 sick. So you have to just have some patience here. So everybody here today, were you all here on Tuesday? You all got the syllabus? Anybody? Okay. There's nobody that ran? Okay. So I will not say anything about that. You heard my spiel on Tuesday. I was a little incoherent. So I had a high fever. I didn't even know. But hopefully I didn't, I made some sense. Okay. So just to remind you real quick, the first assignment, I just want to reiterate this, just to make sure everyone's totally clear on this. On the syllabus it says that it's due on Tuesday. The first assignment will be due on Thursday. So one week from today. You're just going to have 1.1. I'm going to give you the assignment today at the end of class. And then on Tuesday we'll just, probably we're going to start some new material, but I'll definitely save some time on Tuesday to go over some problems and just get you situated and comfortable hopefully with how to do these induction proofs and such. Just out of curiosity before we started, how many of you have seen induction before? I've done induction proofs before. Okay. Somewhat at least. Most of you have, okay. Probably from discrete math, you took discrete math and you probably covered this in discrete. I would hope at some point. So we're going to start from the, sort of the very beginning. I'm even going to start sort of in some sense below induction. So a lot of this is not actually in section 1.1 in the book. It's actually sort of preliminary stuff. So this will be probably a review for most of you, but I figured it's, can't hurt just to cover all the basics before we get into some more difficult stuff. So if I can figure this out. Let's see here. These lectures will be available online. Yes. So they should be accessible through Blackboard. Okay. So there's, yeah. I'll tell you what. So I don't want to spend too much time on this here in class to get through this stuff. I'll send an email out about this. Okay. So I'll just email you all this. And Joe, I got your email. It's, the course is accessible now through Blackboard. Okay. So, okay. So let's see. All right. Let's see if this works here. Okay. Okay. I guess I didn't want to do that. Okay. Sorry. Thank you. I need, I need all the help I can get here. So I think this is, yes, it's on now. Thanks for that. Okay. And let me see if this will work. Okay. Oh, okay. Great. This is, okay. This is, I've never done this before. So this is still going to be, yeah, this is, this is weird. Okay. All right. Let's see here. With this. Okay. Well, unfortunately, I have to do this. So, okay. Let me see if I can, oh man, this is really crazy. Okay. No, I'll get, I'll get the hang of it here in a second. I cannot see it on the tablet. I can only see it on the screen. Okay. All right. This, this, I, I'm praying that this gets better at some point as we go through this. Oh man, this is weird. Okay. Let's see here. Okay. So, yeah. Okay. All right. All right. All right. Yeah. Okay. This is harder than, than it looks. Believe me. Okay. Let me draw on it. Yeah. Okay. Okay. Okay. Wow. This is, this is weird. Okay. This is, stands for definition. Okay. D-E-F. Okay. Okay. I'm just going to go slowly here and then maybe that'll be, make this easier. Set. Okay. You've all seen this word before. Informally is, this is getting better, a collection. Okay. This of course is very fundamental stuff. We're going to be talking about sets. In this case, it's usually going to be sets of numbers, but it doesn't always have to be. These, these objects can be anything really. Okay. So, if is a set B, let's say, is the objects, right. Probably seen this notation before. I'm going to align that. Sorry. That's supposed to, yeah. Okay. That's supposed to be a D. Good. Good. Okay. This will get better. I promise. Okay. Okay. That should be denote. Okay. This, if you've never done this before, trust me, it's not as easy as it is. Because you can't, when you're writing on this, you can't see it. You can only see it on the, on the screen. So it just, it makes it, it makes it very difficult. Okay. So, now I, you guys come up here and try it. I, I promise you, it's not easy. Okay. So have you guys seen this before? This, okay. This Greek letter, epsilon. Okay. Okay. So, all right. So then B is said to be a member of an element of X. Okay. And the thing is, my handwriting, if I was on the board, is so neat. I have the best handwriting you would ever, ever hope to see. It's not a joke. Okay. If I had sloppy handwriting, this would be disastrous. Yeah. Okay. I wish. I wish. Well, that's true. Although I'm going to, yeah. It would be, yeah. The problem is that there's going to be so many, so many, as we go along, there's going to be so many mathematical symbols that it's going to, it's going to be tricky. Yeah, I could do that. I could do that. But I, well, you guys can read this though, right? I mean, it's readable at least. Yeah. Okay. Okay. So let me, let me try to squeeze this in