How did you get on with that massive assignment four? To give you a chance to complete it, this lecture is also fairly short, and this time the assignment is down to a single page. So keep working on that last assignment. It really is crucial to master those ideas and that terminology. Knowing when and why one statement implies another and being able to distinguish between necessity and sufficiency are crucial abilities in today's world. There are two more language constructions that are fundamental to expressing and proving mathematical facts, and which mathematicians therefore have to be precise about. The two quantifiers, there exists, and for all. When we've made these terms precise and are sure we know how to use them properly, our analysis of language will be over. The word quantifier is used in a very idiosyncratic fashion here. In normal use, it means specifying the number or amount of something. In mathematics, it's used to refer to the two extremes, there is at least one and for all. These are all we need to look at because of the special nature of mathematical truths. When mathematics is viewed as a subject in its own right as opposed to being a set of tools used in other disciplines and walks of life, the call of the subject is the theorem, and the majority of mathematical theorems will have one of two forms. There is an object x having property p, for all objects x property p holds. I'll take these one at a time. We're looking at statements of the form there is an object x having property p. For example, the equation x squared + 2x + 1 = 0 has a real root. We can emphasize that this is an existing statement by writing it in the following form. There is a real number x, so it should x squared + 2x + 1 = 0. Or we could write it this way, there exists a real number x, so it should x square + 2 x + 1 = 0. Doesn't matter whether we write it with an is or an exists, it's an existent statement. The symbolic abbreviation that mathematicians use for exists is a back to front E, E for exists of course. So a mathematician would write this as follows, there exists an x, such that x squared + 2x + 1 = 0. This symbol is called the existential quantifier. The simplest way to prove an existential statement of this form is to find an actual x that satisfies the property. And with this example, it's easy. Take x = -1. Then x squared + 2x + 1 = -1 squared which is 1 + 2 x -1 which is -2, and we got a +1, 1- 2 + 1 = 0, which solves this. Well, that was a simple example. Sometimes we can prove an existential statement without finding an actual object that satisfies the property. Let me give you an example of that. I'm going to prove this statement, there exists an x such as x cubed + 3x + 1 = 0. I'm not going to find an actual x that solves this cubic equation, but I am going to show that there is a solution. I'm going to start by looking at the function y = x cubed + 3x + 1. This is a continuous function. Now continuity of functions is actually a fairly deep topic, we'll touch on it briefly at the end of the course. But for my present purpose, all I need to know is that a continuous function is one whose graph is a smooth curve with no breaks and jumps. And you should be sufficiently familiar with cubic equations by now to know that a cubic equation has a nice smooth curve that looks something like that, in many cases. Up and down, two curves, okay. The point is it doesn't have any jumps. If x = -1, This curve has value or this function has value. -1 cubed which is -1 + 3 x -1, which is -3 + 1, which is -3. If x equals +1, The curve or the function, Has value y = 1 + 3 + 1 which equals 5. So the curve lies below the x axis, For x = -1. And above the x axis, For x = +1. So it looks something like this, put the origin here, with the outcome of -1 and we're going to look at +1. It's going to lie somewhere down here, at -1, it's going to lie somewhere up here at +1. And somewhere it's going to cross the axis. Looks like that, that, who knows? The point is, it's going to have to cross the axis somewhere or other, either to the left of the origin or to the right of the origin, doesn't matter where. The important point is, somewhere between -1 and +1, this curve crosses the axis. And when it crosses the axis, the y value is 0. So we haven't said which x solves the equation, we haven't even said whether the x is negative or positive, But we do know there is such an x. We've shown that an x exists that satisfies the equation without finding such an x. This is an example of an indirect proof. We haven't proved it directly by finding an x that solves the equation. We simply showed that there is an x that solves the equation. A lot of mathematical proofs are of this nature. We show that there is a solution to some equation or that there is an object that satisfies some property over there without finding such an object. Pretty cool? Time for a quiz. A simple modification of that last argument yields a more specific result. Which of the following is it? Is it 1? There is an x less than 0, so it should x cubed + 3x + 1 = 0? Or is it 2? There’s an x greater than 0, so it should x cubed + 3x + 1 = 0? Okay, which one do you think it is? Really, answer is 1, and here’s how we can get that. In the previous argument, I looked at the curve at x=-1 and x=+1. And I showed