's two cases but because they're so similar, they don't require different proofs. Both, why do we use the words without the loss of generality? What that means is we're about to make an assumption, an assumption like in this case, A is even and B is odd but even though we're making an assumption because that was an assumption that wasn't in the statement of the problem, right? We started off with the assumptions that we have two integers with opposite parity. That's the assumptions that this conditional statement gives us but we're gonna add an extra assumption but that extra assumption does not make us lose any generality in our argument. Our argument is still just as general as it used to be even though we're adding an assumption for the sake of convenience. And then this case of the assumption is that A is even and B is odd. Why does it not lose generality? Because the other case was that B, excuse me that A is odd and B is even. That's the other case. And like we've already discussed, addition is commutative. So the fact that the first number is odd versus the second number being odd makes no lick of a difference whatsoever. It doesn't matter. And so as such, since it doesn't matter I'm just gonna assume A is the even one because if A was the odd one I can provide basically the same argument again. So we often use this phrase without the loss of generality. We're acknowledging to the reader that there are multiple cases but I'm gonna only look at one of the cases because they're all so similar and I'm making the choice that A is the even one and B is the odd one even though you could make a different choice and it doesn't really make much of a difference. So no generality of argument is lost by assuming A is the even number when all that we really knew was that they have opposite parity. So if we look at the exact same proposition with a slightly written, a different proof, right? We're taking the proactive style of treating similar cases. The first instance is the same. Let A and B be two integers with opposite parity. Then I use the phrase without the loss of generality. Suppose A is even and B is odd. There are other possibilities but the other possibilities are so similar to this one I only have to consider this one and I'm making the choice that A is even and B is odd. You could do the other one, that's fine. It was just a matter of choice right now but my choice, my assumption doesn't lose any generality in the argument. Then there are exist integers K and L such that A equals two K, B equals two L plus one and then the same calculation follows A plus B equals two K plus two L plus one which is the same thing as two times K plus one plus one which shows that A plus B is an odd number. So it's the same argument structure. It's just the way we communicated the similar cases. I mentioned at the forefront that oh, there are two cases, they're the same. I'm only gonna consider one of them. That's what without the loss of generality means. Multiple cases are being considered simultaneously but I'm gonna use a specific case to illustrate the general argument. This is similar to the retroactive approach we had on the previous one where I just mentioned after case one has proven that case two follows similarly, okay? Now one caution I wanna use, I wanna mention that when you consider multiple cases that are similar if you use phrases like without the loss of generality it's important that when you use these type of phrases it's because the cases are actually similar to each other and a second proof is unnecessary. For many beginning proof writers which are typically students they sometimes have the temptation is like I know there's more than one case I only know how to do one of them or I'm just being lazy and don't wanna write the second proof. So we say that the cases are similar when in all reality they might not be and you're sort of bluffing, right? So there's that bluff argument some people use without the loss of generality as a bluff and you should be careful because I imagine for most students you don't have a very good poker face when it comes to proof writing. Professors like myself are really good at sniffing out when a proof is stinky because it's very apparent to me when you're writing a proof that if there's multiple cases they're like oh the second case is similar and it's like no it's not, no it's not. I know when it's appropriate I know it's when it's not. So you should be very careful don't try bluffing, do the right thing write the real proof and if you have similar cases then by all means you can simplify the arguments in using communication skills that we've talked about in this video but only do it because the cases are similar and not because you're trying to trick your reader that's bad form and honestly will probably hurt your proof and reputation than any benefits it'll provide for you. Now with that said with that warning now declared that we're at the end of lecture 15 thanks for watching. If you learned anything from these videos from lecture 15 please like them subscribe to the channel to see more videos like this in the future and if you have any questions please post them in the comments below and I'll be glad to answer them as soon as I can.