 In writing proofs of mathematical theorems, and heck, even when you're reading proofs, sometimes you have to do this, we should always be mindful and alert to the logical structure and meanings of each of the statements and sentences that appear in this proof. And because of this, it's sometimes necessary and helpful to parse these statements into expressions involving logic symbols, like and or not, those type of things. We did this very much in the previous video when we were considering quantifiers. We were rewriting mathematical statements using the quantifiers in the logical symbolic sense. It was a very good practice and this is something we wanna do in general. Now, when you're reading a writing, you might do this like on some scratch paper before you submit it just to get a better understanding. Sometimes you can just do it in your head or when you have a really advanced proof, you might have to do this to help you organize things. Now, as you get more accustomed to mathematical logic, the need to do this is much less. But of course, for those beginners who are transitioning into advanced mathematics, this is a very useful skill to be able to do. So let's take the statement of the mean value theorem from calculus one. In case you don't remember what that says, it's the following. If F is a continuous function on the closed interval A to B and it's a differentiable function on the open interval A to B, then there is a number C that belongs to the open interval A to B such that the derivative of F evaluated at C is equal to the slope of the tangent line there, F of B minus F of A over B minus A. Intuitively, the mean value theorem tells you that if you take the average rate of change right here, somewhere the instantaneous rate of change is equal to it. That or another way to think about it geometrically is that if you take the secant line between two points on the graph of a function, if the function is appropriately continuous and differential, there will exist a tangent line that's parallel to that secant line. Again, give me the intuition. You don't actually need to know that to understand what's gonna happen right now. What we wanna do is rewrite this symbolically. So what are some key words we're looking for? Well, the very first word, if, and then subsequently there's a then in there as well. I actually broke up the sentence because it was too long to fit in one line on the screen here. I actually broke it up at the if, then part there. That gives us some type of conditional that's happening here. There's gonna be this conditional statement. There's this part right here and then there's this part right here. Now looking at the first part, F is continuous and differentiable on these separate intervals. That and is the key word there. There's an and that has a conjunction in the logic here. So we get something like the following. F is continuous on the interval A to B. That's our first statement. I'm gonna erase my arrow. We'll put another one on the screen in just a second. But we get the and, like so. Then we have this other statement. F is differentiable, whoops, differentiable on the interval A to B. So these are the hypotheses of the mean value theorem. And then this is a conditional, right? Conditional, what does the condition guarantee here? There exists a number C inside the interval A to B. That's a quantified statement there exists. That is the existential quantifier there. So we get there exists some number C in the interval A to B. But there's a condition attached to it, right? So we might get something like the following. Such that, such that then this thing holds, all right? And there's really nothing I need to break up from that. It's already a mathematically symbolic statement there. Like so. So if we rewrite the mean value theorem into symbols, we get exactly what we see here on the screen. Look for those words like if then and quantifiers. These are all important things we want to interpret logically here. Let's look at another example. Let's consider the Goldbeck conjecture, which we considered previously in this lecture series. Now, we're not saying that this is a true statement. It's probably true, but we don't actually know that. But nonetheless, it's still a statement and it could be written symbolically. Remember, the Goldbeck conjecture says that every even integer greater than two is the sum of two primes. So there are some quantifiers in play here. Every integer greater than two is the sum. So there exists some prime numbers that satisfy an equation there. So there's some sets that have to be described. Now, for the purposes of this example here, we'll introduce two sets. We'll call E, the set of even integers greater than two. So this will be like four, six, eight, 10, 12, 14, et cetera. And we're also gonna introduce the set of primes. So we get two, three, five, seven, 11, 13, et cetera. So we have these two sets. So with those two sets defined, then I can turn this into some statements involving quantifiers. For every even integer greater than two, that would say then that this right here is saying that for all n inside of E, okay? That number is the sum of two prime numbers. So with that we have these two prime numbers means there exist primes P and Q that belong to the set of P, such that n is equal to P plus Q. So this one took a little bit more effort to put into symbolic form, but once we've described the sets in play here, what's the universe of consideration? We're then able to describe this in this more symbolic manner here. And so this is a good exercise to practice for those who are new to mathematical statements and logic and such. Now in translating a statement, you always need to be attentive to its intended meaning. Don't jump automatically to replacing every and with a conjunction symbol and every or with a disjunction symbol. Consider the very first sentence you see on the screen right here. At least one of the integers X and Y is even. Now, even though the word and appears here, it turns out this is not a conjunction. If you look at the meaning of this statement, it's actually a disjunction and or would actually be the appropriate thing here. Because look at this, at least one. Now this kind of sounds like it's a quantifier, but we have to be careful of the statement. It turns out what's happening here is at least one. This is telling us that either X is even or Y is even. And so when we write this in symbolic form, we would get X is an even number or Y is an even number, not odd. And that's why the word at least one is used here. It's not used as a quantifier. It's actually used as an or, right? At least one and those together actually mean or believe it or not. Because when we talk about or's in the logical sense, what we mean is this one happens, not this one or this one happens, not this one or they both happen, right? It could be that they're both even numbers when we say at least one of them is even. But it's really is a disjunction. So this is why I mentioned we focus on symbolic representation because it can help us get the real meaning of the statement, all right? And so we don't wanna be run astray by the and that we saw right here. But we also have to be careful about words like but, okay? Look at this sentence right here. The integer X is even, but the integer Y is odd. But in this context actually means and. Like if I just replace this with and, the integer is, the integer X is even and the integer Y is odd. Does it actually change the meaning of the sentence? No, no. The only reason the word but is used here is actually cause it's putting some type of contrast. One is even and the other is odd. So but is an and with an emphasis that oh, there's these two statements, but they're different, okay? So in symbolic form we could write this as X is even and Y is odd. So you have to be careful of the meaning of the words and not the literal sense. And does not always mean and and or might not always mean or. You have to look at the context. You have to look at the subtext. What is the meaning of these words here? It can be a little bit tricky, but with practice you'll be able to do it. And that brings us to the end of lecture 10. Thanks for watching. If you learned anything in these videos, please like them. If you wanna see more videos like this in the future, subscribe to the channel for updates. And as always, if you have any questions, please post them in the comments below and I'll be glad to answer them at my soonest convenience.