 Okay, we've just seen how an equivalence relation divides our set into these partitions and these partitions We can also call these equivalence sets. So I just want to or equivalence classes Just want to show you this notation that you might come across that all these equivalent classes We add all of them together if we if we get the union Of course we get a and the intersection of any pair of them is going to be the m the empty set But you'll see these square brackets around it and that just states that it contains So this subset which is an equivalent set Contains all the elements y such that y is an element of the largest set a and we have this relation Y with the relation with a so you're gonna you're gonna see that and you might be asked to prove a few things the first one Which might one might have to prove that a? the element a isn't The element a is an element of this equivalent set Or equivalence class a now that's trivial because you just say by the definition a is all the elements y such y is element a and y That's this relation with a so a is inside of that a is going to be one of the elements in the set So that's just by the definition much more difficult. It's going to be this proof That be It's not really difficult If B is an element of this set a the subset or equivalent set then actually B equals a or a equals b So remember this a is going to have all of these pairs These relations and a is going to be in there I'm saying if B is an element in there and I've and I make this equivalent set with B in it Then actually it is not it's nothing new. It is actually just a So what are we going to use to prove that but remember we are dealing here with equivalence relations So let me just write that in green because that is what we're going to pull out of our little box equivalence relation Equivalence relation remember they've got to be they have this property that they reflexive that they symmetric And that they transitive So We are going to use at least these two properties. That is what we've decided what they are So we can use them when we do our proof So what were we given? We are given that B is an element of a And that implies you have to write that when you write the archer proof that implies that we can write this relation B R a Okay, that is a relation that exists Now let's take let's take something like X. Let X be an element of this set B If it is an element of that set the equivalent set B, then I can let implies that I can write XRB And if I can do that look at this I have at the at the moment this relation and I have the relation B A but This is an equivalence relation. We are told that it's equivalence relation So by the trap by the transitive property, therefore X I can like XR a But remember X was an element of B. So what does this imply? Well, it applies at the moment that B is a subset of a That B is a subset of a so I've got this So I've got to go the other way this time around I've got to go the other way Again, I've got to write given that B is an element of a that implies that I write all of that B So that I can write this relation But by the submit the symmetric property that implies that a are B Now let's get a new element in a I'm gonna say let Y be an element of a And what does that imply that implies that there is this relation y are a So I've got this relation y are a And I've got this relation y are a but I've also got this relation a are B So I've got this relation y are a and I've got this relation a are B and Then by the transitive property by the transitive property that implies that we have y are B but remember y was an element of a so I've just proven here now that this a is a subset of b and by that other little piece of knowledge that I can pull out of my box the b is a subset of a and a is a subset of b therefore a equals b so we've just shown that if b is an element of this equivalent set then I have created nothing new I'm still set with this equivalency b they are exactly the same thing there might be something else that you have to prove which is quite a long proof which I'm not not going to do here you might look at the proof in your book you might find that you think how on earth that someone come up with this took a long time so for you just to do it in a snap it's not that easy that if we have that this intersection of these two is not equal to the empty set if that is not equal to the empty set then at least let me just see this long proof here then well let me just finish here then at least a equals b and that's quite a long proof that you'll find in many textbooks I don't think it's fair to ask you know for that as a proof in the exam have a look through that but you can you can think about the logic of course behind us because remember we said that if they have got to be the empty set to start off with if you take the intersection of any of the two partitions or equivalent sets inside of a larger set then the intersection should be an empty otherwise it's not a partition so you can start thinking how to put all of this together but it's actually quite quite a long proof