 the files will go over that in class but there's a, I don't know what the official rules are called, but the rules are actually in place for a good reason because it might be the case folks that your face may appear on some video somewhere. I mean if you're sort of face in front it's not likely that will happen but you know if you get up or you're walking late or something you might appear on an archived video for this class and this just says if you're uncomfortable doing that then I don't know wear a mask or something or sit in the back or sit behind the camera or something. I mean we're trying to make it so that especially in a class like this where I know some of you have full-time jobs or maybe you're you know already teaching in schools or something like that and you get off work and your car breaks down or your traffic is too bad or you just got something that you know work takes you out of town. Something of that nature at least this way you have the chance of you know heading on to the internet and watching me yammer around. Whoever is going to be recording this class Nick I know it's going to be you tonight Mr. Student is going to have to be sort of like because I do this a lot. Anyway it'll be fun. Okay let's officially start. My name is Gene Abrams feel free to address me anyway you'd like whatever you're most comfortable with either Gene or Mr. Abrams or Dr. Abrams or Professor Abrams or I mean I don't really care whatever it whatever sort of you know is most comfortable for you. There is a website that'll support this course at that website I'll try to sort of rig it that will link you directly to these video things but there'll also be a direct access to the video stuff as well that I'll talk about on Wednesday. I'm officially in my office before class for essentially a half an hour and then it takes me about 10 minutes to wander over here so we'll if you're in my office we'll sort of break at 420 to walk to class. I'm also in Tuesdays and Thursday afternoons for a couple of sessions I actually teach a class in between those two sessions that's why there's a chunk in there and every day by appointment so if you need to see me please feel free to stop by or probably the best thing to do is send me an email and try to make an appointment and you know then I can guarantee that I'll be in my office at a set time also for what it's worth and this seemed to be popular two years ago when I taught a class in here I'd go right from here to the bus stop because there's a city bus that I take home leaves at 6.05 so there's like 20 minutes or so in between so if you want to sit with me at the bus stop or something just you know grab me after class for 10 minutes right outside here I'm more than happy to answer any questions you might have or point in the right direction on the problem you might be working on so please feel free to take advantage of that this supplemental instruction sessions with Jen Holmes I'll talk about those more especially in the context of homework and exams so let's hold off on that momentarily the text is let's see is this one yeah text by Fraley it's well it's a classic by now actually I think the first edition was 30 years ago or something and he has updated it appropriately I include some stuff about cryptography and various interesting questions about algebra that have come up over the recent years some of which have been solved we'll talk about some of those over the course of the semester there I know is like a international version or something like that or something out of London that's a little bit cheaper someone showed me that actually last year when I taught this course and it seemed to be identical but I there's no guarantees if you try to buy something it's a little bit non-standard plan on three exams in here and here's how they will work there'll be two standard 75 minute exams exams one and two I list the approximate dates that those will be on approximate means you know barring a snowstorm or have to cancel class one day or something like that it should be on those dates you'll know two weeks in advance when the exact date's happening it's not any sort of mystery but so I can't say on day one exactly when they'll be but I'll give you plenty of advance notice so that you can prepare for those exam three will be essentially just another 75 minute exam it'll cover the last third of the course but it would be given during the final exam period in here which happens to be when Wednesday December 12th it's also the case that during the final exam period Wednesday December 12th if you so choose you can also take a second to 75 minute exam during that two and a half hour period that second exam will cover the material that was covered on exams one and two together and the way it works is in the end each exam is worth 30% but I've talked about four exams so that doesn't make any sense you have to count exam three the one that you take during the final exam period and then of the remaining three exams you count your best two so if you do really well on exams one and two there's really no reason for you to take this sort of optional final during the final exam period just walk in take exam three and leave but if you don't do so well on either exam one or exam two you have sort of a chance to make it up at the end I'll talk about this more in a little bit that's you know the nature of this course is sometimes it takes a little while for the sort of general idea to sink in and you know some students stub their toes on exam one and then just start you know the light sort of goes on in week six or week ten or something like that so this sort of gives you a chance to maybe make that up at the end okay yeah and let's see it says here yeah I'll explain this in more detail as the final exam date draws nearer don't miss an exam I mean in effect if you miss an exam folks then you're obligated to take this final exam that's how you make up so I'll just count it as a zero but you get to replace that zero by whatever happens on this final okay the most important thing that you can do in here to prepare for the exams is to do the homework in such a way that you really understand the material so here's how homework and a few short quizzes are going to work in here the way homework or the rhythm of homework will be this I'll give you an assignment essentially every Monday the assignment will include problems that I'll cover that Monday and probably the following Wednesday so that week's worth you should start on those right away admittedly not all of them might be familiar because some of them cover them and then you in effect have a week to do the assignment it'll be do the following Wednesday okay so stuff that I signed today it'll be due a week from Wednesday uh yeah so I've tried this two different ways and this is the first semester I'm going to try a little hybrid so up until a couple years ago it was always do the homework by yourself in the sense that I want you to actually write up your own homework assignment turn it in I always encourage you to work with other people and I'm still gonna you know encourage you to do that and I'll talk more about those details in a minute but if you'd like to work together in a group of either a group of two or a group of three and you'd like to as a group submit one assignment from your group uh in such a way that you know when I graded everybody in your group or received that same homework score that's now going to be fine and you can sort of mix and match groups you just make sure that at the top of the paper you'd write down you know this assignment is from this person and this person this person maximum of three so if you feel like you're better off understanding material by working together in your group more power to you that's good if you feel like you're better off working alone good for you too so you know if you know some friends in here or probably most appropriately if you go to some of these supplemental instruction sessions and you wind up working with someone in there you might say hey let's do this assignment