 In the previous lecture, we had talked about disproving universal statements. In this video, I want to do the dual argument. How do you disprove an existential statement? That is, how do you disprove the statement of existence? Okay, how do you know there's no bigfoot running around the California forest? You know, that's a difficult thing to prove, right? How do you prove that something doesn't exist? Maybe the reason we haven't found it yet is just it's hard to find. It's really good at hiding from us. I don't know. Now, from a mathematical point of view, an existential statement would be something of the following. There exists some element x inside of a set x, such that property p of x holds. Now, if you're going to disprove it, you're going to show that its negation is actually true. So what would the negation of an existential statement look like? Well, if we take not this statement, negation turns existential statements into universal statements. So for all x inside of x, it holds that not p of x is actually the case there. And this is a universal statement now. So the negation of existential is a universal. To prove this, be aware this is equivalent to really just a conditional statement, where our hypothesis comes down to that if x belongs to the set x, this then implies that not p of x holds. And you can prove this statement using direct proof, contrapositives, contradiction, or any of the other methods we've talked about in order to prove a conditional statement here. Let's look at an example of such a thing here. Here's a conjecture. There exists a real number x, such that x to the fourth is less than x, which is less than x squared. Now, when you look at this statement right here, it's not so unreasonable of a statement here. For example, there are many numbers, many real numbers who are smaller than their square, like two squared is four, which is bigger. Three squared is nine, which is bigger. There are also many real numbers, which are larger than their fourth power, like one half is larger than its fourth power of one 16th. And modifications of this statement are in fact very easily to verify are true. Like if I could took the following, is there a real number such that a real number x such that x cubed is less than x, which is less than x squared. Absolutely. You can take something like x equals negative two. The square of negative two is positive four, which is clearly larger, but its cube is negative eight, which is clearly smaller. So, you know, there are some numbers that solve inequalities similar to the one here, but search and search and search, if you try to find an x that's less than its square, but larger than its fourth power, you're going to struggle to find one. And it turns out that one doesn't exist. And so we're going to prove that it doesn't exist. So this is a disproof. And how are we going to end up disproving this thing? Well, we have to prove the negation of this thing. So using the principles we had for before, we need to prove the negation that for all x, you know, so we're going to prove here for all real numbers x, we actually have that we don't, it doesn't hold that x to the fourth is less than x, which is less than x squared, which this inequality is a compound inequality. It's actually two statements together. So it's an and statement. We have that x to the fourth is less than x and x is less than x squared, for which by de Morgan's laws, this is the same thing as saying this is the same thing as saying not x to the fourth is less than x or not x is less than x squared. And then the negations of those things, of course, we're trying to show that x to the fourth is greater than equal to x or that x is greater than equal to x squared. So that's what we have to show that for any real number, either its fourth power is bigger than it or it's bigger than its square. And as we don't have to prove both of these conditions, because both don't have to happen simultaneously, but only one of them has to happen. And so we're actually going to proceed by cases. That was a good proof technique we've seen before. So let x be a real number. If x is less than or equal to zero, then that actually means that x is less than or equal to its fourth power, because the fourth power of any real number is going to be non negative. It's important to remember that a fourth power is just a square of a square. And for real numbers, the square is always non negative. So as the fourth power is a square, it's going to be non negative. And if x is less than or equal to zero, that means x is less than or equal to x to the fourth. So the first situation happens in this case. Okay. So another one, let's suppose that one is less than or equal to x. That's a second case. Now, since x is itself a positive number, if I times both sides of this inequality by x, you're going to end up with x, which is x times one is less than or equal to x squared, which is x times x. Now, because x is greater than or equal to one, this tells us that x squared is greater than or equal to one. And so what we've now showed here is that if you have a number which is greater than or equal to one, then it'll be less than its square. But as x squared is a real number, which is greater than or equal to one, it'll be less than or equal to its square, which of course, the square of x squared here is x to the fourth. Now, combining these inequalities here, this is a transitive operation, a transitive relation here. x is less than or equal to x squared, which is less than or equal to x to the fourth. We get that x is less than or equal to x to the fourth. So again, we fall into the first category there. So if x is negative or equal to zero, it'll be less than or equal to its fourth power. If x is greater than or equal to one, then it'll be less than or equal to its fourth power. So the last possibility then is what happens if x sits in between zero and one equality. We don't have to worry about that because again, we took care of equality beforehand. Well, again, if x is greater than zero, it's positive. So if I times both sides here by x, we're going to get that x squared is less than x. So x times x gives us x squared, one times x gives us x like we did before. And well, I have x squared is less than x. We can swap that with x squared is less than or equal to x. And that's then what we wanted to show that x is greater than or equal to x squared. So we've now disproven the conjecture because we have now proved its negation, which is this statement right here. It was an OR statement. By cases, we always landed in one of these two camps, this one or this one. So that then disproved the conjecture there. So as we end this discussion of disproof, there's two comments I want to provide. First, whether you have an existential statement, you're trying to disprove or whether you have a universal statement that you're trying to disprove, whether it's even quantified at all, whether it's a conditional, a conjunction, a disjunction, a biconditional, whatever. If you want to disprove a statement, you can always, always use the technique of contradiction. Right? Because if you want to disprove something, you're trying to prove its negation is in fact true. And to prove that the negation is true, what you can do is you can assume the statement. Like we could assume this statement is true and then derive a contradiction. And that would then mean that the opposite was in fact true. We could do a similar proof to this conjecture that we just did, but be completely by contradiction. But this is kind of like how we were seeing earlier how you can do direct proof or contra positive or contradiction. But I want to be aware that this method works for all. Like if you want to disprove the existence of Bigfoot, what you can do is you can assume that Bigfoot exists and then you follow those logical paths until you derive some contradiction. If you find a contradiction, that means there are no big feet running around. Of course, while it's unlikely there's a Bigfoot running around because the evidence would suggest that they don't exist. Nonetheless, there are no logical contradictions here. So that's why there are still many people running around believing in Bigfoot. But contradiction is actually a beautiful, valuable tool in the disproving statements. Assume the statement's true, get a contradiction that actually means it's false and then you have disproved it. So feel free to use contradiction very liberally in this discussion of disproof. Now, the last comment I want to offer here is actually sort of a sad one. Oh no. This actually brings us to the end of our logical series with regard to Math 3120. We still have a few more videos that we're going to talk about new mathematical topics. In particular, we're going to be very interested in discussing the notions of functions with regard to the lens of set theory, what is a function. And then we also want to use that to help us better understand infinity, right? Because early on in our lecture series, we talked about combinatorics, which is all about studying finite sets. As we end this lecture series, we want to focus on infinite sets for which infinite sets can be a little bit more funky. And that's what we need to understand better functions about so we can better understand cardinality and understand these ideas of infinity. So while our mathematical discussions will continue for the last few lectures of this lecture series, this ends, and I should say this concludes our discussion of logic in Math 3120. So thanks for watching these videos. If you learned anything from them, please like them, subscribe to the channel to see more videos like this in the future, share these videos with friends or colleagues who might be interested in them as well. And as always, if you have any questions whatsoever, feel free to post them in the comments below, and I'll be glad to answer them as soon as I can.