here. Okay. If, yeah. I'm going to, actually, I may end up doing that at some point, but I'd like to get comfortable with writing on this first. Okay. So B is not an element of X. Then we see if I have enough room down here. Then we write this. Okay. Hopefully I don't screw this up too badly. Yes. Okay. That's one with a little strike through it. Okay. Well, this doesn't look very good, but hopefully this will get better here as we go along. Okay. So you guys can at least, I'm sure it looks much better in your notes. So there we go. Did everybody have that down? I should have asked that first. Okay. I can go back if I need to. So I'm having a little trouble following this because it's coming in pieces. Okay. What you have up there is the definition of set membership. Yes. Set membership. B is a member of a set X. Then you just write B epsilon X. If it's not, then it's B epsilon with a slash through it X. Yes. Yes. Okay. So let's just do a quick example here. And by the way, you'll see what the abbreviations are here in a second. But EX, I'm going to use this a lot. This will just abbreviate example. And I'll try to have them numbered as, you know, sequentially as we go through. Okay. And again, I realized this stuff is all stuff that you've seen before, but I just want to get a really nice slow introduction to everything that we're going to do. Okay. Okay. So in this case, X is just the set, the numbers one, two, and three, of course. And really the only reason why I'm doing this example is just to remind you of what the standard notation is for representing a set, especially just a set of finitely many members. For example, in this case, of course, we just have three members. Okay. And so just with the notation that I gave you before, the definition of membership. One is an element of X. Two is an element of X. And three, okay, no, got it weird again. Okay. It's an element of X. But for example, for example, zero is not an element of X. That's easy enough. So that should be pretty clear. I know I'm probably boring you with this. It's so, so simple, but I just want to make sure everyone's aware of this notation. There. Everyone clear on this? Super easy, right? Okay. Okay. So if a set X is finite, so we'll often represent X the same way as we did in the previous example. Okay. So we'll just list the elements as in example one with the elements separated by commas. Also, and this, you'll see this a lot in the text, too. And a similar convention will apply to sets whose, sorry, whose members a clear path. Okay. So some of the examples I'm going to give you are going to be sets that we're going to be studying in this course. This looks like how I wrote when I was five. Is that right? Okay. But they write like this on the white board, though. Yeah. No. If I was actually writing this on the white board, you would think, oh, my God, this is, I can't even believe that he writes this well compared to this. What's that? Well, no, I mean, it's much, much better than this. Yeah, no, that's, that's, I'm hoping that happens in the next couple of days, actually. Exactly. Okay. I assume I gave you enough time to copy all of that down when we were laughing about how bad my writing is. Okay. Okay. So, let's see here. All right. Second example. So, and this, this you've probably seen before. So we're going to call this, you know, Z with a sort of a line in the middle here. Okay. So we've got the, the left parentheses. So these, we've got some dots just indicating that the pattern continues. Again, you've probably seen this before. So in this case, again, just to remind you, note. Okay. So again, I assume you've seen this before, but these, these sort of these, this ellipsis on both sides just means that the pattern continues. Right. So for example, the number 10, of course, is an element of the set Z. Right. You know, you guys believe this, hopefully. That should be hopefully pretty self-explanatory here. Yes. So in the script for like integers, the Z would like you go like that and then the line would play if we wrote it that way. Oh, I see. So you had the, you had it sort of diagonally instead of horizontally. Yeah. Oh, yeah. I mean, as long as it's clear what, what you're talking about, I mean, the, the, there are, there are some conventions that are not uniform. So, yeah, that's, that's fine. Okay. That's fine. Yeah. Yeah. I mean, the, the standard, at least from, you know, textbooks generally have it as a sort of blocky Z. But whatever is, I mean, as long as it, as long as there's a Z hiding in there somewhere, it's, it's, I'm not, you know, I'll be able to understand what you mean. Yeah. Okay. So, so the set Z is called the set of integers. Right. I assume you were all familiar with this set before I told you this, I'm assuming. Okay. So another example. This is, of course, the set of integers is, is going to be a set that we're going to be studying quite a bit in this number three course. And in this case, which you've also