it looked something like this. At -1, the curve was below the x axis. At +1, the curve was above the x axis. And so it has to cross the x axis somewhere between -1 and +1. But instead of going between -1 and +1, I could go between -1 and the origin itself. Well, note that if x = 0, then y = x cubed + 3x + 1 is 1. So 0 is somewhere up here, Which means it's negative at -1. It's positive at 0, so it's somewhere between -1 and 0 that it crosses the axis. Since the curve lies below the x axis at x = -1 and above the x axis at x = 0, it must cross the x axis between -1 and 0. Or I could prove it a different way. I could simply observe that if x is greater than or equal to 0, then x cubed + 3x + 1 is greater than 0, and in particular, is not equal to 0. So 2 is false, and that just leaves 1. We know that one of these two has to be true, as a result of the previous argument. And so if we can eliminate 2, as I've just done here, then I can conclude 1. So either way, either by refining the previous arguments or by deducing it from the previous results, by this observation, I end up showing that the equation has a root somewhere between -1 and 0. I still haven't found that root, but I have shown that there is a root. It's an indirect proof. Incidentally, in giving that last proof, I was careful to distinguish between the terms negative 1 and minus 1. Chances are, your middle or high school teacher made a big deal of that. Like many professional mathematicians, maybe even most of us, I'm generally far less careful. I tend to use negative and minus interchangeably. It's not that we don't know the distinction. It's just that at university level, we focus on other issues. And since this is a university course, I'll assume you can use your knowledge of arithmetic to determine the intended meaning, just as we do with everyday language. In this course, a similar issue arises with implication in the conditional. And since that distinction is likely new to you, I'm going to try to always use the correct term. But when I'm talking with my professional colleagues, we just talk about implication to cover both cases and rely on our shared knowledge of the concept to disambiguate. In both cases, the distinctions are important, which is why we emphasize them in our teaching. But once we show everyone has fully grasped the issues, we adopt a more relaxed attitude, confident that our understanding will keep us out of trouble. The same kind of argument I just used to show that a certain cubic equation has a real root can be used to prove the Wobbly Table Theorem. Suppose you are sitting in a restaurant at a perfectly square table with four identical legs, one at each corner. Because the floor is uneven, the table wobbles. One solution is to fold a small piece of paper and insert it under one leg until the table is stable. But there's another solution. Simply by rotating the table, you'll be able to position it so that it doesn't wobble. You might enjoy trying to prove this. The solution's simple, but it can take a lot of effort before you find it. It's very much a thinking outside the box question. It would be an unfair question on a timed exam. And I'm not giving it as a course assignment. But it's a great puzzle to keep thinking about until you hit upon the right idea. I'll leave you with it. Even if you can't think of a mathematical solution, you could go off in search of a square wobbly table and confirm the theorem experimentally. Meanwhile, let's get back to our main theme. Sometimes, it's not immediately obvious that a statement is an existence assertion. In fact, many mathematical statements that do not look like an existence statement on the surface turn out to be precisely that when you work out what they mean. For example, the statement that a number x is rational is an existence statement, and here's why. Let's take the specific statement, square root of 2 is rational. As it happens, this is a false statement. But we're just using it as an example of a statement. So on the face of it, this doesn't look like an existence statement. But it actually is, because we can write it as there exists natural numbers p and q such that square root of 2 = p/q. Or using our symbolic abbreviation, exist p, exist q, such that square root of 2 is p/q. Incidentally, when we do come to prove that the square root of 2 is not rational, we'll do it by showing that there are no values of p and q that satisfy this equation, so we'll prove a non-existence statement. Well this expression is fine, so long as we know in advance that p and q denote natural numbers. I mean here, I said it explicitly. Here, I didn't. So how do you know that p and q denote natural numbers? I mean, maybe they should denote real numbers or complex numbers. The answer is, you make it more specific. And you do that by writing it in the following way. There exists a p in N, and there exists a q in N, so it's root 2 = p/q, where N denotes the set of natural numbers. Incidentally, another font that's often used to denote a set of natural numbers is something like this, an N with a double line for the diagonal. In this course, I'm assuming that you're familiar with the set theory with the basic set theory, and you know what the membership symbol means. If you're not, then there is, of course, supplements, and I'm just