Welcome to the second lecture. I hope you're making progress with the first assignment. You should not expect to solve all the problems in an assignment in a single session. Or even before the next lecture. What you should do before the next lecture is attempt each question. That's what I mean when I say, complete the assignments. Remember, the goal of this course is to acquire a certain way of thinking, not to solve problems by a given deadline. The only way to develop a new way of thinking, is to keep trying to think in different ways. Without guidance, that would be unlikely to get you anywhere, of course. But the point of a course like this is to provide that guidance. And the assignments are designed to guide your thinking attempts in productive directions. Okay, let's proceed. As a first step in becoming more precise about our use of language, in mathematical context, we've developed precise unambiguous definitions of the key connecting words and, or and not. The other terms we need to make precise implies equivalence for all and there exist a more tricky and we'll handle them later. Let's start with and. We often want to combine two statements into a single statement using the word and. So we need to analyze the way the word and works. The standards abbreviation that mathematicians use for this is an inverted V, known as the wedge. Sometimes you'll see the familiar & used but I'm going to stick to the common mathematical practice of using the wedge. For example we might want to say PI is bigger than 3 and less that 3.2. We could do this as follows, we could write PI is bigger than 3 and PI is less that 3.2. In fact for this example where we're just talking about position of numbers on the real line there's and even simpler notation we would typically write. 3 less than pi less than 3.2. But as an example Illustrates in the use of the word and, this one is fine. What does it mean? Well, if we have two statements phi and si, phi and si means that they're both true. The official term for an expression like this, is it's the conjunction of phi and si. Relative to the conjunction, the two constituents, phi and psi are called the conjuncts of phi and psi. What are the circumstances under which your conjunction phi and psi is true? Well if phi and psi are individually true, then the conjunction phi and psi will be true. Under what circumstances will phi and psi be false? Well, if either phi is false, or psi is false, or they're both false. This might seem very self evident and trivial, but already this definition leads to a rather surprising conclusion. And here it is. According to our definition, phi and psi means the same as psi and phi. They both mean that phi and psi are both true. In mathematical palettes, conjunction is commutative. But that's not the case for the use of the word, and, in everyday English. For example John took the free kick and the ball went into the net. That doesn't mean the same as the sentence, the ball went into the net and John took the free kick. They're both conjunction and the two conjunct are the same. One of them is John took the free kick. The other one is the ball went into the net. But anyone's who's familiar with soccer realizes that these two sentences have very different meanings. The fact is, in everyday English, the word and is not commutative. Sometimes it is, but not always. Let's see what you make of this one. Let A be the stance it rained on Saturday. And let B with the sentence it snowed Saturday. Question, does the conjunction A and B accurately reflect the meaning of the sentence, it rained and snowed on Saturday? Well, what do you think. Yes or No. Although I can think of situations in which the answer would be no in general I would be inclined to say the answer is yes. A useful way to represent a definition like this is with a propositional truth table. What we do is we list the components statements, in this case it will be phi and psi. And they're going to go together to make conjunction phi and psi. And now we're going to draw a table that lists all the possible truth false combinations for phi, psi, and phi ^ psi. So let me see, phi could be true, and psi could be true. Or phi could be true and psi could be false, or phi could be false and psi could be true, or phi could be false and psi could be false. The next step is to list in a final column t or f according to our definition of what the conjunction means. Why don't you see if you can fit a t or an f in each of those four boxes to represent the definition of phi and psi that we've given. According to the definition phi and psi is going to be true, Whenever phi is true and psi is true. So first rule, so there's a T going to go here. But that's the only condition under which phi and psi is true. In all other circumstances it's false. So the entries for these are all F. So in one simple table we've captured the entire definition of phi and psi. This emphasizes the fact that the truth of the conjunction depends only on the truth or falsity of the two conjuncts. The definition was entirely in terms of truth and falsity. What phi and psi meant was irrelevant. It was only about truth and falsity. That's going to be the case for all the definitions that we're going to give In order to make language precise. They're going to depend upon truth or falsity not upon meanings or logical connections. Now let's look at the combinator or. We want to be able to assert that statement A is true or statement B is true. For instance, we might want to say, a is greater than 0 or the equation x squared plus a equals 0 has a real root. Or maybe we want to say ab = 0 if a = 0 or b = 0. Those are both statements that we get, when we combine two sub statements with the word or. Both statements are in fact true, but there's a difference between them. The meaning of or is not the same in the first sentence as it is in the second sentence. In the first sentence, there's no possibility of both parts being true at the same time. Either a is going to be positive or this equation will have a real root. They can't both occur. If a is positive, then this equation does not actually have a real root. In the case of the second sentence, they could both occur together. To get ab = 0 it's enough if a is 0, it's enough if b is 0, or they can both be 0. So these two are different. In the first case we have an exclusive or, in the second case we have an inclusive or. Incidentally it doesn't matter if you try to enforce the exclusivity by putting an either