seen in discreet probably. Okay. So the set capital in is the set of, okay. What, what, what, see if you remember this. What is this called? Natural. Yes. Natural numbers. Well, yeah, that's something else I'm going to mention here, too. Here's the, the issue that the set of natural numbers, it depends on the textbook that you have. There are many textbooks that include zero as a member of the natural numbers. I'm, unless there's something that is just plain wrong, I generally try to follow the textbook that I have for the course. In this text, the author defines the natural numbers to be the set of positive integers. So not including zero. I would say in this case, in modern mathematics now, I think the majority of authors actually include zero as a natural number, but yes, Z also has the negatives. That's right. But like I said, there's no huge harm being done here. It's just, you know, we're just going to set this being consistent with the, with the author as a convention in this course to not include zero as a natural number. So just something to be aware of. So I'll just say this below here just to be complete. So the natural, the set N, the natural numbers is the set of positive integers. Actually, I think my writing's improving slightly already. I think, yeah, it is a little bit. So hopefully, yeah, hopefully in another week it'll look really good. Okay. Are we okay with this then? You guys got this down there? Yeah. Okay. So third definition, which you should have probably seen in discrete math, or before that, bless you, let X and Y be sets, then X is a subset subset. Okay, we underlined that. Of Y provided. So what's this, what is this definition? So we want to finish this off for me. Every, every element of X is an element of Y, right? Sorry. Okay. You can read that element. Okay. That's good enough. And so you've probably seen this notation before. That's a D by the way. Okay. Denoted like this. Okay. Have you seen this notation before too? Subset. Okay. Okay. So like I said, I'm, I am starting off here with, with just basic, basic principles. Okay. So we've talked about notation for sets, especially finite sets or sets that whose members follow up certain pattern. So for example, set on the left is certainly a subset of the set on the right, because you can just check that all the members on the left are also members of the set on the right, right? Ordering of course makes no difference. When we're talking about sets, the order doesn't matter. The set, if I want to write the set 0 comma 1 comma 2, if I want to write it 2 comma 1 comma 0, it means the same thing. Okay. So there's no sort of sense of order built into a set. There are ways in which you can, you can sort of build order out of sets, but we probably won't get into that in this course. Okay. So another simple example. Okay. So we talked about these sets before, right? Okay. That should be clear. The natural numbers, it's a set of positive integers. So of course every natural number is an integer. So this is, this is pretty clear. Okay. So let's do a couple more examples. This is, you're going to see this notation so much that I just want to make sure that everyone's on board with this, especially, and I know a few of you in here haven't, haven't seen discrete math in a long time. So I just want to make sure everyone's, you know, up to speed on this. Okay. So this is something, I was going to write this out as a definition, but because we got started a little bit late, I'm, I'm going to skip this, but you, I think I've all seen this before. Okay. So what I'm writing now is the set consisting of the natural numbers n little n in capital N with the property that little n is prime. Okay. So this is another way of expressing more compactly a set that if we were to write everything out, of course we wouldn't ever be able to finish because there's an infinite number of elements in the set. So often what we'll do to express a subset of a given set is we will say, we'll look at all the elements of a certain set that have a given property. And so in this case, if we, if we look at the set of all natural numbers n that are prime, okay. And so let me just explain this again. When you see this notation, if any of you aren't familiar with this, this on the left side, what this is saying is n and n, the, the, the colon you should interpret as such that. Okay. So if you haven't seen that before, if you want to write that down, it's the natural numbers n and n such that n is prime. Or with the property that. Okay. That's, that's what that means. And so of course you all know what, what prime numbers are. Now we're going to, this is going to be a, certainly a focal point of study in this course. So for example, in this case, we would have two, three, five, seven. Right. And on down. And the dots mean of course that the pattern is that we're just can, we're just building up the prime numbers this way. Right. So it would be the next number in this set. Seven. Yeah. So 11 would be the next number. Right. 