 as a binary operation, you have to make sure folks that you're allowing any possible pair of things to wind up going through the function, because remember a function has to be defined on all of the possible elements in the domain, so that if I try a non-example, so there's a good place to show up in red, non-example, how about division from, I don't know, Q, let's do this one, Z cross Z to Z is not a binary operation, and the reason is a binary operation is required to allow you or to have it be the case that regardless of the pair that you hand the operation, the appropriate output needs to be in the same set, so even though it's sort of, I mean it's not in bold phase here, the point is that what comes out has to be from the original set, it's not a binary operation, well in order to demonstrate that something's not what we might claim it to be, all you have to do is show me one situation where it doesn't work, for example if I ask you to divide these two things, how about one comma two, I'd get one half and that's not in Z, so it's not a binary operation, I'll be more specific on Z, you might say well, let's see, can I trick this thing to work out, what happened here, division didn't work because if I divide two integers, I might get an integer, but I don't have to, how about if I try expanding the underlying set, how about on R cross R to R, it was also not a binary operation, the reason it's not a binary operation is it is the case that it's possible to plug in something here that doesn't get spit out here, specifically it's possible to plug in something here and then get spit out anywhere because you can't divide, for example, one comma zero is one over zero which is meaningless, so division is not a binary operation on the reels, it's also not a binary operation on the integers, it is however, let's see, divide might be a binary operation on, I'm going to denote them this way, Q bigger than zero, now let me give you the R bigger than zero, I want to deal with Q in a minute, so this notation means folks, look at all the positive real numbers, the real numbers bigger than zero, to the real numbers bigger than zero, this is a perfectly good binary operation, if you hand the division process, any two positive real numbers, well when you divide two positive real numbers you don't have to worry about division by zero, so you can always at least divide them, that's not an issue, the reason you don't have to worry about zero is because I've explicitly excluded zero, I've excluded all the negative numbers too, oh let's see two, yeah and if I take two positive real numbers and I divide them I get another positive real number, so this is a binary operation, so this is okay, okay I should have written it in black, but I got ahead of myself there, okay, is a binary operation, it's on R bigger than zero, alright, folks there's lots of other things out there and for homework you'll look at a few more examples but let me just sort of spew them, multiplication of two by two matrices, we've seen that already, the binary operation on the set of two by two matrices, of course it's got to be a little bit more specific, two by two matrices with entries in what set, we're going to talk about the collection of two by two matrices with entries in the reels, and talk about two by two matrices with entries in the integers, entries in the rational numbers, entries in the complex numbers, heck if you push things even further you can talk about the collection of two by two matrices where each of the four entries are themselves three by three matrices, we in effect can talk about collections of matrices, pick your favorite size maybe two by two where inside each slot you simply put elements from any other system that you're interested in, you can talk about multiplying them, so there's binary operations on those sets, if I hand you two functions each of which is a function from the reels to the reels, and I can ask you to compose the two functions and that will give you a function from the reels to the reels, so there's a binary operation on the collection of functions from the reels to the reels, and there's going to be many, many more examples of binary operations that we will look at over the course of the semester, although as we'll see by the end of tonight, we're not going to be interested in all possible sets and all possible binary operations, we're only going to be interested in those sets and binary operations that satisfy certain additional properties, and so what we're going to do for the next 10 or 15 minutes or so is build up those properties. What we need to do first is open up a can of worms that I introduced on Monday, this seems like it's sort of off to the side that it's not really that central to the notion of binary operation, but it turns out one of the key sets and binary operations that we will look at in about week 7 or so will bring to the surface exactly the issue that we're going to look at now, we're going to look at the issue now in the context of the rational numbers, and the point is we're all familiar with properties of the rational numbers, so if I bring up the issue now and you can sort of think about it in terms of something you already are comfortable with, then maybe when we bring it up again, the hope is 6 or 7 weeks from now, when we bring it up again at least you'll understand what the issue is and then you'll be able to simply attack the issue in that new set rather than trying to get your head around the issue for the first time as well. So here's an example, how about I'm going to define star, no, define, I'll call it star here, from q cross q to q. So let's see, I'm attempting to define a binary operation on q, so let's see what are the elements of the rational numbers look like, they look like a over b where a and b are integers and b can be chosen to be bigger than zero, that's what any rational number looks like. So what would another rational number look like, maybe something like that, here's what I'm going to ask you to spit out, how about a plus c over b plus d. So I'm asking you to hand me any two rational numbers, now I'm going to tell you how to somehow combine them, well the method that I want you to use to combine them is this one, well first you've got to ask whether or not when you take two rational numbers of the standard form that we talked about on Monday where you can assume that the denominator is always a positive number, a positive integer, in other words an integer bigger than or equal to one and the numerator can be any sort of integer, zeros and negatives are allowed in the integers, in the numerators that's fine. So I'm thinking alright, it's a perfectly good binary operation, but what was the issue with the set q, the issue was in this particular set there are many different ways to