 Okay, so we've had two very boring lectures. Let's be on let's be brutally honest about this nothing really exciting yet But we've got to ease our way into this Absect algebra and it's important to know about states the next thing on my list my to-do list here is universal states universal states Well, I should just say universal set now It's not always defined but in this course and really to to understand things and get things done We have to have this universal set and we say that this universal set you which is not an empty set It's defined so that we can define subsets We're just going to define it so that we can define subsets So I might suggest that I have this universal set and I'm defining it as one two three four five So that I can have a Subset yeah, let's make it a proper subset and I'm going to have that equal to one two and three One two and three and I'm going to say that a is a proper subset of my universal set you And that's why we are always going to even if we don't stated We accept that every set that we are going to use in this course Does this part of is a proper subset of some bigger set? Which is what which is never going to be an empty set and which we are going to define as the universal set That means I can define a new thing. I can define the complement of a I Can define the complement of a and that is going to equal the set of all x such that Not and well, that's horrible such that x is not an element of a But x is an element of the universal set So it must be one of these, but it's not an element of a and that is how we define a complement And we can only do that now remember a complement is also a subset of my universal set It's also a subset of my universal set and that is why we have these universal sets so that we can start talking about Sets as proper subsets of something so that they can also have a complement. So this a prime or a complement Is what we're going to call it here is just then four and five four and five and This is very important. This is going to be quite important The complement is the set of all elements x such that x is not an element of a Because that brings us just to getting the union of two sets If I have the union of a one union of a two that is going to be The following you can to define that as x such that x is an element of a One a sub one and it's a union. So x is also an element of a two and It cannot be and can it we have to put something else there So I'm not going to have any kind of notation So I'm just going to use all or all and we know that from Venn diagrams If this is my universal set you that's my universal set you if I have a one and I have a two There's my a one a two the union of these two means that any element x no matter where it is It's in there in there in there anywhere in here It's either then an element of a one or it's an element of a two And that's actually an element of both So it's going to be all over these so it's in one or the other one which brings us to the intersection a Intersection a one a two a one a one a two now the important point doesn't matter which way around it goes That is going to be the elements all the elements x such that x is an element of a one and So I can put my comma there as well x is an element of a two So now it's got to be in both so there is this distinction between all and And if we look at our Venn diagram here my universal set a one a two a one and a two it's going to all elements x Such that it is in both it must be an a one and an a two and it's only this little bit here The intersection here that we're going to be interested in all of this is really simple stuff What is going to be important for us is these definitions these are how we define we as beings of this planet define that that is the union of sets and that is the intersection of sets That means we can use this in a proof because this is how we define it We define it in no other way and if something looks like this it is this if something looks like this It is that that is how we going to define that because now I can use the following I can take an Subset or set at least a one and I can subtract from that a two Imagine what happens here in this Venn diagram This is to another one you can just see what this is my universal set This is a one this is a two a one a two if I take all of a one and I subtract everything from a two Which means I take away all of this I take away all of this. So this is all that is left So that's all that is left and I'm going to define that in the following way I'm going to say all the elements x such that x is an element of a one So it can be an element of a one, but at the moment. It's all of those all of those and It cannot be x is not an element of a two that is my definition of Subtracting one set from another because it can be an a one, but it cannot be an a two So this little intersection is excluded as well a one minus a two that is going to be my definition Now I can also say the following I can also write it in this way a sub one minus a sub two that is going to be a Intersection with B prime a Intersection with B prime that looks a bit weird and certainly will have to prove this and we're going to prove this By these statements. These are our statements. We accept that as a fact that is how we define those are axioms for us Basically, that is how we define that is how we define and We say that these two are equal to each other and we'll certainly have to prove that. Let's just look at the Venn diagram We've now decided that this so intersection first of all The intersection of B prime. So this is B everything that's outside of B Meaning it's all of this and that's why we need the universal set in this course So this is everything outside of a two everything outside of a two So it's all of this all of this all of this all outside of that It's intersection so X can still be anywhere. So the intersection of all of that and a one That leaves that part out because it can't be an a two So on a Venn diagram, we can really see that these two things are equal to each other But we will have to prove that and our first proof is coming up