 We'll introduce one more group of set operations, and the key one here that we'll introduce is the complement of a set. Generally speaking, we think about sets as living in some universe. So, for example, the set A, consisting of the elements A, B, C, and D, I can think about these elements as being drawn from some universe, and in this particular case, I might see that these elements are all lowercase letters. So maybe the elements are drawn from the universe, the set of lowercase letters. Or maybe I'll take a look at the set E, consisting of two, four, six, eight, and again, these are, let's see, well, they're whole numbers, and they happen to be a subset of the whole numbers, and so I can think about the elements of this set as being drawn from the set of whole numbers. And given a set A, where the elements are drawn from some universal set, lowercase letters, whole numbers, real numbers, whatever, I can talk about the complement of A, which I'm going to designate by drawing a bar over the A. And that complementary set is going to consist of everything that's not in A, but still in the universal set. So here, I want to have that universal set so I don't throw everything into the complement, everything that could possibly exist into the complement, but to some extent, what is relevant. Now, again, if we're not entirely sure what a universal set is, if it hasn't been identified explicitly for us, or there's several possibilities, we can integrate the complement verbally just by negating the set definition. We'll see an example of that. So for example, let's take that set E equal to four, six, eight, and let's assume that in this case they are numbers drawn from the set of whole numbers that range between zero and ten inclusive, which is to say including ten, including zero. So I've defined what that universal set is. Where did these numbers come from? Well, they came from the set of whole numbers between zero and ten. And I want to find the complement of E with respect to this universal set. So since I've drawn the elements of E from the set of whole numbers between zero and ten, our complement will be everything except for what's already in E. So let's see what's in E, two, four, six, and eight. What's left from this set of whole numbers between zero and ten? It's going to be the remaining numbers, zero, one, three, five, seven, nine, and ten. Now, if the universal set is not given or it's not entirely clear what it is, we can do the following that'll make our life easier. Let R be a set of things where what we're talking about is a red-colored object. Describe the complement of R. Well, since we don't have the universal set given to us, it's not entirely clear what the universal set is. We can still describe the complement of the set by negating the set definition. And so in this case, R is the set of things that include all red-colored objects. So our complement is going to be the set of things that are not red-colored objects. And so there's our description of the complement. Now, the complement is tied in with another set operation known as the set difference. Given two sets A and B, I can find the set difference, and that's going to be defined as the set consisting of everything that's in the set A except for those things that are also in the set B. So you can think about this as starting with the set A and then removing anything in that set that has something in common with B. For example, if A is as given, B is as given, then my set complement A, complement B, a set difference, A, the difference between A and B is going to be the set A where I'm going to take out anything that is already also in B. So A is in B, so I'm going to remove that. E is also in B, so I'm going to get rid of that as well. And so A, the difference A minus B is just going to be the set B, C, and G. Now I can do the set difference the other way, B take away A. That's going to be the set B, A, E, I, O, U, except this time I'm going to take away anything that is also in this set A. So A is in the set, so I'm going to get rid of it. E is in the set, so I'll get rid of it, and all the rest are not there. And it's worth noting that the difference A take away B and B take away A, these in general will be very different. Well, how about a nice little proof, or a proof that the difference A take away B is the same as A intersect B complement. So again, if I want to show the equality between two sets, what I want to do is I want to make an argument that the two sets have exactly the same elements. So let's take a look at what those sets are. I'm going to describe this set A take away B. So by my definition, this consists of everything in A that is not in B. On the other hand, consider the set A intersect B complement. So in order to be in this set, you have to be both in this and also in this. So what you have to be is you have to be in A and not in B. If you're in B complement, this is the set of things that are not in B. So if you're in this set, you're not in B. So this set A intersect B are things that are in A and not in B. And if I compare my two descriptions of the two sets, they describe the same set of elements. These are things that are in A and not in B, and these are things that are in A and not in B. And since they describe exactly the same thing, then the two sets are going to be equal and there's the completion of my proof.