together and if it's a mutually agreeable situation then great and if not not to worry about it if you decide to work together for one assignment and you turn that one in you decide that just didn't work out very well then fine the next time you can just turn them in yourself so there's no I mean you can sort of mix and match as you see fit over the course of the semester whatever you'd like to do the homework will be graded on a 10-point scale that means that you're going to turn in a certain body of work and there won't necessarily be 10 problems but I will grade it out of 10 points 10 being the maximum when you do the homework and I pass around some information about what I expect to see when you turn in a homework assignment what I'm looking for is complete answers I'm looking for a strong indication that you really understood what the issue or the essence of the question was what the tool or technique to get to the solution is that you've written it up clearly not clearly in the sense that well the instructor will understand what I'm talking about so I'll just sort of scribble something down and let him figure it out clear in the sense that your homework solution should be written in such a way that if your solution was to be included in a student solution manual down the road that it would fit that bill well that some other student who presumably doesn't know what's going on would be able to read through your solution to a given problem say oh now I understand what's going on so I write up and give you lots of detail about you know how to turn it in what it should look like what sorts of things you should be wary of or should be on guard for my philosophy on the homework is anything goes I mean almost literally anything goes you want to work together that's great you want to come and ask me questions about the homework before it's due that's great you got somebody that you know that took the class before that's great you got a solutions manual or something like that that's great I mean whatever it is it's just you know you're gonna have to pay the piper once the exams come around so if you've just sort of done the homework in sort of a slap-dash way and just thrown it together all right maybe you'll get pretty good homework scores but then you'll fall flat and so you know you're all you're all adults in here you know how you best learn and if you best learn working with someone great if you best learn locking yourself in a closet and just bang your head against the wall then some of us do sorry okay anyway whatever it takes for you to get it done get it done what I've included is a little paragraph about who you should be writing your homework solutions for I guess that would also include your exam solutions but more importantly your homework solutions and then I've also included some stuff on the back here that indicates some errors that I typically wind up seeing when students submit homework for this course typically the errors look like you start with what you want to prove or you wind up proving that one equals one and somehow using that to conclude that what you've tried to claim is true is true or you wind up getting through a proof and you in the end have somehow left out one of the hypotheses of the proof that typically means you know warning light should be going off that you've missed something or you've made an inappropriate step or whatever it is so I've listed some things there that you might want to read through before you start turning in homework assignments okay let's see what I want to do is get some sort of idea of what your collective backgrounds are so I'm going to send around a sheet where I'm asking for your name your email address your major whether or not you've taken the number theory course with math 311 course and just you know I know that a lot of you took the number three course for me last spring but either put no or put when you took it and then I list other algebra I guess that could include math 313 although I'm going to assume that all of you have taken the linear algebra course but for example if you've taken the math 413 course the you know the more advanced linear algebra course or if you've taken maybe a common a torques course another university something like that just list that out in this last column here what I'll be doing for what it's worth is I'll make a an email list I know that a lot of you don't either use or check your official UCCS email that's why I'm asking for this thing and then you know if there's a snow day or if I have to change something or something like that then I can send out a blanket email in addition to putting information on the website let's see quick quizzes there'll be about a half dozen quick quizzes over the course of the semester for instance if there's a particularly important idea or theorem or definition that I want to make sure you have immediately here what I will tell you the class session before is there will be a quiz the next class session it'll take the first three minutes of next class session I will tell you exactly what's on it it'll be give the definition of a group or give three equivalent conditions to a subgroup being a normal subgroup or just something like that there's no mystery here it's just do this you have to do it because you need to have this stuff you know in solid and so those will count like homework assignments and again there would be no mysteries there just you'll need to know this stuff so I'll just ask to come in and have it memorized grading system uh when it's all said and done when you count the exams at 30% each you know modulo that deal about allowing you to drop either exam 1 or exam 2 in place of this quote unquote final exam and you count the homework at 10% if you're up over 90% of the points then you get an a maybe to a minus if you're at 90.1% or something like that but you're guaranteed to some sort of a if you're up over 80% you're guaranteed to some sort of b etc uh that's a that's an if then statement but the congress is not necessarily true it might be the case at the end of the semester when I sort of see how you did as a class that you know there's a bunch of you clustered at 90.1 and one of you at 89.9 or something like that well 89.9 would be an a-2 or maybe an 88.7 would be an a- so I could drop it but I can't raise it that's you know it means that everybody could be rewarded in here if you all do well and you understand the material there's no reason for me not to give lots of a's and I would be very glad to do that I mean I look I'd be very glad to do that so if nothing else that it's sort of consistent with my homework philosophy too you're working together on the homework you're working together to understand the ideas in here you're not trying to be in competition with each other I mean I don't know you make a small side bed or something like that before an exam with your buddy but module of that there's no quota of grades or anything okay and then a quick remark about administrative dates if you decide after the first two weeks or so that this isn't the place for you to be then you have to get out by September 6 to get all your money back and all that good stuff and then if by about the mid semester when is that about week nine or week 10 or something like that you decide it's really just not working out then you can still withdraw but I need to sign and there's no class two weeks from today or the Wednesday for Thanksgiving all right questions all right well hopefully well you typically senior level classes where I'm grading the homework oh I grade the homework in here for what it's worth by passing the homework back I typically can learn your names by the end of September or so and I've got a relatively good head start in here okay so I've got about 10 already that's good okay so uh you know if you come in my office or you come up after class do me a favor just introduce yourself hi I'm whoever you are and just you know say hi and once I've associated a name with a face it's typically not too hard for me to keep it in there for the rest of the