13. Right. We just keep going that way. Okay. So this is something that let me, let me give you a slightly more, a longer example here, but okay. So what about this x in z now? So remember what this means z is the integer. So again, I want to, maybe I should have gone a little more slowly through the last example, but this, so far what you wouldn't interpret this as is this, the set of integers such that, and I'm going to list some property. Okay. And so it's, here's the property x is less than zero and x equals y squared or some, wanted to get that on one line, for some y in z. And so I'm going to be a little bit ambiguous here and say equals and then the question mark. Okay. So I, what I really, the point of this example really is just to get you to think about what this is saying. I just want you all, hopefully all to, to just understand what, what this is. Okay. So this is the set of all integers x with the property that x is negative. Right. And it's the square of some other integer. And so can you think of any integers that have this property? Well, okay. Well, let's see. It's going to be, yeah, any, any real number squared is going to have to be bigger than or equal to zero. So what can you say about this set? Well, yeah, so I mean, it's, it's the empty set. It's the empty set. Right. So, right, null set or the empty set both mean the same thing. Right. So this, and then I'm going to give you, of course, the, the definition here in the second two, but, which I'll also define here in a second. Yes. Well, well, why isn't it this, the set of numbers that have integer square roots? Well, because the, well, first of all, this, the definition of the set is that if there was anything in this. Yes. That's what I was going to say. Yeah. It's net x is less than zero. Okay. Are we okay with this? Yeah. Okay. Good. Okay. And so that brings us to the next definition, definition for the set with no members called the empty set. And the typical way to denote this is with, is by zero with a little slash through it. Okay. Is this something you guys have seen? Seen this before? Yeah. Oh, no. I mean, so if you, if you, if you put brackets around it, then it's not empty because the empty set is a member of that set. Okay. So, yeah, you don't want to put brackets around it. If someone told you that before, that's, no, no, no. Okay. No. That is not the standard convention. Definitely not. Because, and that's a good question though, but just with the convention of enclosing a set with finally many elements and brackets, if you put the empty set in brackets, then the empty set is a member of the set containing the empty set. So with the bracket, well, the set, if you, if you put brackets around it, then that's a set with one element, the empty set itself. Okay. Yes. Okay. But no, I'm glad that you asked that. Yeah. The empty set is just, this is it. That's it. That's how you denote it. Okay. So now, how do you know, and this, this is really is a definition. How do you know when two sets are equal? Well, okay. Hang on one second, Joe. I thought you were asking. Oh, well, I was sort of rhetorical, but you, no, but you can answer it. It's okay. Right. Yes. That's, that is exactly right. So two sets X and Y are equal. Oh, well, that did not mean to do that. Okay. Yeah. Let's, let's see. I don't, I don't, okay. Okay. Oh, okay. Yeah. Yeah. I see it. I see it. Okay. Let me, let me try this. Okay. Okay. Which is the, which one is the eraser now? This one? Okay. So then, do you really erase the, no, I do not want to do that. Yeah. Let's see if that, let's see if it works. Let's see. Okay. I don't know if this is going to, no, it doesn't, it doesn't, no. Okay. Okay. Tap all this right here. No, that doesn't work either. Yeah. Okay. I could do that too. Yeah. Where's the, where's the color? Oh, I see it. Yeah. Okay. There's got to be a more efficient way to do it than this, but okay. Well, that's one way to do it. Okay. Yeah. Yeah. No, I got that. I got that. Yeah. Sorry. I still have a fever of like 102 or something right now. So I'm still not totally with it yet today. So that's another reason why it's, maybe I'm not, I feel a little delirious still. Okay. Okay. So there we go. Two sets x and y are equal. I'm glad at least this is, this is readable. So let's see. You guys can still read this, right? You can, I mean, sure. Okay. Good. If they have the same members. So what's another way of saying this? Well, okay. It means every member of x is a member, if they have the same members, then of course every member of x is a member of y. And conversely, every member of y is a member of x. So another way of saying this is that x is a subset of y and y is a subset of x. Okay. So now let's do another example here. Let's see where we add. I think we're on eight now. If I'm not mistaken. Example eight. Okay. So this will make you think just a little bit. Let's let the set x be the set of