 Welcome to a screencast on a new concept, our last concept we're going to see in our study of set theory on a thing called a Cartesian product. Now a Cartesian product sounds fancy, but you're actually very familiar with this idea. Just think back to where you first learned about R2, the X, Y plane here. R2, we call it R2 because it's really two copies of the real numbers and sometimes you'll see it as R cross R. You've got one real number line going left to right and one real number line going up and down. And if we put those two real number lines together, we can create a grid and use that Cartesian plane as it's called because it was first conceptualized by Rene Descartes back in the day. That Cartesian plane can be used to plot ordered pairs of numbers. Okay, so every time you plot a point here, let's assume for example that every grid mark here is one unit. If I put a point right there on the grid, then that's at 1, 2, 3, 4 units on this real number line and then 2 units up on that real number line. So I have 4 out this way, 2 up this way and we call that point 4, 2. So points that are in this plane or in this set here are ordered pairs of points and the first number is a real number and so is the second one. And they do matter. If I looked at the ordered pair 2, 4 instead and that would be a completely different point in the plane. I would go to 2 here and then 1, 2, 3, 4 up here. So that's what 2, 4 is. So there are ordered pairs and each coordinate, we had a first coordinate and a second coordinate was a real number and that's why we called it R cross R. So if you wanted to write this in set notation, the Cartesian plane, the XY plane here, is the set of all such ordered pairs where the first coordinate is a real number and the second coordinate is a real number. So this idea of the Cartesian plane here is a way of grouping 2 numbers together in order. And if you think about it, grouping 2 objects in order comes up a lot in life. I think in thinking about teaching, if you're teaching you might have a class here for example that's got say maybe a lot of students but here are 4 students and here are their grades on the last test. Now if you're entering these grades into a database, say like a blackboard type system, then you're going to want to put these things in order. You want to pair off Alice with her grade and Bob with his grade and Clarence with his grade and Dora with her grade. So really you can think of the data here in our little database as an ordered pair. I could represent each of these data points here as an ordered pair. Let's say Alice, comma 90. And so the first coordinate of this ordered pair is a name of the student in the class and the second coordinate here of this ordered pair is a number between 0 and 100 that would represent that student's grade. Bob would be represented as the ordered pair Bob comma 92. In fact databases really work this way. They pair things off or actually there could be several different data fields, not an ordered pair but an ordered triple or quadruple where you have a bunch of different data collected together in order. So Clarence is ordered pair, Clarence comma 73 and so on. So what we're doing here is we're extending the idea of the Cartesian plane, the xy plane, to include things that aren't necessarily numbers. This is right, Dora's ordered pair out here, 88. So you could go through the entire class roster, maybe there are 20 some odd students beyond just these four and you know every student and their test grade could be represented as an ordered pair say s comma g where s belongs to this set, the set of all students in the class and g belongs to this set right here, the set 0 through 100 that represents a grade. So this notion of ordered pairs actually is pretty applicable to things beyond just numbers. We can pair off text with numbers or any kind of object with any other kind of object. And that gives us to the idea of the Cartesian product. So let's let a and b be any two sets whatsoever. Maybe there are sets of numbers, maybe one of them is a set of names and the other is a set of numbers, maybe they're both names, maybe they're one is a set of happy faces and frowny faces and the other is a set of states in the United States. We don't care just any two sets whatsoever. Then we're going to define the Cartesian product and we're going to read this thing here as a cross b. The Cartesian product is the set of all ordered pairs whose first coordinate belongs to a and whose second coordinate belongs to b. That is if we're going to write this out as a set, a cross b, the Cartesian product is the set of all ordered pairs where x belongs to a and that should say y, y belongs to b. And so this is just a generalization of what you already have learned about the xy plane and it also comes in very handy to describe things like the student grade database that we just constructed here, the Cartesian product, set of all ordered pairs with the first coordinate and the first set and the second coordinate and the second set. So let's look at an example here and I'm thinking about kids. I have three kids of my own and they often fight over things at home. So let's suppose I have three kids Jack, Bill, and Annie and put those in the set k. And let's let t be the set of these three toys, a ball, a car, and some blocks. Now I would be wondering if I were taking care of these kids, what are all the possible ways that I could pair off a kid with a toy? Well that would, the set of all possible pairings there would be the Cartesian product k cross t. This is going to be the set of all ordered pairs, let's say little k, little t, such