semester okay uh books books are available not available okay we're all right on books I don't need to call the bookstore and tell them to order more if that happens that they run out let me know and I'll try to make sure that they get more in soon this well okay so uh twice in the past maybe yeah twice in the last five times that I've taught this course the students have voted to do a note taking co-op where one person per day is designated as the official note taker that person then brings their notes to me the next day let's say uh after they've polished them up a little bit I look through them to make sure they're mathematically sound and then we post them in the library on reserve so that if somebody's missed the class they at least know what's going on and we've got 20 something is students in here so if there is a note taking co-op essentially you'd have to volunteer just to do once per semester but we have this video archiving now so it's not clear whether that maybe minimizes the need for note taking co-op but I'll say this if you'd like to have a note taking co-op we'll take a vote on Wednesday and if at least half of you that means at least 14 of you vote to do a note taking co-op then I'll ask the entire class to do the note taking co-op and we will then you know assign you a date we'll just pass a sheet around or something like that but hey you know if you vote not to that's fine too just just think about it for a couple days and we'll do that at the beginning of Wednesday just to see if there is interest in that there's you know it's there's plus and minus you only have to put in one day's worth of it's not really much work if you're going to take notes anyway you just have to polish them up after class and get them to me that way at least there's a written version of what goes on those one slight disadvantage with this videotaping is I mean you see what I'm going to write here but you don't really have any access to a place on the net where you can just print off three or four pages of notes or anything like that is that right Nick? yeah so oh well oh well okay so what the heck is this course algebra so you've been in an algebra course since I don't know since sixth grade you probably saw your first algebra course I'm guessing that most of you why not most of you I'm getting guessing that a significant percentage of you have actually taught algebra in the high school so you think well I know what algebra is yeah you do you've at least seen some pieces of what we're going to do in here this semester probably the best way to intuitively think about what modern algebra is or some people call this abstract algebra but after a while it becomes very concrete so I don't really like the title of this text we'll call it modern algebra the the real idea is to look at certain systems where it makes sense to somehow combine things in the system and produce another thing in the system you know a lot of those systems you take two real numbers you add them together you get another real number subject to certain conditions you take two whole numbers integers and together you get another whole number there's one you take two sets and you form their union you get another set so you know we've seen in your life lots of different systems where you can talk about adding things or combining these to get another thing in the system hey there's nothing special about adding how about multiplying you take two whole numbers multiply them together you get a whole number take two real numbers multiply them together you get a you take two four by four matrices and you multiply them together you get another four by four matrix you take two vectors in a vector space and you add them together you know the vector in the vector space so there's lots of systems out there that at their heart really are just systems where you've got things I don't know what the things are they might be functions or real numbers or complex numbers or matrices or vectors or but in all of those systems you talk about combining two things in the system to get another thing in the system and that's really what algebra is about now I mean the the the types of systems that I just described seem pretty general yeah there's lots of systems out there where you can talk about taking two things in the system and combining them to get another thing in the system not all of those will be captured by what we're going to study this semester we're going to study systems that well in which you can combine things but that have certain additional properties let me intuitively give you one if I talk about yeah let's just do yeah if I talk about the positive whole numbers the positive integers one two three blah blah blah blah okay so there's a system one two three blah blah blah let me give you an operation a way to combine two positive whole numbers again they're positive whole number multiply them take any two positive whole numbers multiply them together you get another positive whole number so there's a system of the flavor we're interested in it's also the case that there's a special positive whole number called this with the property that if you take this and you multiply it by anything in the system it doesn't change the other thing we'll call that an identity element for the system okay under multiplication right well let's talk about another system how about the positive real numbers so the real numbers that are bigger than zero and the operation there is multiplication that's another system you multiply any two positive real numbers you get another positive real number and that system has a special element it's also called this of course you could call it 1.0 depending on what the form of the things in the system is that you're asking me so anything that looks like a positive real number you multiply by this thing you get this thing back all right so the two systems at least at the level that we've looked share similar properties the positive whole numbers to close system under multiplication and it has a special element in it that acts as an identity and the positive real numbers has the property that's closed under multiplication has the special thing in it that acts as a multiplicative identity but here's where the two systems differ if you hand me something in the first system like the number three is there something else in the system that you can combine with three to get the multiplicative identity can you find a positive whole number so that when you multiply it by three you get one well of course not if you try to take a whole number you multiply it by three you're certainly never going to get one you might say well how about one third good choice but one third's not a whole number it's not in the system on the other hand if i'm looking at the positive real numbers if i hand you any positive real number i can always find another positive real number so that when i multiply the two things together i get the multiplicative identity of the system so there is a property or a piece that is true in the second system that isn't true in the first system and if you want what you've just seen as a glimpse of the type of the type of analysis that we're going to do on systems here's a system yeah it's a system here's an operation you combine two things in the system to get something back in the system but certain systems differ from other systems some systems have multiplicative identities some systems don't some systems have the property that if you look at something in the system you can always find something else so that when you combine it you get the multiplicative identity some systems don't and what we're eventually going to do over the course of the semester is the following there are two very important types of systems that we will study one is called a group that's the first type of system and the other is called a ring the notion of a group we'll look at for eight or nine weeks or so the notion of a ring will look at for the remaining six or seven weeks of the semester in effect these two things are simply nice systems that have the