 Welcome back to our lecture series Math 4220, Abstract Algebra for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. This is the first video for lecture two in our series and it will be continuation of section 1.2 about sets and equivalence relationships from Judson's textbook on Abstract Algebra. And so our main goal for lecture two here is to discuss what we mean by a function. In the previous lecture, we had talked about the basics of set theory, just very, very naive set theory things. And then we want to continue building on this because Abstract Algebra is all about taking sets and adding structure to it, structure, structure, structure, algebraic structure specifically for an algebra class, but algebraic and combinatorics, we love adding structure to sets. Topology geometry is included in that. This happens all over mathematics. Now before we're ready to find functions, which we'll actually do that in the next video. In this video, I just want to quickly define one more operation on sets, which is referred to as the Cartesian product. So we have two sets A and B, then we define the Cartesian product A cross B to be the set of all ordered pairs. So for example, if you have A cross B, this would equal the set of all ordered pairs A comma B, where little A comes from the set A, and the little B comes from the set B. Now we can make this an iterative process, right? If we want to take the direct product, I should say Cartesian product of A1 times A2 times A3 up to An, we can iterate this process in the following manner. We take the product from I equals one to N of the AIs. Now in this situation, they're going to use a capital Pi to denote products. This is actually common that we use a capital Sigma when we want to do sums. Sigma is like the Greek letter for S, and therefore is a mnemonic device on sum. Pi is the Greek equivalent of the letter P, and so it's P for product here. And so if we have a larger product, we don't, we typically think of these as intuples, right? So if you take A cross B cross C, we think of this as the set of ordered triplets, A comma B comma C, as A is inside of A, B is inside of B, and C is inside of C. It's typically how we do this thing. And so if you take the N, infold Cartesian product, we think of the elements of that will be this, is this intuple, right? Just, just sort of as convention there. And then another thing to mention, oh, I don't need to erase all of that, we'll keep that on the screen. One special case we do care about is what if all sets are the same, right? What if all end of the sets are just A all over itself? In which case in that situation, you can take A cross A times A times A times A, whatever you want, this is often referred to as A to the N. So if you see a set raised to a superscript, unless there's reasons to think otherwise, that'll typically mean the infold product, Cartesian product of that set with itself. So for example, like in a multivariable calculus setting, you might have talked about the set R3, sometimes referred to as three space, right? This would be the set of all three tuples of real numbers, R3. And so we're just using that notation in general. So if you were to take these two sets right here, take the set A, which is just the letters X and Y, whatever those things mean, take B to be the numbers one, two, and three, then the direct product of A times B, what you're going to do is you're going to take every possible combination of something from A with something from B. And so since A has two elements in it, and B contains three elements, when you take the Cartesian product, we can anticipate this Cartesian product A cross B, it's going to have A times B many elements in there. That is, you're going to get six elements because you look at every possible combination. So you're going to get all the ordered pairs that involve X. You're going to have an X1, X2, X3, there's three of those. You're going to get every ordered pair that's associated to Y, in which case you get Y1, Y2, Y3. You get the six possibilities right there. And this would work in general for finite sets. Similar constructions also work for infinite sets, right? Of course, in that case, the cardinality be infinite. It's also possible that your iterated product could be an infinite, like we could take accountable product or even an uncountable product of sets. We won't have much occasion to do that in this course. One important example I want to mention is that if you ever take the Cartesian product of the empty set, so if you have any set A and you times it by the empty set, it doesn't matter what set A is, the product will always be empty. And that's because as you come along to try to fill in the second slot with something, there's nothing to put it in there. And so you can't create any, let me say that correctly, you can't create any ordered pairs if there's nothing to go in the second slot or the first slot as well. If you take the empty set cross A, this is going to be the empty set. So whenever you take the Cartesian product of the empty set, you're going to get the empty set. Now I should mention, kind of erasing this off the screen here, I should mention that as sets, if you take A cross B, this does not equal the same thing as B cross A in general. I mean, of course, if A equals B, then that's not the case. But in general, the Cartesian product are not equal to each other as sets. Because for example, the first guy here, A cross B, it would contain the element AB. This other one would contain the element B comma A. But in general, the order matters for an ordered pair. So if you switch up the order, that's not equal as sets. Now oftentimes, that's okay. And we'll talk about that in the future, specifically when we talk about isomorphisms of like direct products and things, in that case, we don't care so much about the order. But in terms of sets, the order does matter. If we're thinking of, for example, ration numbers as ordered pairs of integers, right, we could identify the ordered pair one half, maybe with the fraction one half, and therefore two comma one could be identified with the number two, two over one. We wouldn't want those things to be equal as rational numbers. And so Cartesian products involves ordered pairs. Therefore, if you switch up the order, you no longer have equality.