 So a fairly typical thing we do in mathematics is we define something, and then we start to ask, well, how does it relate to each other? And so sets is a good starting point, and so we might begin with the notion of set equality. And so given two sets, I can form the following definition. Two sets are going to be equal if and only if the two sets contain the same element. And so if that occurs, we would write the two sets A and B are equal to each other using our standard equality symbol. Now, this type of set equality is most obvious if we express the sets using list notation, so we can actually see what the elements in the set are. So for example, this can, this can occur in the following ways. I have a set, maybe I have a set like this, and I'm going to define the set using some sort of set builder notation that describes the membership rule for the set. And then I have another set where I'm just going to list what the elements are. And I want to either prove or disprove that the two sets are equal. And so since I want to prove the two sets are equal, I want to show that they have the same elements. And in this case, it's easy enough to use list notation, set B is small enough, that I can verify or not the equality of the two sets. So A, X is an odd-hole number less than 10. Well, I can figure out what those are. The odd-hole number is less than 10, 1, 3, 5, 7, and 9, and that's it. And I have my listing of the elements in set B, 1, 3, 5, 7, and 9, and I see that they are in fact the same set of elements. And since they have the same elements, then my two sets are equal. Now, if the two sets aren't equal, we might have a subset relationship. And so in this case, we say that one set is a subset of another, if every element of the first set is also an element of the second set. If that occurs, then we indicate the subset relationship using this particular notation. A is a subset of B. Now, in this case, if we're looking at a possible subset, listing the elements might not be convenient, because that means verifying that everything in here is someplace in this set B. And B might be a much larger set than A. So trying to find any particular element could be a little bit challenging. And so we want to have some more efficient way of doing that. And in this case, we might want to look at the description of the set element and make some sort of logical argument based on what that description is. For example, I might have the two sets. Let S be the set of things where what I'm talking about is a U.S. state. And let Y be the set of things where what I'm talking about is a U.S. state west of the Mississippi. And I want to prove that W is a subset of S. So let's think about this. So I want to show that every element of W is also an element of S. That it meets whatever requirements S has for set membership. So let's take some element of W. And I want to think about the definition of those two sets. Because I know that Y is an element of W, I know that Y is a U.S. state west of the Mississippi. Now, I want to show that this is also in S. Well, the membership requirement for being in S is that you have to be a U.S. state. And the observation I make is that if you happen to be a U.S. state west of the Mississippi, then you're also going to be a U.S. state. And what that means is that any element of I of W meets the requirements for being in S. So whatever Y is, if it's in W, it's guaranteed to also be in S. And that tells me that since every element of W is guaranteed to be an element of S, then W is going to be a subset of S. Well, let's take a look at another example. So let's prove that if A is a subset of B and B is a subset of A, then A and B are equal. Well, I'm trying to prove A conditional if something, then something. And because I'm trying to prove A conditional, I can always assume we have the antecedent, we have the if part. So I know that A is a subset of B, and I also know that B is a subset of A. So I suppose that's true. Well, I can use my definition of the subset. Since A is a subset of B, then I know that every element of A is also an element of B. And in fact, I have two subset statements. So I can then say that every element of B is also an element of A. And putting them together, that says that A and B do in fact contain the same elements. Whatever is in A is also in B, and whatever is in B is also in A. And so A and B contain the same elements, and so by my definition of set equality, I know that A and B are equal.