 So let's define some operations on sets. So the first one we'll define is the complement of a set, written with a bar over the set name, is the set of all elements, not in A. Now, complement does have a little weird relationship here. If A is a set, then the complement of A is everything in the universe that isn't in A. To make it easier to work with, we define a universal set from which we draw the elements of A. And sometimes the universal set is implied and not explicitly stated. For example, let E be the set of things where thing is an even number. Find an appropriate universal set and then find the complement of the set. So first, we can express the complement of E. Well, that's the set of things that are not even numbers. Now, if we don't specify the universal set, then the complement of E could include the number 5, which is not even, the number 1 fourth, which is not even, and what else is not even? Well, how about a bull elephant? And in fact, our complement would include anything that's not even, and that's why it's so important to specify what the universal set is. Now, unless it's explicitly identified, identifying the universal set is a matter of drawing a reasonable conclusion. Now, let's see. Since all even numbers are integers, it seems reasonable to suppose that the universal set, the set from which we get these even numbers, is the set of integers. So if the universal set is the set of integers, then the complement is the set of non-even numbers among the integers. In other words, we might call it the set of odd numbers. Let's define a few more set operations. Let A and B be two sets. The union of A and B, written this way, is the set containing the elements in A together with the elements in B. The intersection of A and B, written this way, is the set containing elements that are in A and also in B. And finally, the set difference A minus B is the set of elements of A that are not in B. So let A and B be these two sets. We'll find A intersect B, A union B, A minus B, and B minus A. So A intersects B. That's the set of everything that is in A and also in B. And so here, we see that A is in both of them. B is not. C is not. D is not. A is in A and also in B. So it should be in the intersection. And so A intersects B is the set consisting of just A and E. How about A union B? So that's the set of things that are in A, along with the set of things that are in B. So we'll throw in our elements of A and we'll throw in our elements of B as well, except remember that in this notation, order doesn't matter and repeated elements are not listed. So we've already included E and A, so we don't need to include the extra copies. A minus B, well those are the things that are in set A that are not in the set B. And so we might start with A and then remove anything that's also in B. B minus A is similar. We start with set B and remove anything that's also in set A. And we'll introduce one more important idea, Cartesian products. Suppose we have two sets A and B and we form an ordered pair, X, Y, where X is one of the elements of A and Y is one of the elements of B. This gives us what we call a Cartesian product. The Cartesian products of A and B is written this way and it's the set that includes all ordered pairs where X is from A and Y is from B. So for example, let A be the set, natures, soup, salad, and B be the set steak, chicken, tofu. Let's find the Cartesian products A times B and B times A and then let's ask the question, does A times B equal B times A? So the elements of A times B are the ordered pairs X, Y where X is from A and Y is from B. And we can just go through the list. We might pick natures and then steak, chicken, or tofu. The thing we pick from A could be soup and the thing we pick from B could again be steak, chicken, or tofu and the thing we pick from A could be salad and the thing we pick from B could be steak, chicken, or tofu. Now if I want to find B times A, the first component of the ordered pair has to be from B and the second has to be from A. So maybe I'll pick steak and then natures, soup, or salad and similarly I can find the other elements of B times A and so the question is, are these the same set? In order for them to be the same set they have to have the same elements and here it's important to remember these are ordered pairs and because they're ordered pairs the ordered pair steak first then natures is not the same thing as the ordered pair natures first then steak and since the elements of the two sets are not equal then A times B is not equal to B times A.