property that they look relatively general so when we write down the the underlying characteristics that describe what a group is not every algebraic structure is a group but certain things are enough things are groups that merit us discussing or somehow studying those in their own right we'll look what sort of properties they have we'll look at a lot of examples and we'll see what sort of structure we might be able to conclude from that we'll then look at the notion of a ring a ring in effect folks is an algebraic system that has two types of operations defined on if i hand you the collection of all integers makes sense to adam also makes sense to multiply similarly with real numbers similarly with square matrices similarly with functions so there's lots of systems that have more than one operation systems with two operations that behave nicely those systems will be called rings and then why analyze those systems so that's the general overview of what we'll be doing in here what i'll try to do is is at least point in the right direction if you're interested in how these particular structures might i don't know be applicable in the real world that's totally great question and i'll at least try to you know point to places on the internet or places in the text where they talk about that but what i'm more interested in doing because i know that there's a lot of you in here that are potential teachers or are already teaching in the high school or the junior college level maybe is i want to make sure that you take a lot of the ideas that you've seen that you're familiar with certain properties of algebra typically algebra the real numbers and show you what's really driving those behind the scenes what is it about the real numbers that makes it so interesting or not what is it about the the whole numbers the integers that somehow both makes it interesting but makes it different from the real numbers what is it about the complex numbers what is it about that etc etc so there's lots of algebraic systems out there that you're already familiar with that you're probably talking to or teaching to about you know in your classrooms and what i'm hoping to do is just make sure that some of those ideas become a little bit more solidified in your mind as to where they come up and why they're useful etc so that will be a main goal of this semester is to try to give you some tips or some good overviews for those of you that are going to be teachers okay so let's start we before we get to the first main idea which is this we'll probably get to it on wednesday what i want to do is review some ideas that you have seen before either in the discrete math course or in the number theory course or in the linear algebra course if these are new to you i'll give you enough of the detail that you can pick it up here for the first time and we'll be doing some problems for homework on those but for most of you i think this is going to be a review so a review of some important concepts the first is what are typically called equivalence relations on sets equivalence relations when i teach the discrete math course i like to present these as what i'll call generalized equality the idea is this you have a set and i don't care what the set is it might consist of numbers but it might consist of functions or matrices or people or license plates or whatever it is it doesn't matter what the underlying set is the idea is let s be a set then we define a relation a relation on s is simply a way of connecting elements in the set set i'm purposely being nebulous here if the set is license plate of the two numbers is even if the distance between them is some even number that's a way to connect all right typically the symbol that's used for one thing in the set is related to another thing in the set is either this till the symbol or sometimes the letter r and here then is what we mean when we talk about an equivalence relation and equivalence relation on a set relation on a set s is a relation i don't know what you want to call it how about let's call it r for relation that has the following properties where the following three things are true first of all for every element in the set let's call it little a in capital s every element is related to itself so whatever the determination of how it's supposed to be decided whether or not one thing is related to another in order for the relation to be an equivalence relation it has to be the case that every element is deemed to be related to itself so for example if the relation on the collection of license plates is deemed one license plate to be related to another if the first symbol of this one is the same as the first symbol of this one folks if you hand me the same license plate twice then obviously the first symbol of this one is the same as the first symbol or if the relation is on the collection of people and we deem two people to be related if they have the same eye color well if you look at the same person twice is that person of of the same eye color as that person. Well, of course, it's the same person. So that's the idea behind this particular stipulation in an equivalence relation. Secondly, for every pair in the set, we'll call them A and B in the set capital S, if it happens to be the case that A is related to B, so if one thing is related to the other, then necessarily B is also related to A. So for example, on the set of real numbers, if you deem one thing to be related to the other in case the first one is less than the second, if the first one is less than the second, then you don't have the second related to the first because we're talking about one thing being related to another, if this one's less than that one and if you've told me A is less than B, then it's certainly not the case that B is less than A. So this is something that might be true for some, might not be true for others. Let's go back to the license plate example. If you deem two license plates to be related in case the first letter of the first one is the same as the first letter of the second one, well, folks, if the first letter of the first plate and the first letter of the second plate are the same, then the first letter of the second plate is the same as the first letter of the second plate. In other words, there, it doesn't matter which order you hand me them in. And the third stipulation, if we're gonna call a relation, the equivalence relation is the following. If it's the case for every three elements in the underlying set, if A is related to B and B is related to C, then A is related to C. And these three conditions are typically given the three names. This is called the reflexive condition. This is called the symmetric condition on a relation, and this is called the transitive condition on a relation. So the punchline is this. If we have a relation on a set, which is reflexive, symmetric, and transitive, then we call the relation an equivalence relation. There are numerous examples of equivalence relations. Example, oh, let's just give an easy one. Maybe the set is the collection of cities in the United States, in the U.S., and maybe the relation is deem one city to be related to another city in case the two cities A and B are in the same state. Okay, so Collar Springs is related to Denver. Collar Springs is not related to Chicago, because the definition of what it means to say that one city is related to another is you simply ask whether or not two cities. All right, now we can just quickly click through this. Let's see, if I hand you the same city twice, is A city related to itself? Of course it is, it's in the same state as itself. If A city is related to another city, is that second city related to the first one? Yeah, because if you tell me that one city is related to another, you've told me that the two are in the same state. So does that mean the second one is related to the first? Yeah, because they're in the same state. And finally, this transitive condition is easy as well. If city A is in the same state as city B, and city B is in the same state as city C, is city A in the same state? Sure, so