write or to name a specific element of the set, so the thing that we sometimes call one slash two can just as equivalently or correctly be called three slash six or five slash ten or and here's the point if you hand me a function that somehow includes, well here are the rationales but any set that has this property that the elements can be named differently, you have to make sure that the description of the function that you've given doesn't depend on the name that you've given each of the elements of the set, that's a long way of saying look if you and your friend put in the same pair of rational numbers it had better be the case that you and your friend get the same output for those two rational numbers regardless of how you call them, so that for instance it better be the case if this is a good function, if this function makes sense that if you decide to put in one slash two and three slash seven and your friend decides to put in five slash ten and six slash fourteen, well folks the two of you have put in the same inputs, it better be the case that you each get the same outputs, but it turns out you don't here, this is not a well-defined operation, and if you think well why didn't he worry about that in the previous examples as well-defined stuff because in the previous examples presumably the elements of the set can only be written in one way, if you tell me that you've written the integer and you've called it A, okay you've called it A, here though if you tell me you've written a rational number as A slash B it might be the case that you can also write it as C slash D where C and D are different or E slash F or E and F are different, so I'll put an exclamation mark here, here's why, well in order to convince me that it's not a well-defined operation all you need to do is produce one situation where you and your friend put in the same rational numbers simply written in different form and convince me that you and your friend produce different outputs for this function, for example let's see what happens if we do this to something like one half comma three sevens, I'm just making up a pair, let's see what's the definition of this function, ask me to do it, ask me to add the numerators and add the denominators and what do we get four-ninths, be more explicit what that is, we get one plus three over two plus seven, in other words we get four-ninths, so if you choose to input these two, this pair of rational numbers you'd get this out, on the other hand if your friend chooses to input three slash six and what's another way to describe this, how about thirty slash seventy, here's what your friend would get, thirty three over seventy-six, folks these two numbers are not the same, thirty three seventy-six is not equal to four-ninths, so it's the issue, the issue is that it's possible to take the identical rational numbers simply written in a different way and view them as two different input values and wind up with different outputs, so just because I found one place where it's bad, it's bad you're done, walk away, so now your confidence is maybe a little bit shaken about working with the rational numbers, things could go south, yeah things could go south, so the question might be this, if I hand you a proposed function from Q to something, like I did on Monday, or from Q cross Q to something, maybe from Q cross Q to Q, so that we have the makings of a binary operation on Q, the question is how do you go about determining that it is well defined, it's easy to show something's not well defined, just write down a specific counter example, but to show that something is well defined, here's what a proof of that might look like, so let's try another example, star from Q cross Q to Q, the idea is take star A or B comma C over D and define this to be A times C over B times D, think well that doesn't look much different from what I just did, the only difference is in that previous example I added numerators and denominators, here I'm asking you to multiply numerators and multiply denominators, and what we're about to show is show that this function star is well defined, in other words at least show that this thing gets off the ground, and then we can ask the question, alright once we've verified that it's an honest or goodness function, then is it the case that all possible input values Q comma Q can be put in, is it the case that what comes out is in Q, etc. So this is what we need to do to get it off the ground, and again the reason that we didn't have to fret about any of that before is because in these other sets descriptions of functions are, I'm sorry, descriptions of elements are in a sense unique, if you call something Z inside the integers then that's it, I mean if you want to call it W that's fine then W equals Z, the point is though in the rationals we have this sort of, well technically as we saw in Monday we have this equivalence relation on expressions and we're going to make sure that the function satisfies or is somehow consistent with the equivalence relation, so how to do this, well here's what we need to do, we need to prove the following, that if you choose to write a rational number that way, but your friend chooses to write it this way, and secondly if you choose to write the rational number C over D in that form, but your friend chooses to write it in this form, you have to show that if you input the pair of numbers in this form that that equals what you'd get if you'd input the pair of numbers in the other form, C prime over D prime, that's the goal, so you have to convince me that regardless of how you describe the input rational number, whether you describe it as A over B or A prime over B prime, and then regardless of how you describe the second input rational number, C over D or C prime over D prime, you have to convince me that either way you get the same thing out, and folks this is pretty easy to do, look, but let's see, if I do star of this symbol A over B comma C over D, what do I get? By definition this function says you multiply the numerators, then you multiply the denominators, oh but let's see this is properties of rational numbers that's the same as A over B times C over D, just arithmetic, oh but that is, wait a minute, A over B by hypothesis, this is what we're assuming, is the same as A prime over B prime, oh and C over D is C prime over D prime, that's given information, that's what we're starting with, oh but that's nice because look, that's precisely what star of A prime over B prime, C prime over D prime is, so check, this is a well-defined function, so here's an example of a function from Q cross Q to Q that's not well-defined, here's an example of a function from Q cross Q to Q that is well-defined, the ones that are well-defined you technically have to do some proving, you're thinking well of course