 So before I begin the topics of this video, I wanna make mention that in the world of mathematics, there is a wide range of opinions when it comes to proof by contradictions, okay? Now, as a reminder in the previous video for lecture 23, we were talking about how you can prove conditional statements P and plus Q. And I mentioned there's a lot of options you have there. There's direct proof, there's proof by contra positive and there's proof by contradiction. And honestly, the previous video focused a lot on contradiction, but in particular, the video was focusing on, the main topic of the video was how you can combine these techniques to prove statements, conditional statements, of course, being an important example of that. Now, in that video, I also had made mention that while we did prove some things using contradiction, we could have actually proven it better using contra positives. And what makes a proof better? This is actually the heart of what I'm trying to get here. And this is why contradictions, proof by contradictions can be a divisive topic for many people. So let me give a personal story here. When I was an undergraduate student, I was taking a introduction to logic class offered by the philosophy department. This is very common class at universities. And actually it's a class I highly recommend to mathematical students. If you're watching these videos, but you haven't had a formal logic class like from your philosophy department, I would suggest take that as an elective. It's a fantastic class. You learn a lot about logic and the philosophical perspective is a little bit different from mathematics. And that actually makes us better when we work with logic there. But anyways, in the class, there was a time that we had learned about reductio absurdum, an RAA argument, which of course is just proof by contradiction. And I remember there was a classmate in that class that at one moment had his epiphany and he was basically like, wow, I can prove everything using RAA. I can prove everything using proof by contradiction. And he just said this in excitement as we were working on some problems in the class. And the philosophy for professor who was teaching our logic class, he basically acknowledged that the student was right. You can prove everything with RAA and sort of encourage the student to go ahead and do that. So that's one opinion of proof by contradiction. You can prove everything with proof by contradiction. On the other hand, I know of another mathematics professor who has basically forbidden his students from ever proving anything by proof by contradiction. I mean, he lists proof by contradiction amongst the seven deadly sins. I mean, you have gluttony, you have pride, you have proof by contradictions and that's how he ranks them. And I had never quite understood why there's so much hatred to proof by contradiction there. I talked to some of these students who were in this class. I myself never actually took a class from this professor, but talking to some of his former students, they were afraid to write proof by contradiction because he ingrained in them so much that you don't prove by contradiction. I would like to suggest we take a happy medium here. While I'm not gonna advocate that we prove everything by contradiction, I am certainly also not of the camp where we prove nothing by contradiction. Proof by contradiction is a very valuable proof technique and it is a valid proof technique. So we can use it when it's appropriate. So the decision that comes down to when do we use it and when do we not? What is the deciding factor? It's not logic that decides it because logically proof by contradiction is a valid argument. Truth is truth. If you can prove it with a valid proof, even if there's another valid proof out there, it doesn't really matter. It's still true. It's not more true if you write it with a better proof. It's just as true as it was before. And so then what makes it a better proof? Really what we should be using as our golden standard here to decide, do I prove this by direct proof, by contradiction, the golden standard should be clarity. Okay? Which proof techniques makes the proof most clear for the reader? Would it be a direct proof? Would it be contra-positive? Would it be contradiction? Which of those three methods is gonna be most clear to the reader? And oftentimes you start off by saying what's most clear to you? Like if you're writing the proof, like this is a homework exercise or something, if you're trying to write the proof, you're gonna write the proof that kind of seems most clear to you. Now for that classmate I had in the logic class, RAA was most clear to him. He was struggling to understand a lot of stuff in the class. At least he said that to me. He sounded like he was struggling, but he had that epiphany where RAA became clear to him. And therefore he wanted to prove everything with RRA because nothing else in the class was clear except for this one. So for him everything became clear using RRA. So he wanted to prove everything with that. And the logician professor, understanding that truth is truth, one valid proof is just as good as the other one. He encouraged the student to do that, to write using RRA proofs because that was what was clear to the student. And it was still just as true. Now of course we wanna be better than that. We wanna understand all of the topics in play here and we want our readers also to understand us. So we might choose one proof over another because it's more clear. Oftentimes in mathematics we say this proof is slicker, right? Oh, that was a slick proof. And typically what we mean is it was clear, but it was also short, right? When you look at these two proofs on the screen, I'm gonna read them in just a moment. I intentionally put them line by line and actually use very similar language. In fact, I actually copied a lot of the text here and went over here when I rewrote the contrapositive proof. So in many cases it's verbatim. And you can see that there's a noticeable gap here. This is a