that k belongs to the set of kids and t belongs to the set of toys. So what is that Cartesian product? What does it consist of? Well let's start listing out the ordered pairs. This is going to be a large set consisting of ordered pairs. Well one ordered pair would be Jack and ball. Any ordered pair where that has two coordinates with the first coordinate being a kid's name from the set k, and the second coordinate being a toy from the set t is an element of k cross t. So there are probably, you can think ahead there are going to be nine of these things. I could pair Jack off of the ball, I could give Jack the car, I could give Jack the blocks. I'm running out of room here so let me just tuck that in there. I could also, I have a pairing where Bill has the ball, Bill has the car, or Bill has the blocks. And then finally all the pairings that involve Annie, and that would just be Annie could have the ball, Annie could have the car, or Annie could have the blocks. And that set would close off this last ordered pair and then put a giant set brace up here to pair it off with this set brace that we started with. That set of nine things is the Cartesian product. I could even plot them if I wanted to, I guess, kind of like an R2. We don't often do that. We just sort of think of these set of pairs. And those pairings mean something, okay? The pairings are important. They do come in order. I need to have a kid's name first and a toy second. And the kid's name has to actually come from k. The toy's name has to come from t. The set of all such pairs would be my Cartesian product. So let's do a couple of concept checks here to see how well we're understanding these basic ideas. So take a look at the ordered pair that consists of two numbers, the square root of two comma the number one. What set or sets could that ordered pair be considered an element of? Here are five choices and I want you to pause the video and select all that apply. Well, let's mark off some things that are definitely incorrect to begin with. A can't be right. A is just the set of real numbers. And an ordered pair is not a real number. It's a couple of real numbers. It's a pair of real numbers. So the only thing that's in R are individual real numbers like radical two by itself would be an element of R. But not an ordered pair with radical two in it. I need to be in a Cartesian product somewhere. Now B could work because two one, radical two one is an ordered pair and the first coordinate is a real number. So it belongs to this and one is a real number and it belongs to this. So that would certainly work. R cross Z also works because it's an ordered pair. This is an ordered pair where the first coordinate belongs to the real numbers and the second coordinate belongs to the integers. So that would work as well. D does not work and why is that? D does not work because the first coordinate here is not an element of the rational numbers. As we've proven and seen before, square root of two is an irrational number. So although one does belong to the integers, square root of two does not belong to Q and so this ordered pair does not belong to Q cross Z. Here are on the other hand some things that do belong to Q cross Z. Anything that's an ordered pair where I have a rational number in the first position and an integer in the second position. So something like 3 halves comma 17 would be an element of Q cross Z. But radical two comma one is not. And finally this last set here doesn't work either. Even though this belongs to R cross C, it does not belong to Z cross R because the order matters. This is a square root of two comma one. Square root of two does not belong to the integers. And so this ordered pair does not belong to that Cartesian product. So the only correct answers here are B and C. Now speaking of this last issue here that we just discussed, here's another concept check. What's the relationship between A cross B and B cross A? Similar Cartesian product but I'm reversing the order here. Is A cross B a subset of B cross A? Is it equal to B cross A? Or is there just really no relationship between that set and B cross A? Well the answer here is essentially C. There's not really any relationship in terms of subsets or equality between A cross B and B cross A. And I think we saw that in the last concept check. We have this ordered pair of square root of two comma one that was an element of R cross the integers. Okay? Because two is an element of R and one is an element of the integers. But if I keep the same ordered pair, that is not an element of Z cross R. Because or square root of two is not an element of the integers. And so that fails. So certainly there's no subset in relationship and therefore there's no equality relationship. Now there is a sort of symmetry. There is a sort of relationship between those two sets that we're going to explore a little further in chapter six when we talk about functions. They are very similar to each other. The only difference is that the order is reversed. For example, the ordered pair of one comma radical two would belong to the set Z cross R. So maybe there's some sort of a operation or a transformation I can do on an element of one Cartesian product that gets me into the Cartesian product I have by reversing the order. But for now there's no relationship that's really evident there in terms of subsets of equality. So that's the idea of the Cartesian product is generalizing the notion of R2 in your old fashioned algebra XY plane. And it's very useful and very handy. And in the next video we're going to look at how do we prove things about Cartesian products. So thanks for watching.