Welcome to the second lecture, I hope you're making progress with the first assignment.I'll post my answers to some of the assignment questions later. Consult the course schedule on the web site and check regularly for announcements of any changes. 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 assignment. 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 contexts, we will develop 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, are 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 standard abbreviation that mathematicians use for this is an inverted v, known as the wedge. sometimes you'll see the old familiar ampersand 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 than 3.2. In fact for this example where we're just talking about the position of numbers on the real line there's an even simpler this one is fine. but already this definition leads to a rather surprising conclusion. Question. The other one is the ball went into the net. if either phi is false. if phi and psi are individually true. then the conjunction phi and psi will be true. phi and psi means that they are both true. Let a be the sentence It rained on Saturday. Let's see what you make of this one. According to our definition phi and psi means the same as psi and phi. Relative to the conjunction. and let b be the sentence it snowed on Saturday. or psi is false. if we have 2 statements phi and psi. What are the circumstances under which a conjunction phi and psi is true? Well. Under what circumstances will phi and psi be false? Well. But anyone who's familiar with soccer realizes that these two sentences have very different meanings. What does it mean? Well. This might seem very self evident and trivial. in everyday English. Sometimes it is. he ball went into the net and John took the free kick. They're both conjunctions.2. The official term for an expression like this is it's the conjunction of phi and psi. the two constituents phi and psi. are called the conjuncts of phi and psi. And here it is. Does the conjunction a and b accurately reflect the meaning of the sentence it rained and snowed on Saturday? . John took the free kick and the ball went into the net. the word and is not commutative. For example. conjunction is commutative. and the two conjuncts are the same. are they both false. In mathematical parlance. They both mean that phi and psi are both true. The fact is. But as an example illustrating the use of the word and. but that's not the case for the use of the word and in everyday English. but not always.notation. We would typically write 3 less than pi less than 3. That doesn't mean the same as the sentence. One of them is John took the free kick. and psi could be true. in general I would be inclined to say the answer is yes. They're going to depend upon truth or falsity. So let me see. What we do is we list the component statements. This emphasizes the fact that the truth of a conjunction depends only on the truth or falsity of the two conjuncts. Or phi could be false and psi could be false. what do you think? Yes or no? Although I can think of situations in which the answer would be no. and phi and psi. The next step is to list in the final column T or F. What phi and psi meant was irrelevant. according to our definition of what the conjunction means. psi. But that's the only condition under which phi and psi is true. We want to be able to assert that statement a is true or statement b is . That's the first rule so there's a T going to go here. In all other circumstances it's false. Why don't you see if you can fill in T or F in each of those four boxes to represent the definition of phi and psi that we've given. in this case what would be phi and psi. And now we're going to draw a table that lists all the possible truth false combinations for phi. That's going to be the case for all the definitions that we are going to give in order to make language precise. Or phi could be true and psi could be false. Phi could be true.Well. we've captured the entire definition of phi and psi. So in one simple table. phi and psi is going to be true whenever phi is true and psi is true. The definition was entirely in terms of truth and falsity. not upon meetings or logical connections. Or phi could be false and psi could be true. and they're going to go together to make the conjunction phi and psi. Now let's look at the combinator or. A useful way to represent a definition like this is with a propositional truth table. It was only about truth and falsity. According to the definition. So the entries for these are all F. to denote the inclusive or is a v symbol. In mathematics it's different. Those are both statements that we get when we combine two substatements with the word or. . In the case of the second one. it turns out to be more convenient in mathematics to adopt the inclusive use. The meaning of or is not the same in the first sentence as it is in the second sentence. if a equals zero or b equals zero. is ambiguous. In the second case we have an inclusive or. Both statements are in fact true. you could say. To get ab equals 0. The mathematical symbol we use. if you say either this or that then what happens is that the either simply reinforces an exclusive or. They can't both occur. and we rely upon the context to disambiguate. In the first sentence there's no possibility of both parts being true at the same time. Incidentally it doesn't matter if you try to enforce the the exclusivity by putting an either in front of it. We simply can't afford to have ambiguity floating around.true. In other words. or the inclusive or. Or maybe we want to say ab equals 0. Either a is going to be positive or this equation will have a real root. It's enough if b is 0. we might want to say. If a is positive. And in fact that doesn't enforce the exclusivity at all. ab equals 0 if either a equals 0 or b equals 0. they could both occur together. but there's a difference between them. or they can both be 0. it's enough if a is 0. So these two are different. if one happens to be there. if you look at the way the word either operates. in in everyday English. In the case of the second sentence. or the equation x squared plus a equals 0 has a real root. we just accept the fact that they could both be true. For instance. We have to make a choice between either the exclusive or. then this equation does not actually have a real root. the word or. And for various reasons. a