 Well, that last video was really long, and I appreciate your patience with it. So welcome back to one more screencast for this section. And this time we're going to introduce a brand new idea that's related to some old concepts, old meaning we learned in the last few videos, about disjoint sets. So what does that term mean? So we're going to define two sets A and B to be disjoint if their intersection is the empty set. So two sets are disjoint if they basically have nothing in common. Their intersection is completely empty. For example, the set 1, 2, 3 and the set 4, 5, 6 are disjoint because there's no elements that are common to both. The sets 1, 2, 3 and 3, 4, 5 are not disjoint. They don't have much overlap, but they do have some overlap. There is at least one element that belongs to their intersection. In fact, in this case, there is exactly one element that belongs to their intersection. So these guys are disjoint. These guys are not disjoint. Another example, the set of even integers and the set of odd integers are disjoint. There's no integer that belongs to both of those sets at the same time. So what we're going to talk about here is how to prove the two sets are disjoint. And what this involves is proving that two sets are equal to each other, namely that the intersection is equal to the empty set. But it's going to be a little bit different this time. I'm going to give you a common strategy, not the only strategy necessarily, but a common strategy for proving two sets are disjoint. Oftentimes, two sets are proven to be disjoint by contradiction. So to prove that A and B are disjoint sets, remember that means that A intersect B is the empty set. That's kind of a statement that's phrased in the negative. A intersect B is empty. There is nothing in it. And as with many statements that are given in the negative, we often use contradiction. So one common way, again, to prove the two sets are disjoint, is suppose that A and B are not disjoint. Suppose for a contradiction that A intersect B is non-empty. That there actually is something in the intersection. So we're assuming the negation of the thing we're trying to prove, to say something is not disjoint means that this intersection is not empty. So choose an element that's in it, in other words. And then what that would mean is that A belongs, or X belongs to A and X belongs to B at the same time. And then we would just work forward using valid mathematical steps to derive a contradiction. Thereby coming to the conclusion that the assumption that A intersect B is non-empty is false, and so the intersection must be empty. So let's put that to work for us with another example that involves integer congruence. So let's let A be the set of all integers that are 0 mod 4, and B is the set of all integers that are 1 mod 2. And we're going to claim that A and B are disjoint. That is that their intersection is empty. So let's get the proof going here. This is, although we're going to be proving that one set equals another, we're not actually going to do it using the double inclusion method that we talked about last time. Let's go with contradiction. So for contradiction, contradiction, let's assume that A and B are not disjoint. So assume A and B are not disjoint. And what that means is to say A and B are not disjoint is to say that their intersection is non-empty. And if an intersection of two sets is non-empty, that means there's something in it. So choose it. So let's let X be the item that belongs to both sets at the same time. And what we're going to do is play with the supposed element X and show that if X exists, then we run into a contradiction. So if X is in A intersect B, that means that X belongs to A and X belongs to B at the same time. So let's kind of work out what that means. Well, if X belongs to A, what that means is that X is congruent to 0 mod 4. And if X belongs to B, then that means that X is congruent to 1 mod 2. Okay, so let's just keep working out with these things to mean. If X is congruent to 0 mod 4, that means that X is equal to, not congruent to, but equal to some integer, let's say Q1 times 4. And if X is congruent to 1 mod 2, then that means that the very same X is equal to 2 times an integer, let's say Q2 plus 1. Now let's kind of work out what those things mean. Well, let me change a color here so we can see a little bit better. This statement right here means, let's just work out what that means. X is equal to 4 Q1. That means that X is even because it's 2 times 2 Q1. So X is even in that case. But at the same time, from this statement right here, what that is telling you is that X is odd. Because it's 2 times another integer plus 1. And here we have a contradiction because I can't have X being even and X being odd at the same time. So that's a contradiction. So where do we do with that contradiction? We trace it back and see where it came from is what we do with it. So where did this come from? It came from just assuming that we had an X that belonged to both A and B at the same time. So in other words, what we have to conclude is that there is no X that belongs to A and B at the same time. So what we conclude is that therefore A intersect B doesn't have anything in it. That is, A intersect B is empty. Therefore A and B are disjoint, which is what we're trying to prove. So that's one strategy for proving the two sets of disjoints. Suppose that they aren't disjoint and pick an element out of their intersection, use the criteria for A and B that define the elements and then derive a contradiction, and that forces the intersection to be empty. Thanks for watching.