 Welcome to our lecture series Math 4220 Abstract Algebra 1 for students at Southern Utah University. My name is Dr. Andrew Missildine, and I will be your instructor for this course. Before we jump into the content, let me just tell you a little bit of what to expect for this course. Math 4220 is the first of a year-long sequence in Abstract Algebra, the sequel being 4230 Abstract Algebra 2. Take a look for the links to that course if you want to if you want to view the videos for that as well. Abstract Algebra this year-long sequence is an undergraduate sequence for students at SUU where students will be exposed to the basic ideas of groups, rings, and fields with their accompanying proofs, theorems, exercises, examples, etc. What have you. This is an undergraduate course, so we'll take a much more elementary and slow approach to introducing these things as opposed to those in a graduate course, of course there. And so also mention that our lectures are going to be primarily based upon content from the open source textbook Abstract Algebra Theory and Applications by Tom Judson, for which you can find a link to his textbook in the description of all of the videos in this series. Now Judson's textbook will take the same approach that we do that our coverage of Abstract Algebra will be a group's first approach, meaning that we're going to start with the most fundamental Abstract Algebra objects and add axioms on top of that over time, which means primarily we'll be talking about groups this semester and we'll end the semester with the introduction to rings and fields. And then in the second semester, 4230, we would continue our development of rings and fields, other related objects, like vector spaces, we'll say a little bit more about groups as well. And that's the basic structure of this course. Now starting with the most fundamental of all objects here, section 1.2 of our textbook entitled Sets and Equivalent Relationships. With this section, we're going to start off with the idea of ace. Now in Judson's book chapter one, entitled Preliminaries, is supposed to get us warmed up to many of the objects that we have to be aware of in the Abstract Algebra setting. Section 1.1, entitled, what was it? It's a review of some ideas of logic and proofs. It's very, very short. And we will talk about all those ideas spread out through section 1.2. We'll actually spend a couple lectures, the first three or four lectures, I believe, in section 1.2, as we review a lot of these fundamental ideas, because this will be important review for students of Abstract Algebra. So the most fundamental of all structures, really from a mathematics point of view, is going to be the idea of a set. Now, formally speaking, our notion of a set will be that notion derived from the ZFC set theory axioms. If that doesn't mean anything to you, that's perfectly fine. We aren't going to dive into that deep, deep, deep set theory whatsoever. Admittedly, in the next semester, we'll talk a little bit about the axiom choice in its context, mostly to Zorn's lemma. But for class, really, all we need to know about a set is that a set is a well-defined collection of distinct objects. Those distinct objects, we typically call the elements. And this is a chemistry analogy that the elements are the things that make up the set. And they really are the minimal objects. You don't really break down the elements of a set anymore. They belong to the set, and that's what it is. What do we mean by well-defined? This is actually what got set theorists into problems a long time ago. Things like Russell's Paradox come into play if we're not very careful about what it means to be well-defined. So the classic example about Russell's Paradox, as a professor explained it to me once upon a time when I was an undergraduate student, was the following. If you have a crazy man who runs into a barber shop and he decides to shave off everyone's hair who's not shaving themselves, you can imagine it's most people inside the barber shop. But what about himself? If he's in the barber shop, if he shaves off his own hair, then he shouldn't be cutting off his own hair, so okay. But on the other hand, if he doesn't cut off his own hair, that means he's not cutting off his own hair and therefore he should cut his own hair off, right? There's a logical paradox with our crazy barber there. And so it's not well-defined whether he should slice off his own hair or not. And so it turns out that in early very naive set theory, sets were often defined in this ambiguous way and therefore to be a well-defined object meaning that there is a clear rule that enables one to determine whether a given object is an element of the set or not, like with our crazy barber who wasn't well-defined whether he should cut off his own hair or not. And there has to be a clear rule, right? Definite rule. Absolutely it is or absolutely it's not. Now that rule doesn't have to be an easy rule, right? It could be like a three-hour computation to decide whether you belong to the set or not. It could be a year-long computation. There's many open mathematical problems where it's not certain whether an object belongs to a well-defined set or not, right? People are still actively looking on it. But the fact that the rule of containment is easy or hard is beside the point. To be well-defined means there is a rule that we can decide if there are not. Now one important set that we're going to see a lot this semester that I'll introduce now and we're going to see it a little bit later is the idea of the empty set. The empty set is the set that has no elements. So no elements belong to the empty set. And so we define this typically using it's like a circle with a slash through it. I usually draw it like this. The latex symbol here looks more