it's well-defined, it's just multiplication, you know you're right, that's all going on, all right, questions there, comments, what I'm, you know as I mentioned I'm part of the goal in here is to try to give you some idea of how things work and a question that comes up a lot for junior and senior, well typically math majors is, all right well if somebody just drops one of these in front of you, here is a possible function, star from Q cross Q to Q, how the heck do you know whether it is or it isn't, in other words, you know Mr. Instructor, how the heck did you know to look for a counter-example for that one but to try to prove that it was, okay well, because I came up with the examples, that's the answer, you know the better answer is this, presumably the work that has gone on behind the scenes that I haven't shown you just in the interest of time is, all right I don't know if it is or it isn't, so I'll play around with it a little bit, and the first thing to do is to ask yourself do I recognize this as something familiar, and I typically don't recognize situations where I'm adding in denominators, so I'm not familiar with the process that's being described here, so I don't know, so let's just do some examples, and I happened to pick one out of thin air and right off the bat I found that the two were different so I was done, it might have been the case here folks just by bad luck or good luck whatever you want to view it as that I might have written down two things that were different and they actually did spit out the same thing, that's not enough to convince me that the function is well defined, it only shows that it works for that particular pair, but of course then if I try another pair and it doesn't work then I'm done, it's not well defined, if it works for that second pair maybe that's growing evidence that maybe the function is well defined, and if the function is well defined then you go about proving it, all right, well so you'd say well maybe I can prove it's well defined, and you'd start about proving it, of course if you tried to prove this well defined you'd get stuck somewhere which might then lend some evidence that it's not well defined and so you know if you don't know to begin with you sort of try both ends and quite honestly and this is the heart of what a lot of mathematics research is, you write down some sort of statement you just don't know, you sort of get experimental evidence on one side, maybe it is true, you try to prove it maybe you can't get there so you know there's this sort of middle that you move to maybe that's a good thing to try to get across to the students in your classes as well, you know if you're handed a statement how do you know whether or not it's true you might be able to luck out and prove it right away, this wasn't too bad or you might be able to luck out and find a counter example right away like we did over there but a lot of times it's not so easy you might have to do three, four, a hundred or you know significantly more examples okay, questions? Alright let's look at a few more examples of binary operations, so more examples of binary operations, examples of binary operations on sets, here's a good set, the set S is the set that two by two matrices over R so this is going to be notation that will be standard throughout the semester so it's the collection of square matrices each size two by two where the entries are taken from the real numbers and the operation star, so star of well what do we typically call two matrices, I don't know A and B or M and N or something like that is simply A times B matrix multiplication and this makes sense the point is if I hand you a pair of two by two matrices and I ask you to do matrix multiplication I get another two by two matrix back there really is a binary operation is a binary operation on M2R, now there's absolutely nothing special about two and there's also really nothing too special about R here, so we can generalize in fact here's another set the N by N matrices over R you pick your favorite N pick your favorite N so maybe the three by three matrices or the ten by ten matrices or the hundred by hundred matrices I don't care and the same operation star equals matrix multiplication and you get a binary operation on the set of N by N matrices hey you don't want to use real numbers that's fine use instead of entries from the real numbers use entries from the complex numbers that's fine so maybe M and of the complex numbers or the N by N matrices over the rational numbers or the N by N matrices over the integers so each of these sets has a binary operation in it and we call it binary operation matrix multiplication so I'm viewing these as distinct or separated sets this is maybe the set of three by three matrices whose entries are from the integers there's an example here's another example the set of seven by seven matrices whose entries come from the rational numbers there's another example of a set with a binary operation all the binary operations are somehow they look the same they're all called matrix multiplication sort of interesting here is a notational convenience this gets really cumbersome star of a pair equals something so this notation folks in a binary operation if we talk about star of a pair how about s comma t that's typically what the inputs to a binary operation will look like is usually denoted usually means I think every time for the remainder of the semester simply by taking the star thing and putting it in between I mean when we talked about you know plus of a comma b equals a plus b and why don't we just call it a plus b so star might be plus it might be times it might be vector cross product it might be matrix multiplication it might be a circle denoting function composition it could be a lot of different things because we know there's lots of different ways of hammering things together again another thing heck here's another example it might be the case that the underlying things in the set are subsets of some given set so our elements of the power set of some set and the way I'm asking you to combine them is to form the union of the two sets or the intersection of the two two so this symbol could I mean it could stand for a myriad of things alright that's what binary operations are now it's promised we're not going to study all possible binary operations on all possible sets we're going to focus down a little bit I'll give you some of the verbiage you've seen some of it before the I mean I'll get to the definition of what a group is that there's nothing mysterious here that's the goal tonight is to write down what a group is I'll give you all of the information you need to understand what it is and then after the fact and after we looked at a bunch of examples I'll try to give you some sort of intuition as to why