 At this point, we have seen how to prove lots of different types of logical statements, particularly as we allow it to range over all of our different connectives. So we've talked about AND statements. How do you prove that? Well, you just prove P and then you prove Q and then put that together. How do you prove an OR statement, a disjunction? Well, in that situation, you can actually just be like, oh, if not P, then Q, that you can prove an OR statement in that way. That's a nice little disjunctive syllogism. We've put a lot of emphasis on P implies Q. Conditional statements have been the one we've done the most. I mean, I just mentioned how you can prove an OR statement as if it's a conditional statement. We've proven by conditional statements. P is equivalent to Q. That's just, you just prove two conditional statements. And we've talked about even like these, the following are equivalent type proofs where you have lots of equivalent statements and how that's just a bunch of implications and how you can do that in an efficient manner, okay? So we've talked about all those different type of statements with the logical connectors there, but how do you prove a statement that involves some type of quantification? Like if you have a statement like for all X inside of some set X, the property P of X holds. How do you prove that? Or what about the existential quantifier there? There exists an element X inside of X such that the property P of X holds. How do these things work out? Now, it turns out that we've essentially already taken the universal quantifier, that is we've already taken care of that because of the following observation. The proof of a universal statement is essentially just a conditional statement, right? So if you take that statement again for all X inside of X, we have that P of X holds. This statement right here is logically equivalent to the statement that if X is inside of X, then P of X holds, okay? So a universal, universally quantified statement is really just a conditional statement about sets that you're one of your hypotheses is that your element belongs to the set. And as this was an arbitrary element, that means then you'd improved it for all of them. So yeah, like I said, our coverage of conditional statements has taken care of how do you approve a universally quantified statement? Now, in this video, we're gonna focus on existential statements then. This one deserves a little bit of attention, but let's be honest, that's all the attention we're gonna offer here for the following reason. To prove an existential statement, such as there exists an X inside of X, such that P of X holds, all that it requires, all that is needed of us, it's completely sufficient to show that an element satisfies the property. That is to say, all you have to do is provide an example. Of the thing, you just have to provide an element that has the property. It doesn't need to be any more complicated than that to prove that something exists, just show it to me. That's all, that's all that you have to do here. And so let's look at some examples of this via proof. So let's show an existence proof here. There exists an even prime number, okay? So this is an existence statement, there exists. To prove it, all that I have to do is show you an even prime number. So note that P equals two is an even number. How do I know that? Well, two divides P because it's two times one, all right? On the other hand, P is a prime number, right? Because the only divisors of two are one and two. And therefore we have an even prime number. Two is an even prime number. That's all that you have to show to prove it. I've now provided the object and we know there's an even prime number now. Now, admittedly there's no more even prime numbers. Two is the only even prime number, but the theorem didn't ask us to show there's only one even prime number. There's just, we just have to prove there exists one. That leads more towards this idea of an uniqueness proof that we'll talk about in the next lecture. But for now, let's continue with existence. Here's another one. There exists an integer that can be expressed as a sum of two perfect cubes in two different ways. There's a scene in the movie, The Man Who Knew Infinity where Monochon and Hardy are talking about this as they're looking at taxicabs. That's what makes this problem so famous here. Consider the number 1,729. Notice that 1729 is equal to one cube plus 12 cubed because one cubed is one and 12 cubed is 1728. Likewise, 1729 is also equal to nine cubed plus 10 cubed because nine cubed is 729 and 10 cubed is equal to 1,000. So here is in fact a number, an integer, which can be expressed as a sum of two different, it can be expressed as a sum of two perfect cubes in two different ways. Now, both of these proofs illustrate the existence of such numbers that have the properties, but it doesn't suggest at all how one found these. Coming up with two as an even prime number is pretty straightforward, but like the fact that Monochon was able to pick up this fact very quickly when Hardy didn't notice it at all was somewhat astounding at the time. It's like, how did he know that, right? Where did he come up with this? And so was he just a brilliant coincidence or was he a brilliant mind? Probably a little bit of both, right? Monochon of course was a very brilliant mathematician who died very young regrettably, but for an existence proof, it doesn't actually matter where it came from. We just demonstrate that it exists. It doesn't give us any clue on why it exists. All you have to do is provide an example. It also doesn't tell us if there's any more examples. Is there another number that satisfies this property? Maybe, maybe not. The existential proof, you only need to provide one example. One example is in fact enough.