 Welcome back to another screencast about Cartesian products. In this video, we're going to work a proof of a set identity that involves Cartesian products. In your textbook, you don't see a lot of set identities about Cartesian products, unlike some of the earlier sections. So we're going to have to sort of build our own when we need them. And this is a good example of how we're going to do that. So just to remind ourselves what a Cartesian product is, Cartesian products are a real simple idea. You start with any two sets, A and B. And A cross B is the Cartesian product that's the set of all ordered pairs whose first coordinate is the element of A and whose second coordinate is an element of B. And I need to make that correction one more time there. So just a set of all ordered pairs where the first coordinate belongs to the first set A and the second coordinate belongs to the second set B. Just like you learned back in Algebra 1, except now we're using general sets that could contain any sort of objects whatsoever. So let's work on this proposition here that says for all sets A, B, and C inside a universal set U, we have A intersect B cross C equal to A cross C intersect B cross C. So this Cartesian product over here that involves an intersection is the intersection of two Cartesian products. So to set this up, we would like to go back and use Algebra of sets if possible, but in this case it's not really that easy to do because we don't have a lot of pre-existing set identities for Cartesian products. So we're going to use the choose an element method for this. It's kind of the old fashioned way by showing that first of all that A intersect B cross C is contained in as a subset of A cross C intersect B cross C. And then we will show the converse that A cross C intersect B cross C is a subset of A intersect B cross C. We just need to keep track of all the definitions of these concepts here, particularly the subset Cartesian product and intersection and this actually works out to be not that big of a deal. So let's make sure I can change my color here. So let's work on part one here where I'm going to show this direction of subset inclusion. So I'm going to choose an element. I'm going to choose an element inside this set here. Now pause and think about what that element will look like. That element is going to be in a Cartesian product, so it's an ordered pair. This is a very important concept here. If I'm choosing an element inside a Cartesian product, it has to be an ordered pair because that's what Cartesian products are. They're sets of ordered pairs. So I'm going to let, let's say, X comma Y be an element of A intersect B cross C. Now, again, if I'm going to let an element, I'm going to choose an element to be inside a Cartesian product, that is not just an X, it's an ordered pair. So I have to have two coordinates. Now where do those two coordinates live? Well, that's going to be the next line of the proof. Since X, Y belongs to this Cartesian product, what that means is that X belongs to the first set there. X belongs to A intersect B and Y belongs to C. And that's just the definition of Cartesian product. And I think at this point it might be helpful to write down what I want to show here. So we're going to write down that X comma Y belongs to A cross C intersect B cross C. Let's just keep that in our pockets and maybe box it off like we have been doing before. This is something you could say right there in the proof, but just mentally box it off if nothing else to keep yourself from assuming it. Because we don't know that's true yet. Well, let's just follow what we have here. So I know that X belongs to A intersect B. So what does that mean? That means that X belongs to A and X belongs to B. That's the definition of intersection. And I also happen to know, by the way, as a reminder, that Y belongs to C. So let's think of what we have here now. I have these three things that are joined by and so they're all true. Notice that what I have here on the one hand is that X belongs to A and Y belongs to C. So what I can conclude there is that X comma Y, the ordered pair, belongs to A cross C. Okay, that's the stuff that I have circled in the light blue means exactly this bottom line. And by the same token I see that X belongs to B and Y belongs to C. So I could say that X belongs, that the ordered pair X, Y belongs to B cross C. So what I have now here is that the same ordered pair belongs to these two Cartesian products at the same time. And that is going to give me what I want. So I can now conclude, squeeze it in here at the bottom, that X comma Y belongs to both A cross C and B cross C at the same time. And that's exactly what I was hoping to show up here in the red. So that does it for this half of the proof. Again, the key step here is this realizing that when I do a choose an element sort of thing here with a Cartesian product, the element that I choose is actually a pair of elements. And I know something about the first coordinate and the second coordinate. Now let's move on to prove the converse of this statement here. And we'll start up a new page for that. So I'm going to now choose or let, let's call it say U comma V this time. And this is going to be an element of A cross C intersect B cross C. Again, I'm letting this, this is a, this is an ordered pair here. I know it has to be because it's going to be in two Cartesian products simultaneously. It's in the intersection of this Cartesian product and that Cartesian product. Since it's in two Cartesian products, it has to be an ordered pair that I choose, not just an X or a U or a V, but a pair of things. So what I want to show here, again let me do this in the red, so I will know that I'm not allowed to assume it. I'm going to show that U comma V belongs to, is an element of A intersect B cross C. That's what I want to end up with. So let me just box it off. Don't want to touch it. Just want to end up there. So let's proceed through the proof. Now I know that U comma V belongs to this and to this. And so let me just say what that means. That means that U comma V is an element of A cross C and at the same time U comma V is an element of B cross C. Just using the definition of intersection because I'm intersecting this set with this set and U V belongs to both. Well let's unpack what those two things mean. Individually, this statement here would mean that U belongs to A and V belongs to C. That's the definition of Cartesian product. When I have an element, a pair of elements that belongs to a Cartesian product, it means that the first coordinate belongs to the first set and the second coordinate belongs to the second set. And I'm just writing that out here. Let me say the same thing or similar thing about the second statement that would mean that U, and there's an and in here as well. U belongs to B and V belongs to C. Now let's see what we have here. I know that V belongs to C. That's a, I've said that actually twice. But I've said two different things about U. U belongs to A and U belongs to B. Well let's kind of pull that in and what does that mean? Well that means that U belongs to A intersect B. Because it belongs to both U and B simultaneously. And I also knew from above that V belongs to C. Well now what does that mean? If I looked at the ordered pair U V, what Cartesian product does that belong to? Well the first coordinate of that element belongs to this intersection. So this belongs to A intersect B. Almost wrote the wrong thing there. A intersect B and the second coordinate belongs to C. And that is exactly what I wanted to show up here at the top. And so that proves the other direction of subset inclusion. And so that proves the entire equality because I've done the two subset inclusion arguments using the same old choosing element method. Key concept here is just knowing what kind of element you are choosing. When you're choosing an element that is known to be in a Cartesian product, it's got to be a pair of things. But other than that, it's just mainly chasing the definitions around. Thanks for watching.