all, so I'm going to remind you what this means. The set of all natural numbers such that, right? That's what this means. Little n epsilon capital N colon. Set of all natural numbers such that there exists real number or such that n equals, okay, n equals x squared. Okay. So here I'm going to ask you two questions. Sub question one, if you will. This shouldn't require too much thought. Why is this true? What's the question? Is x a subset of n? Is x a subset of the natural numbers? The answer doesn't come to you right away. It's because it's so easy. It's flying over your head. It's a subset of the natural numbers just by definition because it's a set of all natural numbers with a certain property. So anything in there by definition is a natural number. Okay. That's just how it's defined. So the answer to this question is definitely yes. And all I really need to say here is this is true just by definition. Okay. So we talked about the natural numbers. I haven't explicitly talked about the real numbers, but with your experience with discrete math and calculus and such, I'm sure you're roughly familiar with what these are. Okay. Second question is the set of natural numbers, a subset of this set x. Okay. So now the question becomes, remember what this notation means, subset means is every natural number an element of x? Okay. The definition of x? The x is a set of all natural numbers with the property that it's a set of all natural numbers n with the property that there exists a real number x such that n equals x squared. Anyone want to, what do you think? Yes? No? Not sure? No? Some people say yes. Some people say no? Okay. Yes. If n is a subset of x. Well, let's see. Let's see. Okay. Let's let n be just any natural number. Okay. Hopefully, I'm going to convince you of this. What the right answer is. Do you guys buy that? Okay. Let n be any natural number. Then n is certainly equal to the square root of n squared. Okay. Now the square root of n is in general not going to be a natural number, right? If n's a perfect square, then it's going to be a natural number. But what can we say about, but we don't need that. That's not what we say. And the definition of x doesn't require that. Right? So to be an x, all we need is for n to be some real number squared. Right? That's, if you look at the definition of x, some real number squared. And since every natural number n is the square root of n squared and since the square root of n is real, right? Because n's not negative. Remember the definition of the natural number is 1, 2, 3, 4, 5. Square root of n is real, so every natural number, since it's a square root of n squared, is equal to some real number squared. And the square root of n is real again because n's not negative. Make sense? Okay. Thus, I'm not going to write all of that out, but thus n is an element of x. The only part I didn't write down is the fact that the square root of n is real, and that again comes from the fact that n's not negative. That's the capital n. That's capital n. This is little n, this is little n. That's little n. But the one over there is capital. So what can we say? What's the relationship then between capital n and capital x? What can we say about these two sets? They are equal. Okay? x and capital n are the same sets because they're both contained in the other. So I'll write both of those things down. So we just showed that n was contained in x, hence n capital n equals capital x. Okay. Are you okay with that? Okay. And time's flying here. Okay. All right. What's that? I know. Yeah, I know. Okay. It'll get faster hopefully. Everyone has this down? Yeah. Okay. So now we are going to start talking about the last 20 minutes or so, because what your homework is going to be based on, and I realize I'm not going to get very far into this, but we'll do plenty of this on Tuesday. Okay. Induction. So now this is really where the text starts with this concept of mathematical induction. Okay. So before we get into this, let's do an example. Example nine. Let's let, and this is, again, this is going to be very easy, but seven, eight. Okay. So we're going to do the set containing seven, eight, nine, four, 10, 11, 12, 13. And we'll say we're just going to continue this pattern on down. My, these dots, I mean, I'm not going to skip back. I'm not going to go back down. It's going to be 11, 12, 13, 14, 15, 16. It's going to just continue to go forever that way. Well, sort of, and this looks like kind of a weird set, but I'm doing this on purpose. Okay. Do you, do you guys believe that this is true? You buy that? Okay. And the reason why I threw the four over here is just again to stress the fact that order does not matter here. Okay. Okay. So what number is the smallest number in this set? Yes. I realize this is very simple, but I'm just setting up what we're going to do here in a second. No, no, no, no, no. No. Yeah. I mean, that's, yes, and yes, you're correct about that. Well, that, that could