it is that this particular type of structure is sort of natural or become so important I'll try to give you some sort of intuition as to why it is that the particular properties that we wind up listing to give a binary operation as a group turn out to be of interest alright so there are various properties that a binary operation may or may not have operation on a set may or may not you've heard of some they probably these words probably were handed to you back in third grade or fourth grade or something like that for instance some binary operations are what we call commutative some binary operations have the property that if the binary operation if this function is handed a pair a comma b and you figure out what gets spit out from that pair and then the binary operation gets the same pair of numbers handed to it but in opposite order and you see what gets kicked out is it always the same that throwing in the pair a comma b spits out the same thing as throwing in the pair b comma a the answer is sometimes yes and sometimes no for instance if the binary operation is plus the answer is yes if the binary operation is minus the answer is no because you know you do minus two that's not the same as two minus three so that's sort of the typical example of a property commutativity some do some don't here's another example associativity the associative law so this is some magic law that was dropped on you many years ago early in your mathematical career you know the addition of whole numbers is associative you were told that you're not at your head and you had no idea what that meant multiplication of real numbers is associative well all that means it has nothing to do with the order that you write things in all that means is that the order that you group things in isn't relevant so that if you somehow multiply a times b times c it doesn't matter if you first do a times b and then multiply that in turn by c or whether you first do b times c and then multiply by a you get the same answer and lots of binary operations on sets are associative and some binary operations on sets are not associative associativity I mean I can formally write down what it means for a binary operation to be commutative I can formally write down what it means for a binary operation to be associative that'll be easy let me talk you through a couple of more properties that various binary operations on sets may or may not have again what I'm trying to play up here is the idea that we can talk about binary operations but then there are specific properties that we can ask whether or not those binary operations have. Commutativity is a good one to keep in mind because you know a lot of operations that are commutative and you know a lot of binary operations that aren't commutative like well let's see we looked at one the vision on the positive real numbers is not commutative subtraction on the whole numbers whole integers not commutative heck multiplication of matrices is not commutative so there's lots of for associativity there are some examples of binary operations that aren't associative I'll say most of them most of the natural ones are although subtraction turns out to not be if you do a minus b minus c it's not the same as doing a minus b minus c because then the minus c winds up with a plus sign on it so there's some that aren't here's another property think of the following yeah this is a good one the example is the set of even integers so positive negative and zero so any the collection of integers that can be viewed as multiples of two two four six eight zero minus two minus four if you multiply any two even integers get an even integer so it turns out multiplication on the set of even integers is a binary operation take any two even integers multiply them together get another even integer but that particular binary operation doesn't have a very important property one that we looked at on Monday it's missing this in other words it's the case that in the collection of even integers where the binary operation is multiplication there's no special element that somehow well I'll say acts as an identity there's no special element that has the property that when you multiply things in the set by that element that the thing in the set remains unchanged this is just a fancy way of saying one's not in there so in that sense the collection of even integers with multiplication is different than the collection of all integers together with multiplication because the collection of all integers with multiplication has this special element happens to be called the number one here but you know what this set has such an element in it too if we're looking at the collection of two by two matrices over r there is a special element in that set with the property that when you multiply it times anything else in the set you get that other thing back in that situation it happens to look like one zero zero one the identity matrix so it doesn't necessarily have to look like this but in this set there is a element that somehow behaves like the number one does when you multiply so some sets have such a thing and some sets don't have such a thing so I'm going to list that out as existence of an identity element example existence of identity and maybe what I should have been doing here actually how about yes no so for each of these properties let's give a set together with a binary operation where the set has the property and then a set with a binary operation that doesn't here how about the integers together with addition yeah that's easy addition on the integers is commutative give me an example of binary operation on a set where the binary operation is not commutative how about the two by two matrices over r with multiplication is not commutative an example of a set where the operation is associative let's try this one again I'll use z plus a lot it's a pretty famous one addition on the integers is associative subtraction on the integers is not associative now to convince me something doesn't have the property all you have to do is convince me that in one particular case things don't work out and it's easy to write down a pair of two by two matrices you multiply them in one order to get an answer multiply them in the other order you get a different answer and then you're done similarly for subtraction of integers I can convince you that this isn't associative why because if I do something like one minus three minus seven and I compare that to one minus three minus seven let's see what is this this is minus two minus seven so I get minus nine there and this is one minus four and so this is three and these obviously aren't equal so there's a proof by counter example that subtraction on the integers is not associative existence of an identity how about the integers under multiplication I'm going