in front of it. If you look at the way the word either operates, if you say either this or that, then what happens is that the either simply reinforces an exclusive or if one happens to be there. In the case of the second one, you could say ab = 0 if either a = 0 or b = 0. And in fact, that doesn't enforce the exclusivity at all. We just accept the fact that they could both be true. In other words, the word or in everyday English is ambiguous. And we rely on the context to disambiguate. In mathematics it's different. We simply can't afford to have ambiguity floating around. We have to make a choice between either the exclusive or or the inclusive or. And for various reasons it turns out to be more convenient in mathematics to adopt the inclusive use. The mathematical symbol we use to denote the inclusive or is a v symbol. It's known as a disjunctive symbol. So given two sentences phi and psi, Phi v si Means phi or psi or both. This sentence phi or psi is called a disjunction of phi and psi. And relative to the disjunction, the constituents phii and psi are called the disjuncts. Remember Phi or psi in mathematics means at least one of those two is true. They could both be true. For example, the following rather silly statement is true. (3 < 5) v (1 = 0). I can't imagine a mathematician writing actually writing that down except as an example as I'm doing right now. Silly examples like this are actually quite useful in mathematics because they help us understand what a definition means. This thing is true even though one of the disjuncts is patently false. So, this emphasizes the fact that for a distinction to be true, all you need to do is find one of the disjunct which is true, doesn't matter if one or more of the other disjunct is apparently false. Okay, let's see how well we do understand that. Here is a quick quiz, let A be the sentence, it will rain tomorrow and let B be the sentence, it will be dry tomorrow. Here's the question. Does the disjunction A v B accurately reflect the meaning of the sentence, tomorrow it will rain or it will be dry all day. Well, what do you think? The answer is clearly no. If that comes as a surprise to you you need to think about the definition of R a little bit longer, and see what's going on here. I'll leave you to that one. To wrap up this discussion of disjunction, let's see if we can complete the truth table for phi or psi. Okay, if you got this one right, your truth table should look like this. True, true, true, false. The disjunction is true If both are true, if one is true, or if the other is true. The only time when a disjunction is false is when both disjuncts are false. Okay? So now we've sorted out the meaning of the word all. The next word I want to look at is not. If psi is a sentence, then we want to be able to say that psi is false. So given psi, we want to create the sentence not psi. The standard abbreviation mathematicians use today is this symbol, which is like a negation symbol with a little vertical hook. Older textbooks you'll find will use a tilde. That's not the one I'm going to use, I'm going to stick to the modern notation, this sort of negative sign with the hook. And we call this the negation of psi. If psi is true, than the negation of psi is false. And if psi is false, the negation of psi is true. We often use special notations in particular circumstances. For example, we would typically write x ≠ y, Instead of not (x = y). But you have to be a little bit careful. For example, I would write not the case a less than x less than or equal to b. You might be tempted to write something like a not less than x not less than or equal to b. I would advise against that. That one is better than this one. This one is completely unambiguous. It means that it's not the case that x is between a and b in that fashion. This one, well, it could mean, you could agree that it means that but it's really ambiguous as to exactly what's going on here. So I would say, avoid things like that, use something like this. We should always go for clarity in the case of mathematics. Remember, the whole point of this precision that we try to introduce is to avoid ambiguities to avoid confusions. Because in more advanced situations, all we're going to have to rely upon is the language. And then, we need to make sure that we're using language in a non ambiguous and reliable way. Negation might seem pretty straightforward, and in many ways, it is. But it's not trivial. If we took something like not the case that pi is less than 3, then that's pretty straight forward. That means pi is greater than or equal to 3. Okay, that's easy, no problems there. Let me give you one that's not quite so obvious. Look at this sentence, all foreign cars are badly made. What's the negation of this sentence? Let me give you four possibilities. Possibility one, or possibility a, all foreign cars are well made. Possibility b, all foreign cars are not badly made. Possibility c, at least one foreign car is well made. Possibility d, at least one foreign car is not badly made. Well I'm not giving you this as a quiz but I would like you to think for a minute as to which one of these you think is the negation of that original sentence, or maybe you think it's something else. Maybe you think it's something to do with domestic cars, domestic is after all the negation to follow. So what do you think it is? Well let's look at them. A is actually a very common one for beginners to pick. I've been teaching this material for many years now, it's one of the examples I've always give, it's in the textbook I've written for this course, and previous textbook I've written, and this one is a common answer that I often get. But if you think about what the sentence really means, it's obviously not this one this is not the negation. Why? Is the original sentence true? No, of course it's not. There are many good cars that are foreign made. Okay, so it's not the case that all foreign cars are badly made. So this sentence is in fact a false sentence. We know that, just by our knowledge of the world. So if that sentence is false, then its negation is going to be true. But this isn't true. It's not the case that all foreign cars are well made, it's false so that can't be the negation. What about b? Same reasoning, that can't be the negation because it's simply not the case that all foreign cars are not badly made, okay, those are false statements. These are false so they can't be the negation of a false sentence. The negation of a false