 Now if you've made it this far, congratulations. I told you in the beginning this first stuff is absolutely If I say boring I think many people are gonna be angry at me because certainly they you know Some of it is interesting, but you've got a way through these things and you've got a wonder to yourself Where is all this going? Does it is going to mean something? This is abstract algebra So when do I get to groups? But this is all the foundation that we are building. I mean this is one way to study abstract algebra You just build this foundation until you get to groups rings fields all of these things or you can just jump in there And then you have to attach all these things Onto it so that you can understand it properly. So in this series We're just building it from the ground up, which means in the beginning. It's awkward You don't know how to do it. You don't know why you're learning these things is this these You know definitions and abstract things that you've got to get together What I do want to say is congratulations You've made it here because we're gonna actually get to one of the more interesting things although it is still part of the building blocks and This thing that we are going to get to is called Operations and I don't just say operations. That means surgery and surgery is what I do for a living It means operations are actually actually quite interesting and you actually know operations before The addition of two numbers the multiplication of two numbers. Those are operations and operations are Nothing other than this product set onto itself. So if I have a set S and I take its product set with s with itself what I'm doing is I'm mapping s into s So, you know, give me one element of s I do some operation with another element of s what I'm ended up with is another element that is just inside of s I mean if you looked at the set of If you look at the set of real numbers and I told you well, let the operation be addition and I'm going to say well 2 plus 3 and that equals 5 Well 2 is an element of R if the real number was my s here 3 is an element of that I do this operation to them and I get 5 and 5 is also an element there So we say that these operations are or these sets are closed under a certain operation So remember that term you're going to come across it many times it is closed under the operation under the operation That means that if I do that operation on a set that I get to another element that's also inside of that set That's also inside of that set. So I'm just mapping a set to itself Product set see how we build all of those things up. They mean something to us now It means we have a better understanding perhaps of what is going on of what is going on here now the specific operations that we are really going to deal with More proper term for that is going to be a binary operation a binary operation Okay, binary in as much as we're going to take this operator on two elements So what we are going to then have basically is this Mapping of two of of a pair of those elements So I could have said that this mapping here under addition is going to give me this operation Two plus three and that is going to map to another element in that set, which is five now you'll come across This idea that we move away from plus and multiplication and we built this generic operation And we have signs for that either it's a little circle or it's a square block You'll see many of these stars sometimes It just means that this is a binary operation on that set so that we can get away from we are we that arithmetic and Everything that you learned before functions. Remember, let me talk about that. I mean we had a function f of x equals three x Other words y equals three of x all I'm doing is I'm building these Product sets I'm building these product sets because you give me One and three and I say well that maps to three So this is an r. This is an r. This is an r But I want to move away from this being a function I want to move to something more generic which is a mapping because that allows us to map all sorts of brilliant things. I mean Mapping of divide is divisible by For instance or just arbitrary mappings between between elements of a set That's just so much richer than just sticking to functions But you were told what functions are you work with functions But just see that as part of something a much bigger and a much more exciting world So let's just look at a few operations. So one that I've listed here is The Addition let's take let's take the even natural numbers under addition even Natural numbers I'm just going to write that they are closed Closed under addition Why is I can take any two elements In that set so two plus four that's going to give me six and six is also in that set If I look at the odd Natural numbers under addition that is sitting not closed under addition one plus three gives me four one Isn't that set three isn't that set but four definitely is not is definitely not in that set Let's have a look at a set a and my set a is just going to be the set zero one two three and four And I say is that set closed under addition Think about it. It is not closed under addition because I can say Two plus three equals five and that is not an element of a Is it closed under multiplication? No, it's not closed under multiplication because two times three Equals six and that is definitely not an element of a So you see where these binary operations where these binary operations are going I think in the next example, I'm going to show you quite an abstract Set and the operations on that set and you'll get an even deeper understanding of this very intuitive Way of looking at it. So operations First bit of excitement as far as I'm concerned in abstract algebra