 All right, so let's define a few set operations. So given two sets A and B, I'm going to define the union. A union B, using this union symbol, looks like U, is going to be the set containing all elements of A together with all elements of B. Meanwhile, the intersection A intersects B. That's going to be the set that contains the elements that are in both A and also at the same time are also in B. The symbol we use for that is the union symbol turned upside down. Now, these are very clear mathematically. Unfortunately, natural languages tend to have a very bad habit of muddying the distinction between unions and intersections. So we need to be a little bit careful when we try and translate from a natural language back into mathematics. Mathematics into a natural language is pretty easy. We can agree upon a fairly standard and constant set of terms we're going to use. Going the other direction tends to be a little bit more problematic. For example, let's consider the following. Let L be the set of people who bought lottery tickets and R be the set of people who bet on horse races. And I can express mathematically the two concepts L intersect R, L union R. But what happens when I try to translate these into non-mathematical terminology? What if I try to explain what these sets are to somebody who doesn't speak math? Well, we can start off as follows. The set L intersect R is the set of elements that are in both L and also in R. And what this means is that this set is going to include people who are both have bought a lottery ticket and also have bet on a horse race. Now, I can describe that fairly easily in English. The set L intersect R consists of all persons who bought lottery tickets and also have bet on horse races. If I bought a lottery ticket, I'm in L. If I bet on a horse race, I'm in R. If I've done both, I'm in L intersect R. So there's my set L intersect R, not a problem. Intersections tend to translate fairly easily between English and mathematics. Unions are a little bit more problematic. So the set L union R is the set of elements of L together with the set of elements of R. And again, I'll consider what this set looks like. The things in this set, well, if I'm in L, I'm somebody who's bought a lottery ticket. Meanwhile, if I'm in R, I'm somebody who's bet on a horse race. So if I want to describe the set L union R, that's going to consist of all persons who've bought lottery tickets together with all persons who've bet on horse races. Now, if you look very carefully at how we've described these two sets, while there is a difference in our phrasing, persons who've bought lottery tickets and bet on horse races, persons who've bought lottery tickets and all persons who've bet on horse races, there is a difference in the phrasing of these two descriptions. But it's a very subtle distinction. And it requires a careful reader to note that there is a distinction. In fact, most people, if they were to read these two descriptions, would not see the difference between the two of them, because most people don't read things carefully enough. To exaggerate the difference between the two, to make it easier to distinguish the non-mathematical description of L intersect R and the non-mathematical description of L union R, we prefer to describe the situation using the phrase or using the conjunction. So we might change our second description. This is the set that consists of all persons who've bought lottery tickets or who've bet on horse races. And so this gives us a nice distinction between the and and the or. The thing to note here is that in general, when we give a sentence in English, when we describe something in English, we may use and when we really should be using or. And again, if you speak carefully, you can make that distinction easily. For example, who's at the movie? Well, everybody, the people at the movie include are everybody who has bought a ticket and everybody who works at the movie theater. What we really mean in this particular case is the people at the movie are all those who bought a ticket or those people who work at the theater. But we can give that sort of statement using an and conjunction, but it does not correspond to an intersection. I'll take a look at another simple proof. We're going to prove that A intersect B is a subset of B. So we want to show a particular set A intersect B is a subset of another set. And so for that, we might want to go back. We should go back to our definition of subset. We have a subset relationship every time every element of the one set is an element of the second set. So we want to show that everything in here A intersect B is also going to be in B. Well, we might want to start by considering what we know about the elements of A intersect B. So every element of A intersect B, well, let's go back to the definition. A intersect B is the set of all things that are elements of both A and also B. So these things in A intersect B are in A and also in B. And what that means is that everything in A intersect B is also an element of B. And so that tells me that everything in A intersect B is also an element of B and that's what guarantees our subset relationship. And that concludes our proof.