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Universal set and absolute complement

Video transcript
What I want to do in this video is introduce the idea of a universal set, or the universe that we care about, and also the idea of a complement, or an absolute complement. If we're for doing it as a Venn diagram, the universe is usually depicted as some type of a rectangle right over here. And it itself is a set. And it usually is denoted with the capital U-- U for universe-- not to be confused with the union set notation. And you could say that the universe is all possible things that could be in a set, including farm animals and kitchen utensils and emotions and types of Italian food or even types of food. But then that just becomes somewhat crazy, because you're thinking of all possible things. Normally when people talk about a universal set, they're talking about a universe of things that they care about. So the set of all people or the set of all real numbers or the set of all countries, whatever the discussion is being focused on. But we'll talk about in abstract terms right now. Now, let's say you have a subset of that universal set, set A. And so set A literally contains everything that I have just shaded in. What we're going to talk about now is the idea of a complement, or the absolute complement of A. And the way you could think about this is this is the set of all things in the universe that aren't in A. And we've already looked at ways of expressing this. The set of all things in the universe that aren't in A, we could also write as a universal set minus A. Once again, this is a capital U. This is not the union symbol right over here. Or we could literally write this as U, and then we write that little slash-looking thing, U slash A. So how do we represent that in the Venn diagram? Well, it would be all the stuff in U that is not in A. One way to think about it, you could think about it as the relative complement of A that is in U. But when you're taking the relative complement of something that is in the universal set, you're really talking about the absolute complement. Or when people just talk about the complement, that's what they're saying. What's the set of all the things in my universe that are not in A. Now, let's make things a little bit more concrete by talking about sets of numbers. Once again, our sets-- we could have been talking about sets of TV personalities or sets of animals or whatever it might be. But numbers are a nice, simple thing to deal with. And let's say that our universe that we care about right over here is the set of integers. So our universe is the set of integers. So I'll just write U-- capital U-- is equal to the set of integers. And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z. And it's Z for Zahlen, from German, for apparently integer. And the bold is this kind of weird looking- they call it blackboard bold. And it's what mathematicians use for different types of sets of numbers. In fact, I'll do a little aside here to do that. So for example, they'll write R like this for the set of real numbers. They'll write a Q in that blackboard bold font, so it looks something like this. They'll write the Q; it might look something like this. This would be the set of rational numbers. And you might say, why Q for a rational? Well, there's a couple of reasons. One, the R is already taken up. And Q for quotient. A rational number can be expressed as a quotient of two integers. And we just saw you have your Z for Zahlen, or integers, the set of all integers. So our universal set-- the universe that we care about right now-- is integers. And let's define a subset of it. Let's call that subset-- I don't know. Let me use a letter that I haven't been using a lot. Let's call it C, the set C. Let's say it's equal to negative 5, 0, and positive 7. And I'm obviously not drawing it to scale. The set of all integers is infinite, while the set C is a finite set. But I'll just kind of just to draw it, that's our set C right over there. And let's think about what is a member of C, and what is not a member of C. So we know that negative 5 is a member of our set C. This little symbol right here, this denotes membership. It looks a lot like the Greek letter epsilon, but it is not the Greek letter epsilon. This just literally means membership of a set. We know that 0 is a member of our set. We know that 7 is a member of our set. Now, we also know some other things. We know that the number negative 8 is not a member of our set. We know that the number 53 is not a member of our set. And 53 is sitting someplace out here. We know the number 42 is not a member of our set. 42 might be sitting someplace out there. Now let's think about C complement, or the complement of C. C complement, which is the same thing as our universe minus C, which is the same thing as universe, or you could say the relative complement of C in our universe. These are all equivalent notation. What is this, first of all, in our Venn diagram? Well, it's all this stuff outside of our set C right over here. And now, all of a sudden, we know that negative 5 is a member of C, so it can't be a member of C complement. So negative 5 is not a member of C complement. 0 is not a member of C complement. 0 sits in C, not in C complement. I could say 53-- 53 is a member of C complement. It's outside of C. It's in the universe, but outside of C. 42 is a member of C complement. So anyway, hopefully that helps clear things up a little bit.