like a zero in some fonts, but the empty set would be a circle with a line through it. Sometimes it's denoted by two curly braces with nothing between them. That notation will make a little bit more sense in a moment. That gives us our empty set. Now talk about general sets because the set itself is this collection of things. It's a family of elements. We talk about the set by describing who belongs to the set. So a set's usually depicted either by listing all of the elements inside of a pair of curly braces, kind of like you see right here, a little bit of notation. We typically will denote sets by capital alphabetic letters. We might say things like A, B, C, X, Y, Z, just as examples of sets, typically capital letters. And the elements of the set are going to be typically denoted lowercase letters. And if there's no confusion, we might try to use mnemonic devices like little x belongs to capital X. That one can be a little bit more confusing because little x and capital X look very similar in font style, but we might say something like little a belongs to A. And this is usually meant as a mnemonic device to help us keep track of which element belongs to which set when we're talking about multiple sets simultaneously. One way of describing a set is simply just to list all the elements that belong to the set. So for example, set A equals the numbers one, two, and three, something like that. And be aware that the elements that belong to a set can be anything. They could be numbers. They could be people. They could be Pokemon from the Galore region or whatever, whatever we're trying to talk in the in the current context, right? When it comes to listing all the elements of a set, this works out really well when you have a finite set because it might just be a finite list of numbers. But even if the set's infinite or it's just a very large finite set, it can be difficult to list all of the elements of a set. So one might say something like the elements of my set are going to be dot, dot, dot, comma, negative one, zero, one, two, three, dot, dot, dot. Sometimes we use things like ellipses to indicate that there's some pattern associated to the elements of this here, describing the set of integers. We want all integers. And so I listed a few so you can kind of predict the pattern. Dot dot aka ellipses approaches isn't the most well-defined way of describing a set. Because whenever someone uses dot, dot, dot, there's an inherent pattern that the author is referring to. But that pattern could very well be ambiguous. For example, if we describe a set in the following way, c equals one, two, dot, dot, dot. Well, what's the next element of the set? You might be like, oh, it's three, four, five, six, it's doing the, you know, it's just doing the usual counting sequence of positive, like one, two, three, you might think is the next number. But why not like one, two, four, maybe eight? Like it could be a geometric sequence for all we know. Or it could be like one, two, six, it could be the sequence of factorials. The next number is three factorial. The next one's four factorial. Or it could be like one, two, three, five, eight. Maybe it's the Fibonacci sequence. We just forgot the first number one or something. You know, if you don't get enough information, the sequence in play can be very ambiguous. So we try to avoid that when it's possible. But again, context, we often play on that into concepts. A much better way to describe a set is in the following idea that we describe the set using some type of criterion, some type of criteria rule. So imagine that P is a statement of some kind. And so we might say that X belongs to the set if X satisfies some statement, some property associated to that element. And we'll see some examples of that in a little bit. This is often referred to, you see the green circle there, set builder notation. We describe the elements of a set according to some property. And this is going to be the way we describe sets probably most commonly in this mathematical setting to avoid some of the ambiguities that can happen when a set is too large to list all the elements in there. Now the set is really determined by the elements that belong to the set. And so because elements in a set are themselves distinct, we do not, we don't let repetition occur inside of our sets. So when you look at sets like this, one, three, two, one, three, two, we make no distinction between these sets. Now when I say we don't allow repetition, that doesn't mean elements can't be repeated. What that means is when elements are repeated, we make no distinction there. It'd be like saying like my favorite pizzas, let's see, I love pepperoni pizza. I love Hawaiian pizza. I love pepperoni pizza. I love cowboy pizza. I love pepperoni pizza. I love meatlover pizza. Oh, and I love pepperoni pizza. The fact that I listed pepperoni four or three times how many was doesn't change the fact that I like belongs to the set of pizza flavors, pizza toppings I like. So we distinguish between repetitions. Now in this situation where we list the elements one, three, two, two and one, three, two, it might seem silly, why do you list two twice? But it could turn out because of the set builder notation, the property, the rule and play, might actually very naturally produce the same element multiple times. And it just might not be clear from the beginning. If it does produce the element multiple times, we don't distinguish those. Also, the order, so, so in this regard, these two sets are considered the same set even though one of the elements is repeated more than once. The order in which elements are arranged also is irrelevant. That is the set one, two, three, one, three, two and three, two, one are all considered the same set. When it comes to being a