 So welcome to this course on abstract algebra. Now we're going to start with the sets. Now sets are not always the most exciting thing in the world, yet if you open a textbook on discrete mathematics, on real analysis, on group theory, and indeed on abstract algebra, chances are it's going to start off with sets. So we've just got to become familiar with sets. Although as I say it's not the most exciting thing, it's actually quite important and it really will help us as we ease into proofs, and of course proofs is a very important part of mathematics. Let's have a look. Now we're going to define sets in a certain way in this course. So let's do that. Let's say our sets and how we're going to define sets is this, we're going to say it's a well-defined, so well-defined collection of elements. So that's one way to define a set. Now there are various types of sets and you can read up about that. For the purposes of this course, we're going to know exactly what goes inside of our set and what is not inside of our sets. So if that is a set, just another word that is sometimes used as a class, but we're going to use this term set, is what is an element? What is an element? Well an element is just the building blocks, the building blocks, building blocks of a set. And that is the elements, that is the building blocks, that is how we populate a set. So I'm going to use some notation in this course. So let's just have a look at notation because depending on where you study, what book you read, there's going to be different notations, the notations that I'm going to use here is always a capital letter. It's going to denote the name of my set and if I list the building blocks or the elements, I'm going to use curly braces and let's say that the following r, s and t are elements, they are elements of my set. I might also write something like this, r, s and t, r and s and t, so that's and are elements, they are elements of my set a. And if I have something like p, p is not an element, I'm going to write that p is not an element of a there. So let's construct a nice notation. Imagine I have the building blocks, the elements 1, 2, 3, 4, 5, how can I write that? I'm going to call this my good example a1, so a subscript 1 might also be how I write an element and that is going to equal all the elements x such that, x, so that's such that, x is an element of the natural numbers, my double struck in there and which is going to be a, x is less than or equal to 5. So that is a well-defined set and that is good set notation because a1 now is really going to be the elements 1, 2, 3, 4 and 5. No problem there. Let's construct a set where we say let's take all the natural numbers divisible by 5. I'm going to call that a2, a2 and I can write that as such. All elements of x such that x is an element of, we said we wanted the natural numbers and x is divisible by 5. So here really we're going to have a2 is going to equal, 5 is divisible by 5 without a remainder, 10, 15, 20 and it goes up to infinity and we use the dot notation for that. So that brings us to these sets that we're going to use all the time in abstract algebra. That is the set of natural numbers and that means 1, 2, 3, 4, 5 up to infinity. So we start at the number 1 for natural numbers. Of course that means we must have the integers double struck z and that is just going to go from negative infinity, say negative 2, negative 1, 0, 1, 2 all the way to positive infinity. That brings us to the rational numbers. The set of rational numbers denoted there by double struck q and that is going to be all the elements x such that x is equal to p over q, p over q and x is an element, we should say p there, p is an element of the rational numbers. I should say the integers, q is an element of the integers there and q is not equal to 0. So we've all seen those. The rational numbers is this equation of integers and then of course we're going to have the real numbers and that just falls on the real line. So all rational and irrational numbers, remember an irrational number cannot be expressed as the quotient of integers. So that's really what we're going to use in this course on abstract algebra. Next up I just want to remind you when two sets are equal. So if I have a set A2 here and that set equals the elements 1, 2 and 3 and I have the A sub 3, my second set there and that contains the elements 1, 1, 2, 2, 2, 3, 1, 2, 3, those two are exactly the same. When we have repetitive elements like that we only view them once. So this A3 is actually going to be the same 1, 2, 3 and elements are the same. They are equal to each other if they have the same building blocks or the same elements. So these two sets, although they appear different here, remember by definition we really are going to view them as exactly the same. So just as some of the notation, just so that you can be on board with what's going to be used in this course, it's always different for all sorts of courses and all four sorts of textbooks. But this is what we're going to stick to. Next up we're going to have a look at subsets.