this is an equivalent relation. Then R is, and I'm gonna start abbreviating this by ER, equivalence relation on the set, couple of us. Example, the set S is the collection of Yeah, let's do the example now. Is the collection of, oh, let me give you this notation now. Capital Z with an extra line. Capital Z for the remainder of the semester will denote the set of whole numbers, which I'm gonna call integers, that's the formal word, positive, negative, and zero. Positive, negative, and zero. So this is negative 10, negative 9, so I don't know all of the whole numbers. I'll do a little history in here too. The YZ, take as that. It turns out historically the study of what we now call modern algebra, but I'm just gonna call algebra for the remainder of the semester. I'm never gonna call abstract algebra. As a discipline started in earnest in the German school in Gürgen in the mid to late 1800s, and it turns out the German word for number is Saal, Z-A-H-L, so that's A number in pluralist Saalan. And so the German school started using this as a very natural symbol for the collection of numbers, and it turns out that that school's influence has been powerful enough that this notation is totally standard throughout most of mathematics today. If you look at the set capital Z, or sort of boldface capital Z, that typically is understood to always be the set of whole numbers. Here's a relation, D-M, the relation A is related to B in case if you square the first one, you get the same as the square of the second one. So that's the definition of a relation. I've just sort of cooked this up. I don't say that it's thin air. I mean, I've had this in mind because it's got certain properties to it. So for example, one is related to one, for instance, because one squared equals, one is also related to minus one, because one squared equals minus one squared. So there's zero is related to, well, it's related to zero, but it's not related to anything else. There's no other integer that has square zero, okay? Then it turns out script R is an equivalence relation. I already broke the promise. I was gonna abbreviate that by E-R and just wrote it out. Why? Well, I won't run through all the details, but it's pretty easy to see. First, is it the case that every integer is related to itself? In other words, is it the case that A is related to A? Well, in order to test whether or not A is related to A, you have to decide whether or not A squared equals A squared. Check, sure does. Question, if A is related to B, in other words, if A squared equals B squared, is it the case that B is related to A? In other words, if A squared equals B squared, does B squared equals A squared? Yeah. And finally, if A squared equals B squared and B squared equals C squared, does A squared equals C squared? So that's pretty easy to check out. Let's give a non-example. Non-example, let's let the underlying set again be the integers, although it doesn't really matter. I could let this be one of any number of sets. Define the relation r by a is related to b in case a is less than or equal to b. So here's what it means for the integer a to be related to the integer b. You deem them to be related in case this one's less than or equal to that one. So 3 is less than or equal to, so 3 is related to 7, for example. That's good. So let's see, is this an equivalence relation? Well, is it the case that every integer is related to itself? In other words, is it the case that a is less than or equal to a? Yeah, that's good. How about the second one? Is it the case that if you hand me a related to b, is it necessarily the case that b is related to a? The answer is no. If you tell me a is related to b, then what you've told me is a is less than or equal to b. Does that imply or necessarily yield b related to a? Well, no, if a is less than or equal to b, it's certainly not necessarily the case that b is less than or equal to a. It might be if you've handed me a equals b, but for instance, 1 is related to 2, but 2 isn't related to 1. So this is not symmetric. And now, here's a habit that I want you to get into. If it's the case, folks, that you've got some system or some list of requirements for something to be a, well, here's something to be an equivalence relation. Later on, we'll have a list of requirements for something to be a group, for something to be a rings, for something to be a subgroup, for something to be a normal subgroup. We'll have a list of requirements. Here's what it means for this to happen. If the requirement includes some sort of stipulation that for every blah, blah, blah, blah, blah, blah, in order to convince me that the proposed system isn't one of the flavors that you're looking at, all you need to do is come up with one specific counter-example. Just one, and then you're done. So I can convince you that this relation is not symmetric by simply coming up with just one situation where this isn't true. Why? One is related to two, but two is not related to one. Done. So this is not an equivalence relation. So r is not an equivalence relation. Let's look at a few more that are. Questions? Sorry? Question? I understand the question. Yeah, so if the condition that you're trying to decide whether or not it's true or not consists of more than one property, as soon as you find one where it's done, then you're done. Don't worry about trying to show that other things are true or not. So in fact, that'll come up in the definition of both group and ring and subgroup, and there'll be more than one condition that a thing needs to satisfy in order to call it a whatever we're calling it. As soon as you've found something that doesn't hold, just walk away. I say it's not this, therefore it's not a group, or it's not a ring. All right, an important example that we'll see later. Example, the set is, again, the set of integers. And here's how we're going to deem the relationship. Define a relation r by setting a related to b in case when you look at the difference b minus a that you get multiple of four. This one's a little bit more interesting. Now, I have to define what it means by multiple. When we're talking about whole numbers, the understanding is to say that something's a multiple of another whole number means that you have to be able to take the original number that you're handed and multiply the number four by another integer. So for instance, I don't want seven to be a multiple of four. You might say, well, it is. You take four and you multiply it by seven four, so you get seven. Not allowed. When we talk about integer multiples, we mean that the thing that you multiply has to be another integer. So I'll put in here for now an integer multiple. But that's going to be understood from now on when we're talking about the underlying system being the system of whole numbers. So this is sort of interesting. Two things are deemed to be related in case when you take the second one, you subtract the first one, that the number that you get is a multiple of four. And it turns out this is an equivalence relation. Sort of interesting. Let me briefly go through how you might prove this. Prove. Let's see, we have to show first that this relation is reflexive. We have to show that a is related to a. I'll use this symbol. You'll get used to it for all integers in z. So the upside down capital A is a shorthand that represents the phrase for all or for every or for each. This is the for all quantifier. We have to prove that. That's one of the requirements of being an equivalence relation. All right, well let's see. So a related to a, question mark. So what do we need to determine in order to decide whether or not a is related to a? We need to decide that if we take the second thing, there it is, it happens to be called a in this case, and we subtract the first thing. This just happens to be a also, is zero. So that's what we get when we do the second minus the first. The question is, is that an integer multiple of four? In other words, can I take four and multiply it by some integer and get zero? Well sure, because it's zero times four. So check. A is related