to put the integers under addition here too how about the even integers under multiplication I'm going to denote the even integers by simply writing the number two in front of the integers so that's the even integers multiplication the collection I'm going to put this over here if I look at the integers with addition perfectly good binary operation on the set of integers you add two integers to get another integer back there's an identity element in that set that must be called this this thing has the property that if you take it and you combine it via this operation with anything else in the set you get the other thing in the set that doesn't change okay there's one more example of a property example this fourth example only makes sense in situations where you have a binary operation that has an identity element okay so only makes sense makes sense when the set together with the binary operation has an identity element what do you want to call the identity element well you might want to call it this that'd be reasonable of course in other situations you might want to call it this that'd be reasonable of course in other situations you might want to call it this one zero zero one so what we do is we come up with a letter that in general stands for this special thing called the identity element that doesn't sort of make you always think of this because it might be this because it might be this because it might be a lot of other things we typically call it e for reasons that I really don't know probably because it came out of the Goettingen school and maybe this means identidad or something else made that up anyway it wouldn't surprise me that the genesis of using this letter to denote identity element was somehow from the German alright here is the example look this set has an identity element it's called one here the integers multiplication has an identity element question is it the case that if you hand me anything in the set two or seven or nine I don't care that you can find something else in the set so that when you combine those two things via the binary operation that you get the identity element well in the integers multiplication the answer is no if you hand me the number seven can you find something to multiply time seven to give one no because one seventh isn't an integer on the other hand if I look at the collection of and did a little bit of this on Monday it's sort of opening up the idea for right now if I hand you something like the positive real numbers and I hand you a positive real number is the case that you can find another positive real number so that when you multiply the two together that you get the multiplicative identity the answer in that case is yes so in certain situations each element in the underlying set has a twin or a cousin or another element in the set so that when you combine those two together you get the special one so this is usually called existence of inverses so let's give an example example here is a set the real numbers bigger than zero with multiplication this is not only a binary operation on the set it's a binary operation that has an identity element it happens to be called one and it has the property that every element in the set has a paired element or another element that is with so that when you multiply together you get the special identity element let me give you another example if I look at the integers with addition so the binary operation on the now is addition well this is a binary operation the binary operation on the integers known as addition actually has an identity element it's called this zero you add zero to anything it doesn't change the other thing that's what identity element means question if you have any integer can you find another integer so that when you add two together you get zero sure because if you take any integer just look at its negative which is again another integer and if you add the two together you get the identity element in the set so here are two examples of sets with binary operations each of which has an identity element so that this thing actually gets off the ground and it's the case that for every element in the set things work out nicely things where this doesn't work out so nicely are maybe the set of all real numbers with multiplication and what's up with that is it the case that well first of all is it a binary operation sure just multiplying the two reals and the real back is there an identity element in here you have the number one is is it the case that if you hand me any real number that you can find another real number so that when you multiply it together you get one almost except for zero there is a bad one in here everything else works I admit that but zero doesn't I'll put in parentheses since zero has no and here's a word that we'll use inverse here's another one let's see the collection of two by two matrices over the reals with multiplication here's a binary operation on that set multiply me two two by two matrices over the reals together you get another two by two matrix over the real so it's a binary operation there's a special element in here that acts as the identity to one zero zero one question is it the case that if you hand me anything in here that you can find another thing in here so that when you multiply the two things together get the identity sometimes you do and sometimes you don't I mean if you're thinking well start with zero zero zero zero that one's bad yeah it is bad I agree there's other bad ones too like one zero two zero is bad three six two four is bad like how am I making those up because it turns out folks the question that we're asking here boils down to nothing more than which two by two matrices have inverses and those are precisely the ones that have determined it not equal to zero so the question of existence of inverses turns out to be one even in that situation even the situation multiplication of matrices that corresponds to something that you've seen before is something in linear algebra alright I could I mean I could spend the next half hour listing additional properties that binary operations might have or might not have I've listed out four here I hesitate to give you the commutativity one first and the reason is this in some sense that's the most obvious one that's maybe the one you're most comfortable with because you've heard that word before at least and it makes sense is it the case that if you do things in this order that's the same as having done them in the other order yes or no and there's examples above it turns out though that when we get to this fundamental structure this thing called the group the commutativity does not play a role does not play a role in the definition it'll play a role later on there will be some special groups that happen to be commutative but some are