is greater than 0. In the first case we have an exclusive or. The disjunction is true. is. if one is true. And let b be the sentence. even though one of the disjuncts is patently false. is called a disjunction of phi and psi. because they help us understand exactly what a definition means. And relative to the disjunction. Remember. Except as an example. false. True. as I'm doing now. So given two sentences phi and psi. This thing is true. 3 is less than 5 or 1 equals 0. your truth table should look like this. what do you think? The answer is clearly no. the constituents phi and psi are called the disjuncts.It's known as the disjunctive symbol. phi v psi means phi or psi. true. They could both be true. I can't imagine a mathematician actually writing that down. Silly examples like this are actually quite useful in mathematics. is patently false. phi or psi in mathematics means at least one of those two is true. true. it will rain tomorrow. it will be dry tomorrow. which is true. So this emphasizes the fact that for a disjunction to be true all you need to do is find one of the disjuncts. Let a be the sentence. tomorrow it will rain or it will be dry all day? Well. The only time when a disjunction is false is when both disjuncts are false. Okay? Let's see how well we do understand that. Here's the question. It doesn't matter if one or more of the other disjuncts. I'll leave you to that one. This sentence. Does the disjunction a or b accurately reflect the meaning of the sentence. or both. phi or psi. i both are true. If that comes as a surprise to you you need to think about the definition of or a little bit longer and see what's going on here. Okay? . If you got this one right. For example. Okay. or if the other is true. Here's a quick quiz. the following rather silly statement is true. To wrap up this discussion of disjunction lets see if we can complete the truth table for phi or psi. I. because in more advanced situations all we are going to have to rely upon is the language. the negation of psi is true. Older textbooks will. will use a tilda. You might be tempted to write something like a not less than x. yeah. And if psi is false. And we call this the negation of psi. could mean. So given psi we want to create the sentence not psi. but what you.So now we've sorted out the meaning of the word or. The standard abbreviation mathematicians use today is this symbol. I mean you could agree that it means that. to avoid confusions. And. well it's. Negation might seem pretty straightforward and in many ways it is. we should always go for clarity in the case of mathematics. So. you'll find would be. I'm going to stick with the modern notation. than the negation of psi is false. I would. I would say avoid things like that use something like this. But you have to be little bit careful. If psi is a sentence then we want to be able to say that psi is false. We often use special notations in particular circumstances. This one. This one is completely unambiguous. it's really ambiguous as to exactly what's going on here. That one is better than this one. We. If psi is true. and then we need to make sure that we're using language in a non ambiguous and reliable way. but it's not trivial. I would write not the case a less than x less than or equal to b. it means it's not the case that x is between a and b in that fashion. this sort of negative sign with the hook. Remember the whole point of this precision that we're trying to introduce is to avoid ambiguities. which is like a negation symbol with a little vertical hook. . The next word I want to look at is not. If we took something like not the case that pi is less than 3. instead of not x equals y. that's not the one I'm going to use. For example. we would typically write x not equal to y. not less than or equal to b. I would advise against that. For example. it's obviously not this one. Maybe you think it's something to do with domestic cars. We know that just by our knowledge of the world. So this sentence is.Then that's pretty straightforward. Okay? That's easy. Or maybe you think it's something else. Look at this sentence. All foreign cars are badly made. It's in. A is actually a very common one for beginners to. a false sentence. at least one foreign car is not badly made. to pick. Possibility c. at least one foreign car is well made. It's not the case that at all foreign . There are many good cars that are foreign made. so it's not the case that all foreign cars are badly made. What's the negation of this sentence? Let me give you four possibilities. This is not the negation. all foreign cars are not battery made. let's look at them. Possibility d. And if that sentence is false. this isn't true. as to which one of these you think is the negation of that original sentence. But if you think about the. Let me give you one that's not quite so obvious. or possibility a. Why? Is the original sentence true? No. is a common answer that I. Well. Domestic is. and this one is. So what do you think it is? Well. the negation of foreign. for a minute. Okay. in the text book I've written for this course and a previous textbook I've written. that I often get. Possibility one. what the sentence really means. I've been teaching this material for many years now. Well I'm not giving you this as a quiz. then its negation is going to be true. in fact. after all. it's one of the examples I always give. That means pi is greater than or equal to 3. No problems there. Possibility b. of course it's not. but I would like you to think. all foreign cars are well made. But let me stress a point I made a minute ago and. I know why they're saying that because they're saying. Look at the following sentence. . And because this is false. this says something about all foreign cars. All domestic cars are well made. because it's simply not the case that all foreign cars are not badly made.cars are well made. I've actually had students over the years who have thought that. and this says something about all cars that are not foreign. those are false statements. is this one true? Yeah. That's false. These are false. the negation of this? we'll come back to this. So. the. 