happen, but, but yeah, I mean, it's, it's not going to be something. I'm not going to trick you and say, no, no, you know, 56, 57, and then three, you know, no, I wouldn't do that. No. Okay. So this is, this is so obvious. I think I'm not going to say anything about this. Okay. Now here is a very fundamental property of the natural numbers, which is called, and again, those of you who've taken discreet note know what this is. This is called the well ordering principle. Okay. And it says this every non empty. Okay. I'm going to underline that. Of course, it's clearly false if you have an empty set, but every non empty subset of the natural numbers capital N has a least element. Okay. So because of time, I really want to get through this. I may not write this down. So writing is really slow. So I'm trying to make it legible, but this, if you think about it, is something that should be intuitively obvious if you, if you just think about it for a second. Okay. And instead of writing the sound, I'm just going to say it. All right. So take your set S to be some non empty subset of the natural numbers. What I mean by non empty, I mean there's just a, there's at least one member in S. Okay. One element, at least one element of S. Okay. Well, if, if one is an element of S, then one has to be the least element of S, because there are no smaller natural numbers than one. So one's certainly the least element of S. If one's not, I'll assume one isn't an S. Well, suppose two is an S. Well, then two's the smallest one. Right? If two's, and one and two are both not an S. Well, three is an S and three has to be the smallest one, because one and two aren't there. And then you just keep doing this, right? And so eventually you're going to have to come to an element of S. The first one you come to, there's the least element. Okay. And again, you have to come to something at some point, because S is non empty. Right, right, right. I mean, yeah, it doesn't, so there's no, it doesn't sort of make sense to say that, you know, you have this set, for example, if I were to write, I'm not going to write this down, but left, you know, bracket two comma two comma two right bracket, that is just the set containing two. Okay. So, yeah, I mean, this kind of stuff doesn't, doesn't happen. Yeah. Okay. So, hopefully that you guys buy, buy that. I don't think that's too hard to wrap your mind around. I think that's pretty clear. Right. Okay. So, here's what the last thing I'm going to get to today is just called the, I think I'll go ahead and do this on a, on a new page here. And this is a, this is actually a theorem that we can, we can prove from this well ordering principle, theorem one. And so the book calls this the first principle of finite induction. That's kind of a non standard terminology, but that's okay. Okay. Oops, sorry. Yeah, that's all. There's something in, I've been working on, there's something in mathematics called a principal ideal domain. And I've been working on these all day long. That's why, that's why, that's why I spelled it this way. I know how to spell it just, it's just the other mathematical object is a L instead of LE. So I was just on autopilot there. Sorry about that. We're not going to be dealing with these things in here. No, no. Okay. Now, principal ideal domains are great, but no, no, but now these things come up all the time in abstract algebra. If you go into abstract algebra, you'll learn about these things. Okay. And here's what it says. So suppose that s, and this is why I kind of did the background with sets and subsets, because now you know what all these things mean. s is a subset of the natural numbers. And suppose that s has the following, so first property is very easy. First property is just that one is an element of s. Second property is that, and I'm going to write it out in a really, well, lengthy way, but I'm going to underline the relevant words here for all natural numbers. If, if n is an element of s, then n plus one is an element of s. Okay. So that's the hypothesis. Okay. So again, I'm guessing the majority of you have seen this before. What's the conclusion? If s is a subset of the natural numbers with these two properties, what can we conclude about s? Then anybody remember this? Yes. Then s is equal to a set of natural numbers. That's right. And so what I'm going to do, and unfortunately, I may only have time to do this today. At least I'm talking about the principle of finite induction. I'm going to try to give you a sort of a informal heuristic argument to again convince you or to get a feel for why this is true. I probably won't get to this now, but next week I certainly will. I'll give you the formal proof of this from the well-ordering principle, and then we'll talk about how to write induction proofs and we'll start talking about how to prove things about the natural numbers. That's going to be part of your homework assignment that'll be due next Thursday. We're not going to get