not so we've got to be a little bit careful that's why I hesitate to give this one first but it's just an example these other three things are also examples of properties of binary operations on set so here finally not further ado is the definition of the most important property of well the three most important properties of binary operations that we're going to group together and call any binary operation that has all three properties a group so the definition is this let star be a binary operation on the set s a set s then s with star and that's the standard notation tell me what the set is tell me what the binary operation is that's the notation that I've been using all throughout here I've told you what the set is and I've told you what the appropriate operation is is called a group group in case three things are true in case first star is associative this we know what associativity means but you know what we can write out associativity formally in other words i.e. for any three elements elements I'll call them a b and c in the set capital s if you hand me those three elements and you perform the binary operation between a and b and then take the result and combine it with c that what you get is the same thing as if you had grouped them not necessarily by changing their order but by simply grouping the pairs differently what this makes sense if I've got a binary operation on the set it means that a star b is in the set so it makes sense to somehow take this element of the set and combine it with that one and get an element of the set and the requirement is that whenever you do that for any three elements of the set that you get the same output as if you had done the binary operation by grouping differently I mean you've seen this before with pluses and you've seen it before with multiplications and you've seen it in matrix multiplication Richard question oh good question yes so the question is well what if you don't have three elements in the set and the answer is a b and c don't necessarily represent different elements they might be the same element taken a a and b so I'm not precluding the possibility that I've handed you some overlap so the answer is yes it can be because then the situation simply boils down to well take any two of them and then just put them in the appropriate slots and see what happens but typically we'll be looking at I mean not always but typically we'll be looking at sets that one for any elements sorry alright that's the first requirement on a binary operation that we will need in order for us to call it a group here's the second property that we will need in order to call a binary operation a group that ass contains an identity element for star for the operation in other words well what does it mean to be an identity element it means that there's a special element in there I'm going to call it e as I mentioned before there exists an element we'll call it e in ass with the property that the property that for every element in the set let's call it a in ass if you do e star a in other words if you combine the special one with the other one I don't care what the other one is that you get the other one back and if you combine them in the other order as well a star e in other words the second property that we're going to be interested in is this property of the existence of an identity element and all I've done in both the case of associativity and existence of identity is simply written out using this general star notation what the heck the property means associativity just means group in a different order existence of an identity means there's a special element in there with the property that when you combine that element with anything else that it doesn't change the other thing that you get a and here's property three existence of inverses for each element a in the set there is there exists an inverse inverse here's what that means in other words more formally for each element in the set let's call it a there exists an element I'll call it b for now so that what does it mean to be an inverse it means when you combine the given element a with this paired element with this twin when you combine them together that you get the special element this thing that's called the identity in other words when you do a star b you get e and if you do b star a you get there's lots of comments to be made the first comment is why is it the case that I had to write each of these out twice you're thinking well isn't that enough doesn't that necessarily tell you that and the answer is well no if I compute e star a there's nothing in this list that says that e star a or anything star a has to give you the same answer if you turn things around so the stipulation that this thing is called the identity element means you have to check both that it's the case that when you do the operation with it on the left that doesn't change the original given element and if you do it on the right that doesn't change the given element similarly here you have to check in order for this thing to be viewed as or to be called the inverse of the element a you have to make sure not only that when you perform the operation of a with b in this order that you get the special identity element but that you also get the identity element when you perform the operation the other order so that's a mouthful but here's the definition of a group if you have a binary operation on a set and if the binary operation satisfies these three particular properties of binary operations satisfies all three of them then we call the set together with the binary operation so what I want to do is give you know ten minutes worth of examples and then maybe the last five minutes I'll try to give you some sort of gut feel as to why it is that these three particular binary operations turn out to play such an important role David question oh that's a good question ah that's a great question yeah so in fact what I've done in previous semesters I mean that's an absolutely great question so here's David's question look the definition of a group is a set together with a binary operation that satisfies these three properties in some sense you might want to think of the entire situation as we've got four things that we have to check in order to make sure that the thing that we're looking at is a group the first thing we have to check maybe you want to call it step zero is we have to check that the proposed thing together with the proposed set is actually a binary operation on the set and that sometimes requires a little bit of work I mean what we spent the first half hour of today doing was looking at certain situations where we do or subsequently don't have a binary operation on the set so some authors and some instructors view this as a separate condition but typically it's more natural