sentence is going to have to be true. So whatever the negation of this original sentence is, That negation will have to be something that's true. And we know what's true and false in terms of cars being well made. Well, is this one true? Yeah, that's true. Is this one true? Well, these are both true. So these are both possibilities for the negation of that. And this is still not a quiz, but I'm going to leave you for a little while to think about this one. Which one of these do you think actually is the negation of this? We’ll come back to this. I’m going to introduce some formal notation from sort of algebraic notation and eventually we’ll be able to reason precisely, to see which one of these two here or maybe a different thing is the actual negation of this. But let me stress a point I made a minute ago and I didn't write anything down. Look at the following sentence. All domestic cars are well made. I've actually had students over the years who have thought that that was in fact the negation of this. And why are they saying that? Because they're saying, this says something about all foreign cars, and this says something about all cars that are not foreign. So there is a sort of negation going on between these two, but it's not the negation of the original sentence. How do I know it's not the negation of the original sentence? Because the original sentence is false, therefore whatever the negation is, is going to have to be true. Well, this isn't true. This is also false. And because this is false, it can't possibly have been a negation of the original sentence. And in fact, this one really falls a long way of one being a negation of that, for the following reason. The original sentence is about foreign cars. That's what it's talking about, it has nothing to do with domestic cars, it's purely talking about foreign cars. So the negation can only possibly talk about foreign cars. These were good candidates for the negation because they talked about foreign cars. This one, isn't even in the ballpark for being a negation, because it's not talking about foreign cars, it's talking about domestic cars. Negating a word in a sentence, is not at all the same as negating the sentence. So this one, is a really bad choice. Let me finish with a very simple quiz. Let me ask you to fill in the truth table for negation. This one's a much simpler table because there's only one statement involved. Phi, and then we're going to negate it. So. Very simple truth table. What do you think the values are? Yep, this one was an easy one. If phi is true the negation is false, if phi is false, the negation is true. And with that you should be in a position to complete assignment two. That last example, about the negation of the sentence all foreign cars are badly made, should I think, illustrate why we're devoting time to making simple bits of language precise. To figure out what the correct negation is, we relied on our knowledge of the everyday world. That's fine for statements about the everyday world we're familiar with, but in a lot of mathematics, we're dealing with an unfamiliar world. And we can't fall back on what we already know, we have to rely purely on the language we use to describe that world. When we've taken our study of language far enough, we'll be able to look at that foreign cast statement again, and use rigorous mathematical reasoning to determine exactly what it's negation is. Well, that brings us to the end of the first week. How are you getting on? For most of you, this will seem like a very strange course. And certainly won't look much like mathematics. That's because you've only been exposed to school math. This course is about the transition to University level mathematics which, in some ways, is very different. There isn't much material. And as a result the lectures are short. I'm not providing you with new methods or procedures. I'm trying to help you learn to think a different way. Doing that is mostly up to you. It has to be. If you're at all like me and pretty well every other mathematician I know, you're going to find it hard and frustrating, and it's going to take some time. You definitely need to connect to other student's and start working together. If you're able to scan pages of work into PDF or use your smart phone to take good clear photos of your work, I advise you to start showing your work to other students to get their feedback. Send images as email attachments, put them on Google Docs, or upload them to whatever networking site you choose. You should definitely attempt all the assignments that I give out after each lecture. Doing those assignments, both on your own and in collaboration with others is really the heart of this course. Yeah, sure you can watch the lecture several times. But you'll find that it almost never tells you the answer. Or even how to get the answer in the way that you're familiar with from high school. It's like learning to ride a bike. Someone can ride up and down in front of you for hours telling you how they do it. But you won't learn to ride from watching them or having them explain it to you, you have to keep trying it for yourself and failing until it eventually clicks. All right, this is a very different way of learning than you're used to, at least in mathematics. As well as the assignments, there is also a weekly problem set. The problem sets comprise assignment questions that count directly towards your grade. Because this course is designed for many thousands of students, it's impossible to look at everyone's work and provide feedback, so we have to rely on automated grading. This means that the questions are posed in multiple choice format. But these are not at all like the in lecture quizzes. Those are supposed to be answered while on the spot. The problem set questions will require considerable time. This is not ideal. For the material in this course, whether you get particular questions right or wrong, it's pretty insignificant. It's your thinking process that's important. But we can't check that automatically. Asking you to answer multiple choice questions is like checking your health by taking your temperature. It tells us something and can alert you and others that something is wrong, but it's pretty limited. Still, checking temperatures is better then nothing and the same is true for the problem set grading. What I'd like you to do is to try to grade your own work and that of others in whatever study group you form and you should definitely try to get into one.