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. here. That can't be the negation. these are both possibilities for the negation of that. Well this isn't true. in fact. And this is still not a quiz but. therefore. that was. the negation of this. that's true. and didn't write anything down. or maybe a different thing is the actual negation of this. Which one of these do you think actually is the. Is this one true? Well. How do I know it's not the negation of the original sentence? Because the original sentence is false. Okay. So that can't be the negation. I'm going to introduce some. So whatever the negation of this original sentence is. I'm going to leave you for a little while to think about this one. And eventually we'll be able to reason precisely to see which one of these two. it can't possibly be the negation of the original sentence. What about b? Same reasoning. The negation of a false sentence is going to have to be true. these are both true. Well. So there is a sort of negation going on between these two. some formal notation from sort of algebraic notation. but it's not the negation of the original sentence. whatever the negation is is going to have to be true. This is also false. so they can't be the negation of a false sentence. What do you think the values are? Yup. we'll be able to look at that foreign car statement again. phi. And certainly won't look much like mathematics. To figure out what the correct negation is we relied on our knowledge of the everyday world. And then we're going to negate it. This one's a much simpler table. So this one is a really bad choice. And with that you should be in a position to complete assignment two. When we've taken our study of language far enough. this one really falls along with on being the negation of that. I think. it has nothing to do with domestic cars. It's talking about domestic cars. illustrate why we're devoting time to making simple bits of language precise. So the negation can only possibly talk about foreign cars. We have to rely purely on the language we use to describe that world. should. the negation is false. because they talked about foreign cars. That last example about the negation of the sentence all foreign cars are badly made. and use rigorous mathematical reasoning to determine exactly what its negation is. but in a lot of mathematics we're dealing with an unfamiliar world. The original sentence is about foreign cars. That's fine for statements about the everyday world we're familiar with. Let me finish with a very simple quiz. this'll seem like a very strange course. So. . this one was an easy one. For the following reason. If phi is false. It's purely talking about foreign cars. How are you getting on? For most of you. because there's only one statement involved.And in fact. If phi is true. These were good candidates for the negation. and we can't fall back on what we already know. Let me ask you to fill in the truth table for negation. that brings us to the end of the first week. That's what it's talking about. This one isn't even in the ballpark for being a negation because it's not talking about foreign cars. the negation is true. Well. very simple truth table. Negating a word in a sentence is not at all the same as negating the sentence. but you won't learn to ride from watching them or having them explain it to you. If you're at all like me. Yeah sure. Alright this is a very different way of learning than you're used to. You should definitely attempt all the assignments that I give out after each lecture.That's because you've only been exposed to school math. Doing that is mostly up to you. If you're able to scan pages of work into PDF. It has to be. I'm trying to help you learn to think a different way. or else use your smart phone to take good. which in some ways is very different. I advise you to start showing your work to other students to get their feedback. The problem sets comprise assignment questions that count directly towards your grade. Or even how to get the answer. As well as the assignments. the lectures are short. and pretty well every other mathematician I know. Doing those assignments. there's also weekly problem sets. but you'll find that it almost never tells you the answer. It's like learning to ride a bike. you can watch the lecture several times. both on your own and in collaboration with others. Send images as email attachments. is really the heart of this course. You have to keep trying it for yourself and failing. There isn't much material. You definitely need to connect to other students and start working together. clear photos of your work. at least in mathematics. or upload them to whatever networking site you choose. This course is about the transition to university level mathematics. put them on Google Docs. so we have to rely . Someone can ride up and down in front of your for hours telling you how they do it. Because this course is designed for many thousands of students It's impossible for me or my TA's to look at everyone's work and provide feedback. you're going to find it hard and frustrating. until it eventually clicks. Each set has a submission deadline. in the way that you're familiar with from high school. I'm not providing you with new methods or procedures. And it's going to take some time. and as a result. It's called calibrated peer review.on automated grading. whether you get questions right or wrong. in whatever study group you form. It's the method we are going to use to grade the final exam. but these are not at all like the in lecture quizzes. The problem set questions will require considerable time. Asking you to answer multiple choice questions Is like checking your health by taking your temperature. This means that the questions are posed in multiple choice format. it's pretty insignificant. but we can't check that automatically. It tells us something. What I'd like you to do is to try to grade your own work. but it's pretty limited. This is not ideal. For the materiel in this course. It's been tried a number of times. Check out the description on the course website and then give it a shot. and can alert you and others that something is wrong. . And you should definitely try to get into one. It's your thinking process that's important. There's a mechanism for doing this. The advantage of using it now is that it does yield positive learning outcomes. and that of others. Still checking temperatures is better than nothing. So anyone who wants to earn a distinction in this course Is going to have to learn and use it eventually. and the same is true for the problems set grading. and does appear to offer learning benefits. Those are supposed to be answerable on the spot.