going to leave it to you to read that supplement. And of course, there's a course textbook, if you acquired a textbook, that's my own book, Introduction to Mathematical Thinking. In any case, I'm going to assume that you are familiar with set theory, and I'm not going to talk about it in the course itself. You sometimes see mathematicians write it in an even more abbreviated fashion. There exists a p q in N such that square root of 2 is p over q. That's fine if you're confident about the material and you're familiar with the notations. But we're going to be looking at examples where there are many different types of quantifiers coming in together. And then if you start putting them together in this fashion there's a possibility of getting confused and going off course. So I would say at the early stages, keep things distinct, as they are here, and try to avoid that for now. As I say, professional mathematicians write this kind of thing all the time, but just as we're beginning to learn mathematics, it's important that we distinguish things like negative and minus. And then when we master them, we tend to forget the distinctions, or we don't forget them but we don't make them explicit. So here, we're beginning to look at quantifiers. I think it's important to be very explicit. And then once you really understand it, then you can do what we do in the business, we just write things in the simplest fashion. Confident that we know what's going on and the way we write it won't mislead us. Incidentally, see if you can prove that the square root of 2 is not rational. We'll do that in class later but see if you can do it now. What that amounts to is proving that there do not exist p and q in the natural numbers. Such that the square root of 2 equals p over q or that 2 equals p squared over q squared. This is the statement you actually prove in order to show that the square root of 2 is irrational. It's not a difficult argument, but it's rather clever and ingenious. It's fairly short, and the chances are that you're not going to come up with it, but it's worth giving it a shot. Spend a half an hour, an hour or so thinking about it to see if you can prove that statement. And good luck on that one, but we will come back to that in class. >> How are you doing? Do try show that the square root of two is irrational. Not so much to find the proof, but to get used to what it's like to do university level mathematics. One feature you need to get used to in mastering college mathematics, or more generally what I'm calling mathematical thinking, is the length of time you may need to spend puzzling about a problem or even one particular detail. For the most part, without seeming to be making progress. High school mathematics courses, particularly in the US, I generally put together so that most problems can be done in a few minutes, with the goal of covering an extensive curriculum. At college, there's far less material to cover, but the aim is to cover it in more depth. That means you have to adjust to the slower pace with a lot more thinking and less doing. At first this comes hard, since thinking without seeming to be making progress is initially frustrating. But it's much like learning to ride a bike. For a long time you keep falling, or relying on training wheels, and it seems you'll never get it. Then suddenly, one day, you find you can do it, and you can't understand why it took so long to get there. But that long period of repeated falling was essential to your body learning how to do it. Training your mind to think mathematically about various kinds of problems, it's very much like that. Okay, sermon over. The one remaining piece of language we need to examine and make sure we fully comprehend is the universal quantifier, which asserts that something holds for all x. >> This notation means for all x it's the case that something or other. The symbol here is an upside down letter A, Which means for all. For example, if I wanted to say the square of any real number is greater than or equal to 0, I could write it like this. All x, x squared greater than equal to 0, so the short way of actually saying that is, for all x, x squared greater than or equal to 0. Just as we would write things like there exists an x, so should x squared = 0. Notice that I didn't use the word all in the sentence. I use the word any. The same thing happened with exist, there are different words we can use to express an existence assertion, and there were different words we can use to express a flawless assertion. This is the universal quantifier. How do I know what the x means? Well I have to specify what it means. If I want to be explicit, I would have to write something like for all x in the set of real numbers, x squared greater than or equal to 0. Okay, let's take a look at combinations of quantifiers. Most statements in mathematics involved two or more quantifiers combined. For example, if I want to say there's no largest natural number. What I would write is this, for all m in the set of natural numbers, there is an N in the set of natural numbers, such that N is bigger then m. For all natural numbers m, there is a natural number N such that n is bigger than m. That clearly says there is no largest natural number. Note that the order of the quantifiers is important. If I swap