 So welcome back to another screencast on sets, and in this video we're going to look at an old idea that we picked up back in chapter 2 about subsets and set equality, but in a slightly new light here. So let's go back to an example we saw in the previous video where we had these three sets here, and let's try to understand a little more carefully why we got A minus B equal to the empty set with these two sets A and B up here. Let's understand what's actually in A and list it out in roster notation. So A would be the set of all natural numbers that are 0 mod 4, that would be 4, 8, 12, anything that's divisible by 4, it's a natural number, so there would be a roster notation version of that set. And B would be the set 2, 4, 6, 8, anything even, anything that's divisible by 2, and a natural number would be in there. So A minus B is the set of all elements that belong to B A, sorry belong to A, but do not belong to B. But the problem with that is that every element of A belongs to B, and so if you go into A here and start subtracting out, or xing out the things that belong to B as well, what 4 would have to be thrown out, and so would 8, and everything would get thrown out. And so that's why that set difference was equal to 0. In a slightly different way of understanding this, let's draw some diagrams here that might help us represent this. These are called Venn diagrams, and you can read about that in your book or look these things up online. We're just going to make a giant circle, kind of like my social network, that consists of the set B. B is actually a fairly big set, it's the set of all even natural numbers. And A, if you're going to draw, it would actually be a piece of B. It's actually a fairly large piece of B, about half of B would be A. So the 0 mod 4 stuff is in here, and that's contained within or nested inside the 0 mod 2 stuff, as we kind of see in the roster notation. So the fact that A lives inside of B means that if I subtract out everything that's in B, then I'm going to be left with nothing. Every point that I draw, every dot that I draw in here that belongs to A is also a dot that belongs to B. Now the vice versa is not true. If I draw, I can draw plenty of dots outside here in B that don't belong to A. That would correspond to something like the number 6, which belongs to B, but 6 does not belong to A. So understanding this way that sets nest with each other is actually pretty helpful. And we're going to bring this up again in the notion of some old terminology that we've seen here. So back in chapter 2, we said that two sets A and B, A is a subset of B. If every element of A is an element of B as well, we use this notation. We said that A and B are equal if A is a subset of B and B is a subset of A as well. And we use the equal sign to denote that. That means that every element of A belongs to B and every element of B belongs to A. In some sense they are contained within each other and they have the same elements. My new piece of terminology we're going to bring up now is that A is a proper subset of B and use this notation here, which I actually need to correct. I notice here, put a little equal sign under there. If A is a subset of B but A is not equal to B. And we use the notation that A is a proper subset of B if we draw the subset part without the equal part. So if A is a subset of B but is known not to be equal to B, it's said to be a proper subset. So here are some examples here again from the previous video. So with A and B being what they are, we saw before that A is a subset of B. Everything that is 0 mod 4 is automatically 0 mod 2 for reasons that you can work out on your own. It would also be correct to say that A is a proper subset of B because we saw that A is not equal to B. There are things that are in B like the number 6 that don't belong to A, so that makes it a proper subset. It would be correct to say that A is not equal to B. And why is that? Well that's because B is not a subset of A. Not every element of B is an element of A, so that makes those two things not equal to each other. And likewise we could say that A is not a subset of C. Now why is that? If A were a subset of C, then every element of A would belong to C. Now let's just list those things out here again. Remember A is a set of 0 mod 4 natural numbers, so these things here. And pretty quickly if you just look at the definition of C, that's the set of natural numbers that are 0 mod 3, 3, 6, 9, 12. Occasionally you get an overlap, but there are a lot of numbers in A, say the number 16, that you will not find in C. So because there is something in A that does not belong to C, that's why these two guys are not subsets of each other. So let's talk about this as in terms of quantified statements here. Now you might be getting the feel that there are some universal and existential quantifiers playing around here, and you would be exactly right about that. So what we're going to do on this slide here is set up kind of a logical framework for understanding subset inclusion and set equality that will be very helpful for us in later work. So when I say that A is a subset of B, what that means definition wise is that every element of A belongs to B. Now listen to the quantifier in that every element of A belongs to B. I'm going to write that as a quantified statement. That would mean that for every x in A, the following conditional statement is true. If x is A in A, then x is in B. So those two things mean the same thing. To say that A is a subset of B means that for every x in A, if x is in A, then x belongs to B as well. So subset inclusion is really about satisfying a conditional statement. So later work we're going to try to prove that one set is a subset of another, and to do that we'll have to prove a conditional statement. That gets us into pretty familiar territory. Now on the flip side of this, let's suppose I wanted to say something about A not being a subset of B. What does it mean to say A is not a subset of B? Well that would just be the negation of the red statement you see above. And let's just think about how negations of quantified statements work. If I'm negating an existential statement, or a universally quantified statement like I have here, that's going to lead me to an existential statement. So I would say there exists an x in A, and now I would negate this conditional statement. And remember that the negation of a conditional statement is not another conditional statement, it's a conjunction. This would be x is in A, and x is not in B. So that's what it means to say that A is not a subset of B. And this is fairly easy to think about actually, because if you think back to our A set from before, which was the set 4, 8, 12, and so on. And the set C, which was the set of the zero-mod 3 natural numbers, 3, 6, 9, and so forth. It was pretty easy to see that A is not a subset of C, and why was that? Well it's because there existed something inside A, such that that element was in A, but not in B, or C in this case. So you could take your pick, you could just look at the number say 8. 8 belongs to A, but it does not belong to C, and that's why A is not a subset of C. Finally, what would it mean for in this sort of language to say that two sets are equal to each other? Well it means that A is a subset of B, and B is a subset of A. In our logical framework, what that would mean is that for every x that belongs to A, that x, if x is in A, then x is in B and vice versa. So in other words, we'd have a bi-conditional statement, x is in A, if and only if x belongs to B. In other words, being in A is the same thing as being inside B. So this is really important here, and this is a slide well worth remembering, as we move on into proofs that involve sets. This is how we translate from set language into logic languages. This is very, very big deals here. Finally, here's a concept check to see how well we're acquiring these ideas. So let's let U be the set one through nine. A is the set two, six, and B is the set here given in a set builder notation. The set of all n in U, where U is up here, such that n is congruent to zero mod two. Then which of the following can you say? Well, there's more than one thing you could say here. Actually, almost all of these you could say. You could definitely say that A is a subset of B, because for every x that belongs to A, that x also belongs to B, because two is zero mod two, and six is also zero mod two. So that certainly works. It's certainly also the case that A is a subset of itself. That's simply because A is equal to itself, and we're not talking about proper subsets here. We're talking about subset or equal to. You could also say that A is a proper subset of B, because there are certainly some things A and B are certainly not equal to each other. And you can see that very quickly by finding an element of B that's not in A. And for example, the number four. Four belongs to B, but it does not belong to A. So that tells you another thing too, that you cannot choose option B, because the fact that four belongs to B but not to A means that B is not a subset of A. So there are your three choices. So we're beginning to reframe our set language in more logical type language, and that's going to be really helpful for us later. Thanks for watching.