 So what else can you do with a coset to see what you can do? Let's go ahead and take a look at a very familiar group the set of integers with the operation of addition So let's define nz to be the set of integers nz where z is an integer and because we're working with the operation of addition We know that this is a subgroup. You know, this is true because it's well because you can prove it Now let's consider what the cosets of nz look like and so Left cosets my operation is addition So first of all my set itself my subgroup by z looks like well minus five zero looks like all the multiples of five and My group operation is addition so I could take something that's not in this set We'll have about one and add it on the left to everything in here So this is one plus five z everything in here I'm gonna add one to it and I get my first cosets and they look for some of the guts in neither of these two Well, how about two looks like it's not in there So I'll add two to everything in five and I'll get this and likewise. I can find my cosets three and four Now one thing that's worth noticing here is remember all of our cosets have the same size I've taken this set z and I've partitioned it into one two three four five sets of equal size And part of the process of being a mathematician is looking at something and ask yourself where have I seen this before I've taken Something and I've broken it up into five sets of equal amounts in all of them Well, this is really looks like a division and it seems like we've taken this set of integer z And we've somehow divided it by this set of integers five z So I've taken this I've broken it into pieces that all look sort of like This so it looks like we've done that division and so our five sets above form our quotient Thing I don't know what to call this right now, but we'll come up with a good name for it later But it is very similar to what we consider to be a regular quotient now No surprise as this is is an abstract algebra course the question that we do want to ask is can I make a group out of These things so I have a bunch of things here How can I make these into a group and so let's begin by trying to develop some notation? We want a somewhat more convenient way to represent the subgroup and its coset So remember our subgroup in the cosets They were the subgroup five z and the things that we got by adding an element that wasn't in five z to them now We might consider that this added element is what's going to distinguish the cosets because cosets are either disjoint or Identical once I know what element I'm using here. I can use that to represent the entire coset So I'll represent this coset with a one this with a two this with a three and so on Well, actually, that's not really great notation because I am liable to confuse this one Which is really a representing one plus five z with the integer one so to avoid confusion what I'll do is I'll throw these Representatives into brackets now consistency counts. What do I do with this thing? Well, I got this thing by well I got everything else by Something plus what am I going to do to get five z well I can add zero to it and that means the way I can represent the group itself is Zero plus and my group subgroup itself is represented by zero in brackets. So here's my cosets and Using the notation that we have I can say that the quotient thing z divided by five z gives us this set zero one two three four Well, we're not quite there yet The problem is that we claim that the only distinct cosets are these But we can form a coset by adding any element we want to to five z So I could talk about a coset like 31 9 7 8 74 plus 5 z That's going to be this coset and it's got to be one of these Because all of our elements of z ended up in one of these five cosets, but which one is it equal to? And what we want is some easy way of determining when two cosets are equal Well, we can start at the end. So I want to find that two cosets are the same thing Well, one of our previously proven lemmas or theorems said that if two cosets are the same thing It's got to be because the element of the one coset is going to be in the other coset So I know for example that P has to be an element of Q because now these two cosets are not Disjoint there is something in both of them and as soon as they something in both we know that the two cosets are equal so I know that P is in the coset generated by Q and Well, Q is a set so remember we got Q by Adding an element of 5 z to Q So Q plus something in 5 z is going to be this entire set So in order for P to be in this set P has to be Q our representative plus Something from 5 z and the important thing about that is maybe we're not so comfortable dealing with cosets here Because a kind of new idea is maybe even here this still deals us with set With set operations and maybe we're not so comfortable with that But this P Q and K these are all things in the set of integers So I'm perfectly comfortable working with things like that. In fact, I know I can do some algebra P minus Q is equal to 5 K Why this choice? Well, we know that we're going to compare P and Q later on So we want to be able to say if I know what P and Q are then I know something else and I know P minus Q has to be 5k and what that translates back into is if I know that P minus Q is a multiple of five Then I have this first line and I can proceed straight to the end Now it's a good habit to get into to add some notes to make sure you can actually take these steps So let's see P minus Q equals 5k Well, I am assuming that P minus gives a multiple of five and so that says by definition P minus Q has to be five times something I Can do a little bit of algebra make sure that this is actually algebra in the integers We're allowed to do that P Q and K. These are all integers. So we're allowed to do that Our definition of what this coset is is it's Q plus some multiple of five Well, that's Q plus a multiple of five So that says P has to be in our coset and we have our lemma about disjoint cosets Guaranteed that once I have a single thing in there I have everything else in there once you let the camel's nose into the tent the rest of the camel shows up Now if you want to be a good mathematician one of the things that you also look for is any time you can reuse a set of steps and in Particular because we started at the end and worked our way backwards and then worked our way forward to actually complete the proof Because we worked our way backwards the first time we can actually reverse the proof steps So that tells me suppose I know two cosets are equal then I know that P is an element of the second coset and I can get the Similar result to what I started with as a conclusion. So what does that mean? Well, I start here P minus Q is a multiple of five I end here the cosets are the same or I start with the cosets are the same I know that P minus Q is a multiple of five. This is an if and only if proof. I start here I end there I Start where I end it I go back to the beginning because I can prove both sides of the conditional I actually now have an if and only if proof I get the converse and So that gives me the following nice lemma Which is that if I know that P and Q are left or right cosets of Z mod Zn Then I know that they are equal if and only if their difference is a multiple of whatever n is And so far so good. What that means is I now have a way of being able to compare to cosets Now, what do I need? Well my cosets are elements, but I do need that binary operation So if I can introduce a binary operation, I can form a group Well, here's where things get a little bit complicated because introducing that binary operation does involve us in some complexities, but the good news is that is us a way of doing form mathematics So let's take a look at that in the next video