to a. I've shown that if you take whatever the appropriate prescription is, here it's you take the second number you subtract the first, that what came out as multiple of four happens to be zero times four. That's fine. Therefore the appropriate condition is satisfied, and therefore the particular relation is indeed reflexive. How about the next one? Secondly, if a is related to b, do we get, do we get, or can we conclude b is related to a? In other words, this is a symmetric relationship. So what you get to do here is assume a is related to b, so we get to assume what a is related to b means that b minus a is, and let's write it out formally, is a multiple of four. Let's call it four times n for some integer n. That's just the definition of what it means to be a multiple of four. That you can write this number as four times some whole number. And what do we need to do? We need to show, this is what we're trying to conclude, that b, oh, pardon me, we're trying to show that when you take the thing on the right, which here happens to be a, and you subtract the thing on the left, which happens to be b, is four times some other integer for some integer t. This is what we're given and this is what we have to show. Well, how do you get from here to here? Sometimes it's not so easy to see. Here it is pretty easy to see. Take what you know. We start with this, that's what you get to assume. Well, because it's a given equation, you can manipulate both sides of it and get to here. But if we start with b minus a equal to four n, multiply both sides by minus one. Folks, that's legit to do because this equation is assumed to be true. That's given information. Well, if I multiply the left side by minus one to get a minus b, if I multiply the right side by minus one to get minus four n, which doesn't tell me what I need to know yet, what I'm trying to do is convince myself that a minus b can be written as four times some integer. So now rewrite, just use some arithmetic. a minus b is four times minus n. I've written minus four times n is four times minus n. That's just arithmetic. But then this is okay. Equals four times t, where t is minus n. And so, check. So I've concluded the appropriate equation is true. I've shown that a minus b is four times some integer because if n is an integer, then necessarily the negative of an integer is an integer. I guess I officially haven't written out everything. I'd probably like to see a statement like that in your homework. Why is it the case that I've written a minus b as four times t, where t is an integer because I've written it as four times minus n? Where n is an integer and if n is an integer, then the negative of n is also an integer. All right, that was easy. Step three, we'll do it a little bit more quickly. You get to assume what a related to b and b related to c. What you need to do is show that a is related to c. Presumably what we need to show in order to show this thing is an equivalence relation. Can we do it? Yeah, what are we assuming? We're assuming that, oh, b minus a is some multiple of four. Let's call it four times n one. And we're assuming that c minus b is some multiple of four. Let's call it four times n two, where n one and n two are integers. That's what it means to say that a is related to b and b is related to c. That's just the definition of this relation. And what do we have to do? We have to show that c minus a is four times some integer. Let's call it t for some integer t. That's the goal. So this is where we want to land. Quick remark, maybe because it's been a long summer and you haven't done any mathematics for three or four months or something. Folks, you can't start here. If on your homework you say, well, here's what I want to show. Therefore, it's true. And I'm going to start manipulating both sides of this equation. You will see a large red x and you'll just get a kick back at you because you somehow violated this, unfortunately common error that students seem to think constitutes a valid proof. If you're trying to conclude something, you can't start with it. Might say, well, if I start with it and I get to something I want equals one, does that not constitute a valid proof? The answer is no. You can't prove something is true by simply somehow showing that it implies something true. And some of you have heard the story before because if you allowed that as a proof technique, then I'll be able to prove for you that two equals negative two. Here's my proof. Prove that two equals negative two. Prove two equals negative two. That's what I want to show. Square of both sides, four equals four, which is clearly true. And therefore, two equals negative two. So don't, this is where you have to land, folks. I mentioned this in the homework sheet as well. If somehow your conclusion is therefore one equals one, that is not good because I'm not impressed by the statement one equals one. It's not. I know it's true. You know it's true too, but so what? So what? I am impressed by your being able to convince me that C minus A is four times an integer. Let's see if we can do that. Hmm. Well, I mean, I can show you the cute little trick here. That's true and that's true. These are assumed to be true. So I can manipulate with these because those are allowed in the system as true statements to begin with. So that means I can add both sides of these together, which would magically say that's cheating. Well, then the B's canceling. You get C minus A is, okay, I'll do that. Maybe another way would be, look, let's solve. What's C? C is four and two plus B that's from the given equation two, arithmetic on a given equation. So let's see. And A is, oh, if I solve for A, I get what, B minus four times N one. That's also arithmetic. I don't have to do it this way again. I'm doing it the long way, but I'm doing it a way that maybe most of you would approach it. These are legit to write down. I can manipulate both sides of these because these are assumed to be true. All right, now let's compute C minus A. That's what I'm trying to say something intelligent about is what is four and two plus B minus A, oh, A is B minus four and one. Just substitute, I'll put that here if you want substitute, which is by arithmetic. And I'm looking for each line to be at least denoted enough so that I know what you're doing, how are you getting from one line to the next? So this is four times, oh, N two plus N one. I believe you can do the arithmetic at this stage, folks. I don't need you to show me the minus of the minus, that's fine. Plus, let's see, oh, B minus B, that's zero. That's nice, the B's cancel. So this is four times N two plus N one, but this is an integer since it's the sum of two integers and N two individual integers. So we boil it down to, you get four times the sum of two integers, so this is four times T where T is an integer, it happens to be N one plus N two. Check, that's what we're asked to show. Showing that, and I said, how'd you do that? Well, look, technically there should be a follow-up statement, although we're mathematically mature enough that we don't have to put this level of detail in. I've then shown that C minus A equals this and justified Y, which in turn equals this and I've justified Y, and in turn equals this and I've justified Y, and in turn equals this and I've justified Y, so that in effect we've now shown that this equals this. So if you want to put one last line is there for, C minus A is four times T as required and we're done. Now, anytime you have an equivalence relation, equivalence, on any set, we've just seen one, this will be an important one, relations give rise to what are called equivalence classes. Here's how you get them. Once you've determined that you've got an equivalence relation to find on whatever set it is, and I don't care what the set is, I'm going to try to keep you away from thinking all the sets have to be numbers, but that was an important example here. What you then do is you look around in the underlying set and you ask what things are related to each other and you lump all the related things together