shorter proof. And that's what makes it slicker. It's clearer and shorter. These are what we want. We want a clear proof. We also don't want long proofs. Long proofs are actually usually make it less clear. And so with that said, let's compare two proofs of the exact same statement and make a decision, which one we think was better here. Suppose that A is an integer. If A squared minus A plus seven is even, then A is an odd number. Now, this proof is a little bit awkward if you try to do it by direct proof. We've seen some examples like this before to motivate why we would use contrapositive. Because after all, like the expression that you know to be even, A squared minus A plus seven is a lot more complicated as an expression than A. And so we know something about the complicated expression and we're supposed to infer something about the simpler expression. Again, that's sort of an awkward argument because we would love if we could just plug A into this expression and work from there. That makes it very clean. And if I had to prove something like, oh, if A is odd, then such and such is odd or even, you can kind of just plug and play and go with that proof. That's very nice for direct proof. But because we have to, we assume something with a more complicated expression, we have to conclude something about the more simple expression. A direct proof gets a little bit more awkward. An indirect proof actually is a little bit more preferable. And so our two main methods of indirect proof would be contrapositive and contradiction. Let's start with contradiction here. Assume that A squared minus A plus seven is an even number. That is, there exists some integer K, such that A squared minus A plus seven is equal to two K. So suppose to the contrary that A is not odd. That is, A is likewise an even number. Thus, there exists some integer N such that A equals two N. Now let's make a comparison. Let's plug this A into the expression like we wanted to do in the first place. A squared minus A plus seven would be equal to two N squared minus two N plus seven. Simplifying that, that'll become four N squared minus two N plus seven, for which if you factor that, you can write that as two times two N squared minus N plus three plus one. So this shows that A squared minus A plus seven is an odd number. Now this contradicts the fact that we assumed that A squared minus A plus seven is an even number. Since we got a contradiction, the opposite of what we assumed for the sake of contradiction must be true. So it actually must be that A is odd. Now this is a perfectly fine proof, but it turns out we could do better because notice here, we showed that A squared minus A plus seven is an odd number, but we assumed that A squared minus A plus seven is even. So the contradiction, we got two statements that were, we got a statement and its negation both true, but one of the statements was our assumption. We did nothing with this assumption other than prove its negation, which if that's your proof by contradiction, it turns out you can simplify it by writing it instead as a proof by contrapositive. The thing is we didn't need A squared minus A plus seven to be even. I mean, we did for the sake of contradiction, but if we had slightly rewritten it, we could avoid some of the awkwardness and actually make it a more straightforward proof, still indirect proof, but more straightforward using contrapositive. So for contrapositive, instead of proving P implies Q, we're gonna prove not Q implies not P as these are logically equivalent. So assume that A is not odd, that is, it's even. Thus there exists some integer N such that A equals two N. Then we're gonna plug this into A squared minus A plus seven. This gives us two N squared minus two N plus seven, which is four N squared minus two N plus seven, which is two times two N squared minus N plus three plus one, which then shows that A squared minus A plus seven is a non-number, which makes it not even. And so this is the thing is I never needed to introduce all of this for the sake of contradiction. Assume this, because it turns out that this argument right here is basically just the contrapositive argument, but you just added some more bloat to it. If we were to revise our proof from a contradiction to a contrapositive, then this actually makes it for the slicker argument. Now, I don't want you to get the false impression that this is always the case. Sometimes contrapositive might be more awkward than contradiction. Sometimes the direct proof is the best proof, but sometimes direct, sometimes contradiction is the best one. So our metric should be always clarity. What makes for the most clear proof? Now, if you're writing a proof, by all means, you might start off with something like contradiction. That's perfectly fine. But then as you're revising it, you're reading it, you're like, you know, I did this by contradiction, but it actually is a little bit cleaner, if I just do it toward contrapositive. So this might be your first draft, which is a very good first draft, but then in revision, you can then get your final draft over here, and that's what mathematical writing's all about. We have different proof techniques, but we can improve our writing and switch our techniques for the sake of clarity. That is the golden standard. So don't buy into these conversations that don't use contradiction or always use contradictions. Those are baloney. The standard is, do I use contradiction? If it makes the argument more clear, then yes. If it makes it less clear, then no. And so with that, we're at the end of lecture 23. Thanks for watching. If you learned anything about proofs by contradictions or relations or any other topics we've talked about in this lecture, please like these videos. Subscribe to the channel to see more videos like this in the future. Share these videos with friends so they also can learn this type of advanced mathematics. And as always, if you have any questions, feel free to post them in the comments below and I will answer them as soon as I can.