to say look you've been handed a set you've been handed an operation you've done possibly some work to check that that operation is actually a binary operation on the set alright now we can actually start talking about whether or not the thing is a group so if you're handed for a homework question here is a thing a set and here's an operation does this set together with this operation yield a group technically what you have to do first is a step zero you have to convince me that the operation is actually a binary operation on the set then once you've sort of gotten things off the ground then check is it associative is it the case there's an identity element and is it the case that each element has an inverse okay so that's a great question careful might need to be checked need checking but at least in all the examples that we will do here the set together with the proposed operation really will be a binary operation on the set so that it will boil down to deciding whether or not each of these three properties are satisfied alright examples I can spend hours well examples of groups here's a fundamental example the set of all integers together with addition so we've handed you a set together with a well together with an operation I need to mumble oh yeah addition is a binary operation on the integers if you take any two integers and you add them together you get another integer no big deal there's no well-definedness issues there's no dividing by zero issues so check we get off the ground question is it the case that each of these three properties hold in this particular setting well let's see we just got to go through and check so binary operation check associativity I'm gonna say check and here's the deal folks checking associativity on all these is I mean it boils down to you know foundational mathematics or something inside it what you'll get to do in here for all these is say I'm supposed to check that it's associative but I know it's associated by what they told me in third grade check that's all I mean eventually we're gonna get to is matrix multiplication associative yeah it is because they told me that it was and then or is function composition associative yeah it is because it just is so the associativity issue is sort of a non-issue for us resistance of an identity element so I'll just write it as identity element is there a special thing inside this set that has the property that when you combine it with anything else in the set in either order that you get that thing back sure yes zero is an identity element for the set why because zero plus let's call it z equals z and z plus zero equals z for every z in the set so it means to be an identity element I've identified something that works they might say well how did you know that that was in there well I just I don't know got lucky or I knew enough about this set to point to it and say yep this is in there and it works for it existence of inverses it's third on the list but it's the fourth thing that we actually have to do to make sure that we've got a group here existence of inverses inverses sure here's what we have to do we have to take any let's call it a in z show that there is exists something called b in z so that when you add two together that you get zero and when you add them in the other order you get zero that's exactly what this says notice I've used the notation plus rather than star here because that's what the particular binary operation in this set happens to be how you're going to do this where you're just going to hope to somehow have an algorithm that cooks one up for you well yeah here it is but let b equal negative a the point is minus a is in z I'll put a check mark here this step is key folks because in a lot of situations what you'll be able to do is say well sure I know how to cook up an inverse just put its negative on there or maybe if the operation is multiplication I know how to cook up an inverse just do one over it but that operation might not give you something back in the set for example if the set had only been let's say the positive integers or the natural numbers rather than all of the integers it turns out that all of these three operations are all these three properties are satisfied it's a binary operation it's associated with an identity element but existence of inverses if you try to do negative I mean the negative of a positive integer is a perfectly good thing it's just not in the given set so minus a is in z and the point is if I do a plus minus a I get zero and if I do minus a plus a I get zero so check so done we've now run through all of the steps required to verify that the set of integers with binary operation with binary operation addition forms a group alright other examples examples save this for non-examples examples let me sort of an interesting one I'm going to hand you the following four elements one minus one complex number I and complex number minus I so if you want I can actually draw a picture in the complex plane of what this set looks like I'm asking you to take that complex number that complex number that complex number and that complex number so there's the set called s and the operation is going to be complex number multiplication so step zero is this even a binary operation is it the case that this set consisting of four elements has the property that if you take any two elements and you combine them in terms of multiplying you get something back in the set well technically you've got to check a whole lot of things but it's not too hard to see hey if you do that times any one of them that's no big deal if you do that times that minus one times minus one you get that if you do that times that you get that if you do that times that you get that it's a complex I so it's okay yes just check it just check and folks that's going to be a I mean I don't say that physically it's here's a set there only happens to be four things in it I'm proposing that this set has a certain property how do you verify that it does just pound it out see what happens alright now we'll be able to do this a little bit more subtly or sort of more stylistically about two days but for now that's fine alright how about associativity sure because all we're doing here is we're multiplying complex numbers I'm not multiplying all the complex numbers just some of them but hey I don't care if you've got a subset of something that's associative it's by default associative so that's check since on the complex numbers is associative so that's a good deal let's see identity is there a special thing in this set with the property that when you combine it with anything else in the set doesn't change anything sure there it is if you do one times one you get one if you do one times minus I minus one you get minus one sure one works how about existence of