in a subset. So for example, we can ask this question, if I look at this relation that we just described, so on Z, A related to B, if I'm A, I'm sorry, B minus A is a multiple of four, is multiple of four, so the relation, the equivalence relation we just looked at, we can talk about the equivalence classes. For example, if I hand you something in the underlying set, I'll just randomly pick zero here. If I put then square brackets around an element, what I'm asking you to do is write down all the things in the set that are related to zero. Well, zero is related to zero because this is an equivalence relation and anything is necessarily related to itself. What else is related to zero? In other words, what other things can I write down so that when I do it minus zero, that I get a multiple of four? Well, four works, because four minus zero is certainly a multiple of four and eight works, because eight minus zero is certainly a multiple of four and 12 works and oh, a negative four works because negative four minus zero is negative four, which is an integer multiple of four, that's four times minus one, negative integers are perfectly legit here, et cetera. So what we call the equivalence class of zero happens to be all the multiples of four. We can talk about the equivalence class of one, it's those integers that are related to one where this is the definition of the relation. Well, you can always start by writing down the element that you, oh, what other things are related to one? Those things so that when you subtract one, give a multiple of four, 13 minus three, et cetera. For instance, the equivalence class of two, I'll let you write it out, two, six, blah, blah, blah, blah, there's lots of things in there. Equivalence class of three, three of course is in there, seven's in there, and 11's in there, minus one is in there, et cetera, et cetera, et cetera. And it turns out this is all of them. It turns out that every possible integer is included in exactly one of these equivalence classes. Exactly, every integer is related to exactly one of these four things. That's where the number four comes in. And for those of you that have seen the notation, what we've done is we've written down in effect mod four arithmetic, or we've sliced and diced the collection of whole numbers via their mod four remainders. And for those of you that saw the number theory, of course, what we've done is we've collected up all those integers according to what their remainder is in the division algorithm. So if you have remainder zero, you're in the first set. If you have remainder one on division by four, you're in the second set, et cetera. Let's see, let's get now to a set, and an equivalence relation on that set, which is significantly more familiar to you. So here's a set. So a new set. I'm gonna call the set S. It's the set of symbols that look like, let me call them C comma D. So the symbols in this set are inherently pairs, where C is any integer you want, but D is an integer bigger than or equal to zero. And I'm gonna, for this class, use the notation capital N. This will be the set of natural numbers. And for me, this will include only the numbers one, two, three, et cetera, et cetera, et cetera. It turns out that the discrete math books that's used at this university, it's actually a really good one, but they use a slightly different notation. For their natural numbers, they include zero. For me, the set of natural numbers will just be one, two, three, et cetera. It turns out this is a more standard notation than the one that's used in the discrete math book, but we're gonna put it here and there. So there is the set. So I want you to look at all the things that look like ordered pairs, where the first thing is any integer you want, and the second thing is some positive integer, like minus two comma seven is in this set, and the pair 17 comma one is in this set. The pair 16 comma negative one is not in this set because I'm requiring the second thing to be positive. So I'm gonna define a relation on this set. Define on this set. And notice what the things in the set look like. They look like pairs. And here's the relation. If I hand you a pair, let's call it C1D1, I wanna deem that to be related to the pair C2D2 in case when you multiply the outer two ones, when you multiply that times that, you get the same thing as if you would have multiplied the inner two ones, D1 times C2. So we're gonna deem two pairs to be related. In case that product, call it the outside product, is the same as the inside product. So for example, this pair one comma two is related to the pair three comma six. Why? Because when you multiply the two outside numbers, you get the same as if you multiply the set, the two inside numbers. All right, now, it turns out, in the interest of time, I won't prove this for you. It turns out, it's not too bad to do. You'll do something slightly related to it when you're doing homework problems. That script R is an equivalence relation. Relation on the set of pairs consisting of an integer in the first slot and a positive integer in the second slot. So let's look at some equivalence classes. So proof omitted. Omitted. But what I wanna do is look at some specific examples. I'm just gonna pull these out randomly and look at various things that are in the same equivalence class of the particular pair that I happen to pull out. Let's, for example, find some things in the equivalence class of the pair three comma two. So what does this mean? What I want you to do is write down some things that are related to this particular thing via this relation. So I need to somehow write down other things with the property that when you multiply the outside ones, that you get the same thing as if you multiply the inside ones. Well, look, this is an equivalence relation. So if I hand you something that's in the equivalence class, necessarily the thing itself is in there. Let's check that it is. This will be stupid, but it will be. Is three times two equal to two times three? Yeah, it is, okay. So it's no big deal. Let's see, what else? I need to rig something. Let me show you one that's in there. So that, is it the case that this is related to this? If I multiply the two outer things, three times four, do I get the same as multiplying the two inner things? You have two times six. How about, somebody wanna give another suggestion? How about 12, eight? Is that in there? Multiply the two outer things, we get 24, the two inner things, we get 24. So that's in there. How about, no, let's not do that one. Anybody see a pattern? What would the pattern be? What do they mean to be related to the bare three comma two? Yeah, it means if you take the first one, you divide it by the second one, you get three halves. If I take six divided by four, I get three halves. I take 12 divided by eight, I get three halves. So it turns out, turns out, and hey, this follows directly from the definition there that C1, D1 is related to C2, D2 precisely when the fraction, I'll say as fractions, this symbol C1 over D1 is the same as the symbol C2 over D2. Now, why is that? Well, look, you just divide through. These are equal then. And, quick remark, it makes sense to divide by D1 and D2 because we've assumed that the things that sit in the second coordinate are natural numbers. In other words, they're bigger than zero. So these make sense. So here's the point, folks. If I hand you this fraction, the point to be made is that fraction could be represented in other forms that don't just look like three slash two. It could be represented in the form six slash four or 12 slash eight or... So the point is this, the collection of fractions that you're very used to dealing with really is a collection of pairs that come equipped with an equivalence relation on it. As a symbol, three horizontal line two is not the same as the symbol six horizontal line four. This has got threes and twos in it. This has got sixes and fours in it. So as symbols, they're not equal. But somehow you're asked to interpret those as representing the same thing in the