into a lot of that today. Like I said, we're going to run out of time. But what I'm going to do is, oh gee, let's see. I might have to write kind of small here, but I want to leave this up here. So the first thing I want to tell you about is what I'm just going to call an informal argument to just try to convince you of this. And I think I will. If you pay attention to this, I think I'll convince all of you that this is true. And this won't take long. And then I'll give you the homework, and then we'll stop for today. So I'm just going to ask you a few questions, and hopefully I can squeeze all these in here. Question one. Okay, now we're assuming, so I don't have room to write all of this down, but we're assuming that s is a subset of n with the following two properties. And what I'm trying to do is I'm going to try to convince you that s has to be all of n. So we're assuming s has property one and property two. So the first question is, well, what's the smallest natural number? One, right? We talked about this, right? Natural numbers are positive integers. One is the smallest. First question is, is one an element of s? Okay, why? Property one, right? We're assuming these two properties about s. So by property one, one is an s. We're just assuming that s has that property, right? So yes, by question two is two an element of s. Okay, now here is what I want to be very clear about this property two. It may be the intuition may be marred a little bit just by the length of the statement. What is property two really saying? Here's all it's saying. Here's what property two is saying. It says that if you have a natural number that you know is an s, then the next natural number is an s. That's what it says, okay? So if you know something's in there, then its successor is in there too, okay? One is in there. We just established that by property one. So therefore by property two, remember what property two says. It says if we have something in there, then the next guy is in there. So since one is in there, we already know one is in there, therefore two is in there by property two. Yeah, well yeah, exactly, and that's right, and that's exactly what I'm trying to get at now. Right, so yeah, I'm just trying to do this slowly so everybody catches on to this principle. No, no, no, I mean that's exactly what- No, no, no, no, it's fine. I'm hoping that this is something that's becoming more transparent now. So now from the second part, question two, we know that two is an s, and therefore by condition two that we've imposed on s, whenever we know something's in there, the next one's in there, since we know two is in there, three is in there. Now that we know that three is in there, by property two, four is in there. Now that we know four is in there, by property two, five is in there. You see that? So everything is going to be in there, okay? I'm running out of room here, so when I say by two, I mean, really I mean by two, and the affirmative answers to question one and two, therefore three is an s, and then you just continue to carry this down, okay? And so that's what I'm running out of room here, but let me just write that down. Dot, dot, dot, or you can write it vertically if you want to. Continuing this argument, it should be, hopefully I've convinced you that yes, s has to be all of the natural numbers. Does that make sense? Okay. Anybody not okay with this? Okay. Now of course the trick here is using this property now to prove theorems about the natural numbers, which of course, we don't have time to get into today. I'm going to give you the homework that'll be due next Thursday. Again, I'm aware that we haven't gone into the problems that you're going to be encountering in the homework yet. I'm assuming how many of you have actually done proofs by induction? I know I asked about induction before, but you've done proofs by induction. Okay. So most of you are already familiar with how these sorts of things go, and that's really the content of the homework that I'm going to give you. Tuesday, just so you know, Tuesday, what I'm going to do is I'm going to demonstrate how you use induction to prove certain statements about the natural numbers. So if you're still kind of foggy on some of this, Tuesday's lecture should really get you up to speed on the homework, which you'll still have two more days to work on. And I'm not giving you a ton of problems here for the first assignment. So let me give you the homework, and I guess that will be it for today. Okay, so from the text, homework is section 1.1. So I'm going to give you problem 1a. So problem 1 has a lot of parts to it. 1b, 2, 9, 10, and 14. Okay, so this is due, again, I guess that should be what, the 31st, I believe. Okay, so this will be very clear here. This will be due Thursday at the beginning of class. Okay, if I don't want you working on the homework during class, when I'm talking about new stuff, so make sure they have it done by the time you get in to class.