those quantifiers around and write exists in n in N, switch that for all m in N, n is bigger than m. The result says there is a natural number n which has the property that for all natural numbers m, n is bigger than n. In other words, this says there is a natural number bigger than all natural numbers which is false. All I've done is swapped quantifiers around, when in so doing, I've turned a true sentence into a false sentence. Remember that example from the American Melanoma Foundation? In their fact sheet they wrote, one American dies of melanoma almost every hour. Using our quantifiers, we could write that like this. There exists an American, such that for every hour A dies in hour H. There is an American, such that every hour A dies in hour H. [LAUGH] Poor guy, I mean quite amazing. What the writer obviously meant was the following. For every hour there is an American such that A dies in hour H. It will be different Americans for different hours, now as I mentioned earlier In the case of everyday English, these are almost never a problem. Everyone understands the context, we know what's meant. We know that this one is not meant, because this is clearly wrong. We know that that one's the one that's meant. And with a simple mathematical example like this one, arguably, it's not important either, because everybody knows what's meant. We know that there is no largest natural number. We know it must be this one and not this one. So in these examples, there's probably no problem. But we want to use mathematical notation in complicated situations and to talk about things we don't yet understand. That means we can't disambiguate, it's really important that we say things in the right order. Okay, I think it's time for a quiz. So how did you get on? Where the literal meaning of this statement is captured by number 1, that's the literal meaning. It says, there is a license for which there are two distinct states that that license comes from. So, this is the literal meaning, But it's false. Licenses are issued by states, you can't get a license that's issued by two separate states. So the correct answer to the question is number 1, but it's not what the person who wrote that sentence in the form, in the driver's license application, meant to say. Let's look at number 2. That says there are two different licenses, they're different. There are two licenses which are issued by one state Well, I guess it's possible that you could own two licenses that are both issued by the same state if you had two identities or something like that. So this is a possibly true statement. But it's not what the sentence means. Let's look at number 3. That says, there are two licenses and two states, different states, different licenses. There are two licenses in two states. One license comes from one state, the other license comes from the other state. This is the sentence that the driver's license application meant to say. So the literal meaning is number 1, which is false. Number 2, may be true, but it's not what the sentence is about. Number 3 is clear that the intended meaning is the meaning that we would all understand from this, because we know about licenses and about states and so forth. So this is the one we understand, but it's not the literal meaning. It doesn't matter when you're applying for a driver's license. So unless I guess if you get into trouble, you get your lawyer might try to get you off by arguing on the basis of the mathematical meaning. Good luck, if you want to try that. But in mathematics, we can't allow this kind of thing to happen. You get into all sorts of problems. And in mathematics, we can't hire a lawyer to get us out of difficulty. Okay, let's look at the next question in this quiz. Well, the correct answer is number 3. Let's see why. What does the first one say? It says, if there's a license and there were two different states, so that license is valid in one state and valid in the other state. So, it’s says that a license is valid in two states. Well, okay, it’s true, because in America when you have a license valid in one state, it's valid in any state, you can use it anywhere. So it doesn’t say anything particularly interesting. What about number 2? It says, for every license and for every pair of states, which could even be the same state, the license is valid in one state or it’s valid in another state. Well, this is actually false as a statement, because there can be invalid licenses. And if you got an invalid license, that means the whole thing is wrong. This one is the one that captures that sentence. Let’s just read it. It says, for every license, any license, if there is a state in which that license is valid, then that license is valid in all states. So 3 is the one that captures the sentence. One is actually a true statement but of very little relevance, and 2 is actually a false statement. It’s false, remember, because you can have invalid licenses. So it's not the case that for all licenses and for all states, the license is valid in one state or another state. Okay, how did you do on that? You might have had difficulty reading the formulas in that last quiz. If so, check out the supplementary video tutorial called How to Read Mathematical Formulas. You'll find it in the same place you access all the regular tutorial videos.