system. Technically, what you've done is you've taken the collection of pairs where the denominator is not zero and you've defined two things to be related in case this quantity is satisfied. So the punch line is a set that you have great familiarity with happens to be a collection of pairs on which an equivalence relation has been defined and in which you're simply comfortable with viewing things in the same equivalence class as somehow behaving the same way as being equal. Now, that's all well and good in your thing. Well, so what? You know, I know what the rational numbers are. Why is that an issue? This could be an issue and in fact in chapter eight will become a huge issue for us and here's the type of issue that could come up. If it's the case that you're working with a set where the things in the set are actually equivalence classes for some equivalence relation and folks now anytime you're working with the rational numbers that's exactly what you got going on. Think intuitively whenever you're working with a set where the things in the set have different names to represent the same object like the symbol three slash two is meant to represent the same object as the symbol six slash four in the rationals. Whenever you're in a situation like that you are in danger of having trouble when you're trying to define functions from such a set to another set maybe to the set itself. So what today's big punchline is is anytime you're working in a set where the elements of the set could be represented by different symbols like fractions. If you try to define a function from that set to some other set you have to be careful that the function is what we usually refer to as well defined. So on sets for example the rational numbers and I'm gonna start denoting the rational numbers by capital Q with an extra line through it rational numbers positive negative and zero YQ, Q stands for quotients it's quotients of integers for which the elements are actually equivalence classes are actually equivalence classes of an equivalence relation so this is for an equivalence relation there might be what we usually refer to as well defined issues. Issues when trying to define functions from that set. That set which I could have phrased as when trying to define functions having that set as domain. You're scratching that thing off you know, what the heck are you talking about? Here is an example sample I'm gonna ask you to define a function. Define a function f from the rational numbers to the rational numbers here's the definition of the function Hammy any rational number well that's what a generic rational number looks like and the rational number that I want you to spit out is take whatever's in the numerator add one and divide by b not your head saying that's no big deal what goes in is a rational number what comes out is certainly a rational number because this goes in you're assuming a and b are both integers with b bigger than zero what comes out is something in the numerator divided by something that's not zero and the numerator is an integer so you're thinking well it's no big deal it is a big deal and here's why. Note f is not well defined meaning it's possible we can input the same rational number written in two different forms two different forms and the issue is different rational numbers may come out outputs may result here's the issue folks if you've presumably defined a function from the rational numbers to something else here to the rational numbers if you ask what comes out when you plug in three halves alright if you then ask what comes out when you plug in six fourths better be the same output because three halves and six fourths are the same rational number but the issue here is look if I plug in three halves what comes out four halves but if I plug in six fourths what comes out seven fourths so I've plugged in the same rational number drawn or written or viewed from a different form or having assigned a different name to it first the rational number three halves then the identical rational number although written differently six fourths and the issue is for this particular description what comes out is two different things that's not allowed for a function because then you don't know what is it that actually comes out when you plug in the rational number well what do you want to call it 1.5 for a minute I don't know the answer why is it not well defined since well this is another example of a situation where in order for a function to be well defined it has to be the case that regardless of how you describe a specific input value that the same output value always has to come out if I can show you just one situation where that doesn't happen then the function is not well defined since for example if I plug in f of three slash two I get three plus one over two which is four over two which is two but plugging in the same rational number viewed from a different form or the way we usually say it is with a different name attached to it here's another name for the same rational number six slash four we get six plus one over four sorry which is seven fourths and the point is that these are not equal so this function is not well defined so now suddenly you're a little bit nervous like wait a minute you mean every time I define a function from something like the rational numbers I have to somehow make sure that the function is well defined technically yes but this certainly hasn't come up as an issue before I'm playing it up because it will come up as an issue when we're looking at a specific set later on for those of you who have seen this before when we're looking at the collection of cosets of a group by a normal subgroup that collection we're going to want to define an operation on and we're going to need to be careful that the operation that we define on it is actually well defined so here's what we'll do at the beginning of next time that you'll start doing for a homework next time we'll give an example we'll give an example of a function let's call it G from Q to Q that is well defined and we'll show you how to prove it's actually well defined and we'll give a proof so that's the sort of thing that you'll start seeing on the homework that comes up tonight okay so here is typically how I will assign homework it'll be on Monday it'll be due the following week from Wednesday so here's the assignment that's due Wednesday the 29th in section two and those of you that have taken a class from me before know what the notation means for those of you that haven't I'm going to list out a bunch of problems the problems that I list out are doable I certainly don't want you to turn all those in I want you to view these as maybe a good place to look for additional problems to practice before an exam or if you're trying to nail down a concept to do a few more of those the only two problems that I want you to turn in from this section are the ones that I'll circle here just problems seven and nine and in section four problems one through eighteen the ones I want you to turn in are two four and eighteen and I'm going to give you a sheet with two extra problems on it extra problems and I want you to turn those in as well and I'll pass that sheet out now anything that I hand out in class I will also I can't guarantee it will be done immediately but I'll also at least post on the course website so if you're watching the video tape of this class and you're not getting the assignment in person you can just go on what we'll do on Wednesday right at the beginning is vote as to whether or not you want to have a note taking co-op in here let's see did that sheet with emails and I'll take that did everybody get a chance to sign the email roster take that thank you for those of you that might be off campus tonight and watching the video if you could email me abramsatmath.uccs.edu with your email contact information that would be great especially if you're going to perhaps not be in class quite often then I need to be able to get in touch with you and that's it for tonight folks if I don't see you before then I'll see you on Wednesday