inverses of inverses this one's a little subtler and typically is the hardest one of the four and if you want to call it three properties or four properties to verify the binary operation property together with the other three you have to convince me that if you take anything in the set that you can pair it up with something else so that when you combine the two things that you get whatever you've identified as the identity for the operation in step three alright well let's check that it happens let's see if I start with that thing in the set can I find something that I can combine it with to give one sure hey there's no stipulation you have to pick something different you can pick itself that's fine let's start with the second thing in the set can I combine it with something so that when I combine the two things together I get one sure how about minus one so there is an inverse for that element happens to be itself how about this one is there something I can multiply I can combine with that to give one or is that because let's see I times I is minus one but I've got a minus sign and finally if I take the fourth thing in the set can I find something in the set to combine it with to give one the answer is yes so check I've checked the existence of inverses just by and quite honestly folks this is the terminology that's used in mathematics by the brute force method just grind it out did it because I checked all the possibilities I mean you don't always have this opportunity or this method at your disposal because the set might be infinite or it might be just too big to get your hands around but oh so this is a group so this set s one minus one I minus I together with complex number of multiplication is a group pretty important example of a group too so there's a couple of examples what we will do for the first I don't know half hour forty five minutes of next Monday is look at more examples of groups but as promised what I want to do is let's see give me a minute here yeah okay is try to give you as promised some sort of intuition as to why these three properties I'll assume that we've got a set with a binary operation to start with happen to be somehow natural together so this is a sort of final thought or a quick aside about why these three what's so special about grouping them together and I like to view it as this whenever you're in a group you're in a situation where you can always solve linear equations so think of it this way if I start with an equation that looks like two plus x equals five there's a linear equation so the variable in there just appears as sort of order one can you solve that equation yeah you solved the integers x equals three no big deal but can you solve the equation five plus x equals two in the positive integers no because you might not have negative three in there so think of it as the positive integers are pretty good but they don't allow you to solve all possible linear equations even if the things that you're using are positive integers because you can't solve five plus x equals two so what do you do is sort of expand out you throw the negatives in together with zero and what you wind up with is a situation where you can solve all linear equations a plus x equals b you can always solve that equation in the integers how about an equation that looks like a times x equals b so there's a linear equation corresponding to the operation multiplication well you certainly can't always solve that in the integers two times x equals one the answers I have alright so you say well how about if I try to solve it in the rational or in the reals two times x equals one yeah I can solve that linear equation the reals of course I can't solve zero times x equals one so that's unfortunate I mean it is unfortunate so there's a situation of an equation a linear equation in the system that you can't solve it turns out that these three properties associativity existence of an identity and existence of inverses precisely the three operations that you need are the three properties that you need in order to conclude that in the system that you're working in you can always solve a linear equation in other words these three together allow you allow you to solve any linear equation in the system in the system well because we typically don't call the generic binary operation plus or times or matrix multiplication we call it star ie we can solve a star x equals b for any choices of a and b if you're in a group you can always solve this how do you do it well what are you trying to do you're trying to somehow figure out what value of x works so how the heck are you going to solve it well what you do is you take whatever is the inverse of the element a you somehow multiply both sides of the equation by this you then use associativity you then use the existence of an identity and you wind up getting a solution for x let me do that for you the one minute version so I want to show you how to go about solving this equation and I want to convince you that I'm going to use exactly the three properties that are inherent in the definition of a group by definition of a group this thing has some what do we call it inverse what do you want to call the inverse element let's call it c because I've used b for something else here so let c be an inverse for a so I've used property three of a group now multiply I shouldn't use that phrase now perform the binary operation on the left with that element c so if I have two things that are equal then I perform the binary operation that's legit then the equality remains so I've used property three now property one says associativity so by associativity I can regroup this left side I get c star a star x equals c star b but wait a minute these are inverses so what's c star a is e but wait a minute what happens if you combine anything in the system with e you get it I don't care what it is so what I've just done is I've solved the linear equation for x I've isolated x in fact I've given you the answer it happens to be c star b where c is the inverse element of a the point is these three properties together actually are natural they allow you to take anything in the given system regardless of what the underlying set is and regardless of what the appropriate operation is whatever the binary operation is it allows you to solve things that look like either a plus b equals c or a a plus x equals b or a star x equals b or a times x equals b okay questions comments alright yeah that's a good place to stop then so I gave the homework assignment out on Monday if you need to get a copy of that it's on the web you can grab it if you didn't get this information about the si sessions you can come up and grab one now I have office hours tomorrow if you want to come by and if I don't see you before then I will see you on Monday