 Okay, so let's have a look at a very interesting topic and let's cosets. Cosets. The emphasis here is on sets, we can have a set. And there's a few things, this goes somewhere which is quite important but as per usual it takes a while to get there. So we're going to define a few things. Again we're going to have a group it is going to be a group and it contains a set of elements and some binary operation on that. And we're going to have a group H which is a subgroup of group G. I'm going to use that less than sign as subgroup. I'm saying that as a subgroup. I'm also going to have a relation and we're going to have, we're going to use this tilde squiggly line and they are left and right cosets by the way. So we're going to just concentrate on the left for now anything we do you can just do in the reverse and you can show the right as well but let's stick to this relation and the relation is going to be on elements A and B elements of G such that or if and only if A inverse binary operation B is an element in H. That is how I'm going to describe my relation. I'm going to take two elements of of G and I'm going to say that relation is only for if I take two elements there and I take the inverse of A binary operation B that that product is and I use product some group composition remember binary operation can be anything is an element of H and then furthermore I'm just going to state the following. I'm going to state that this is an equivalence relation and if it's an equivalence relation I have these equivalence classes on G and if A is an element of that I'm just going to say I'm just going to write it like that that's an equivalence class on this group G and it contains the element A. Remember I take its inverse binary operation with B and I say that's an element of H that is how I define this and my claim my theorem in the end is going to be that this equivalence class which is a set remember it's a set of elements is equal to the set of A binary operation with H such that H is an element of my subgroup. I'm stating this is a set and I'm stating that's a set and in the end I'm going to call them cosets and in the end I'm going to call this a left coset. Okay first of all before I prove all of this and if I say these are two are equal to each other I've got to prove it in this direction and I've got to prove it in this direction. First of all is this an equivalence relation? This video is just going to be about is this an equivalence relation? For this to be an equivalence relation we have to have the properties the reflexive property. Reflexive property let's see if that holds I'm going to say in other words I'm looking at A and A. Now is that is that true what does this mean? Well it says what I have to do here is then if this is my relation I'm saying A inverse binary operation A. Well I set this up as a group if A inverse is there then definitely A has to be there and this is just the identity element E and the identity element we know is inside of H because it's a subgroup it must have that same identity it must have that identity element so the reflexive property clearly holds E as an element of H. The symmetric property symmetric property does that hold if I have A, L with B does that imply the B equivalence relation with A is that so? Now what I'm asking here is if I have this relation remembers A inverse and B that is an element of H that is how I set it up this chalk is flowing right to my eye anyway if this is in that set then it's if that is an element there its inverse must also be an element of H and remember how we define an inverse we looked at this before I take the inverse of the second one and then the inverse of the first one that must be an element of H and lo and behold B inverse the inverse of the inverse is just A and that's an element of H so what do I have here well I have just shown that if this holds then that holds because this is nothing other than saying B this relation on A so we're shown that the last one is the transitive property transitive property what we have here I'm going to say if A this relation to B and I'm just using set notation and B this relation to C this implies A's relation with C C so can we show that what are we saying here we're saying A inverse B and we are saying B inverse C we're saying that's an element of H we're saying that's an element of H well if they're both elements of H if I take the binary operation between two of them by the closure property that should obviously also be there that should obviously also be there so if I have A inverse B and that with B inverse with C and I say I claim that that's an element of H and by the associative property of groups I can do this first that gives me the identity element which leaves me with A inverse C as an element of H therefore I'm showing that there is A this relation to C and that's an element of H so all of them hold the reflexive the symmetric and the transitive and the transitive property so really I have shown that this is an equivalence relation and you can do it the other way around as well when you do R it's B inverse A everything swaps around it's easy to prove that that's an equivalence relation so I have really shown that this exists it is a set here with elements A in it that's how I've defined it it is an equivalence relation it's an equivalence class and now in the next video we're going to show that this is a set well this is a set but that it's equal to doing this it's equal to this set and I'll show you how to do it in this direction and then you take this and end with that or you take this and you end with that because if you want to prove that two things are equal to each other the best way that we are going to do it is if you want to show that two sets are equal to each other remember these are two sets is to show that A is a subset of B and B is a subset of A and if that is so then A equals B and that perhaps by the way kids you can see if you can if you can set that up very easily I'm just going to show you how it starts you're going to take X as an element of this A and then you continue and the same you're going to do with this side and then get to that side but we'll get to that video first of all we've defined this we're showing that this is an equivalence relation and we're going to end up with this set which we call a coset and then we're going to call this a left coset and then also you get the right coset and the right coset well we also write this as a h h and you also I'll check that out and you also get the right coset let me not get into that now before I get too confused we'll look at all of those once we've proven once we've proven this this is by the way this is the left coset because the A which forms our equivalence this class here is on the left and then it's going to be on the right which we just write as h as h a and this is a h but we'll get to that that's not important we've shown that this is an equivalence relation