set, the only thing that matters is set containment, not how many times we repeat the element or the order in which they're described. And so for the idea of a set, we don't care about those things like order and repetition. If we did care about order, sorry about, if we cared about repetition, we could talk about something called a multi-set, which is something very important and common in torques that we won't worry too much about. If we did care about ordering, we'd have some type of array or be like an intuple of elements, like the splits of elements that we care about order, but we don't, not when it comes to a set. When it comes to a set, the only thing we care about is containment. We have a symbol that describes this relationship when considering if an object is an element of a set or not, we use this symbol right here, kind of looks like an E. Honestly, I think it's, I think the symbol sort of derived from the symbol Epsilon, the Greek letter Epsilon, although it doesn't really look like an Epsilon anymore. The symbol Epsilon, all right, in this case the symbol here that looks like an Epsilon, this means that you're an element of a set. And then if you put a slash that symbol, that would negate the symbol. That's true for basically any mathematical relationship, draw a slash through it and negates that object. That would mean you're not an element of the set. So consider the following example, take a very simple set A equals the number, the set of numbers one, two, three and four. The number two belongs to that set. So we would write two is in A. That's all there is to it. On the other hand, the number five is not inside of that set. So we would write five is not a member of A. That's how we, that's how we use the symbol there. Very simple, basic. There are some important sets that you should know by definition. These are going to have some important, these are important sets. So we're going to attach some symbols to them. If you see this symbol right here, this, when you write it by hand, you'll often just write an N with an extra line on it. This is our blackboard N. That's the official font and latex for this thing. And this is even the set of all natural numbers. So N is a natural number in this context. And by natural numbers, we need the non-negative integers, zero, one, two, three and et cetera, going on and on and on. Now, I should mention that the way we are defining the natural numbers is we will include the number zero. Many mathematicians, when they define the natural numbers, they don't use the number zero. And there are many reasons for doing that. For example, Judson's textbook for which this, which our course is based upon, actually defines the natural numbers as the positive integers one, two, three, four. It excludes zero. And I think, I disagree with that statement. I mean, there are good reasons to exclude zero. But in terms of the, fundamentally, do they do, the natural numbers are counting numbers. They count things. For example, they count the sizes of finite sets. If ever you see a set A and you put what looks like absolute value symbols, this is going to count the cardinality of the set, which this means the number of elements, the number of elements in the set A. And this cardinal number, if A is a finite set, will always be a natural number. So if you take the example we had from before, like, say this is our definition of A, then in that case, the cardinality would equal four. On the other hand, if we take the empty set, well, the, the size, the cardinality of the empty set would be zero. There's no elements inside of that. And so since zero is a num, a natural cardinality of a set of a finite set, we do allow zero inside the set of natural numbers. But like I said, there are many mathematicians who have strong reasons why they shouldn't. And why can't the mathematicians decide? Well, because there's good reasons in both camps there. And so it's kind of interesting to have the two competing thoughts there and having, having you kind of decide for yourself the right approach there. But for our, for our lecture series, zero will be considered a natural number. And also some algebraic benefits that the set of natural numbers actually becomes a monoid because it has an additive identity. Things we'll define in the future. The set of integers we will denote using blackboard Z. This actually comes from the German word for number, which in the German alphabet starts with a Z. The integers are going to be positive, negative or zero, right? So we have positive integers, we have zero, we have negative integers. I won't say too much about that. We have the set of rational numbers, which we're going to denote using blackboard Q. Q here stands for quotient, because a rational number is going to be a quotient of integers. So we look for fractions of the form P divided by Q, where both P and Q are integers. See the notation there. But we also require that Q cannot equal zero. And the rational numbers will have the usual equivalence. Notice that there are different, there are different integers that produce the same fraction. For example, one half and two fourths are the same rational number. So the rules that determine the set of rational numbers can produce multiple representations of the same element, things like this. This is what we were talking about beforehand. These are considered the same rational number. The blackboard R will represent the set of real numbers. And blackboard C will represent the set of complex numbers with the usual interpretation. A complex number Z has the form A plus B I, where A and B are both real numbers. A is the real part, and B is the imaginary part. So this gives us a very basic introduction to sets that will be useful as we study abstract algebra in this course.