 Hello and welcome to the screencast on the elements, subsets, and set equality. Let's take a look at these three sets here. A consists of the numbers 1, 2, and 3. B consists of the numbers 3, 6, 9, 12, 15, and so on. So other things that are in this set would include 18, and 21, and 24, all the positive integer multiples of 3. And then C is just the set of all natural numbers. And remember that means the set of all positive integers. In this screencast, what we want to do is describe containment. What it means for something to belong to a set, and how we talk about sets that seem to have some overlap to them. First of all, let's talk about objects being in a set, like the number one here. The number one is an object that lives inside the set A. It's kind of like I go on a trip and pack a suitcase. I put my shoes and a shirt and a shaving kit inside. The shoes are inside the set. Now, how we're going to say that is to say that one is an element of A. Element, and we're going to use this notation here. It looks like an E for element of sorts. To say that one is an element of A. Other elements we see here, three is an element of A. We can see it right there. It's an object that is in that set A. Three is also in B, it's an element of B. And three is also in C, although we don't actually see it here. It's understood, and we all know that three is a positive integer. And that's what this set consists of. One is not an element of B, it's just not in there. It's like I went on a trip and forgot to pack my shoes. Where is it? It would be here if it were anywhere and it isn't. So one is not an element of B. And five, for example, is an element of C, the entire set of natural numbers. But five is not an element of B, and it's certainly not an element of A. Now, some of these sets have overlap. And you probably noticed that, for example, three belongs to all three of these sets here. And in fact, every element of A belongs to the natural numbers. Every element of A is also an element of C. And so when that happens, we're going to say that A is a subset of C, a subset. And we're going to use this notation here, which kind of evokes the notion of less than or equal to. It's not really less than or equal to, but it does mean less than in some sense. A is contained in C. All the elements of A are also in C, the set of all natural numbers. All the elements in B are also contained in C. And so B is a subset of C. It seems pretty trivial to say that B is a subset of itself. I mean, every element of B is also an element of B, just saying the same thing twice there. And every set is considered to be a subset of itself. On the other hand, if I could just clear some of the writing off here, we can see that A is not a subset of B. And how do we know it's not a subset? Well, it kind of focuses on this word every. Is it the case that every element of A is also in B? Well, no, it's not the case. For example, one and two, those are elements of A that do not show up in B. I mean, there is some overlap. There are some elements of A, one element of A that belongs to B. But there are also other elements that do not belong to B. So A is not a subset of B. So let's have a concept check to see how well you're understanding these notions here. So let's A let A be the set we've seen before one, two, and three. And D is a new set that consists of three, two, three, one, two. Then which of these four statements is true? And you can just select all that apply. There could be more than one statement that is true. So look at those options and pause the video and check all that apply. And we are back. And the answers here, there is more than one, are C and D. It's the case that A is a subset of D and that D is a subset of A, too. But we don't say elements. So let's think about why these two are true and then we'll think about why these two are false. A is a subset of D. This is fairly easy to see if you understand the definition. It's certainly the case that every element of A is also an element of D. You can just visually check them off. One, yeah, we got it. Two, yeah, we got it. Three, yeah, it's over here. So certainly this is true. Now what about the other way around? Let's remove the check marks here. If I go the other way around and start going down the list, well three, yeah, that's over here. Here's an element of D and yeah, there it is over here. And here is another element of D and that's over here. So every element of D is also an element of A. We don't really, so we don't really think about listing elements twice. These are just kind of redundant here. And if you mark those off, you can really see that even though they happen in a different order that every element of D is indeed an element of A and vice versa. Now we would not say that A is an element of D because think about what the elements of D are. The stuff that lives in D are numbers, okay? The only thing that is an element of D is a number. There are no sets that live inside D. They haven't packed a suitcase inside a larger suitcase. So it wouldn't be right to say that A, the entire set, one, two, three, is inside here somewhere because this is not a number. This is a collection of numbers. And so to say that A is an element of D isn't really the right syntax here. Elements of D are numbers in this case and A is not a number. And so that also rules out B. So A is a subset of D and D is a subset of A. And when we have this sort of mutual subset inclusion relationship here, we say something special about those two sets, namely that they are equal. Two sets are equal if they contain exactly the same elements. So that would be the case for the sets we just looked at in the concept check. They contain the exactly the same elements. The order is kind of screwed up over here and we had some doubly listed items. But they have the same elements. There are no new elements introduced on either side of the equation. We would also say although one, two, and three, the set containing one, two, and three is a subset of the entire set of natural numbers. That's certainly true. They're not equal by a long shot. They do not contain exactly the same elements. For example, the number four is in the natural numbers, but not in here. So that's something new, something that lives on the right-hand side set, but not in the left-hand side set. Every set is certainly equal to itself because every set contains exactly the same elements as itself, kind of no-brainer there. And then finally, something that's going to be an important way of thinking about set equality. So kind of put that on your radar screen is that two sets are equal, if and only if two things happen. A is a subset of B and B is a subset of A, just like in the concept check. So if A is a subset of B and B is a subset of A, then that means that they contain exactly the same elements and vice versa. So that gives us a handle on what it means to be an element of a set, a subset of another set, and whether two sets are equal. Thanks for watching.