 Let's continue our discussion about sets for the context of abstract algebra. And specifically in this video I want to talk about the idea of a subset. One can compare real numbers by determining whether one is bigger or if they're the same, right? We often say things like two is less than or equal to three, right? There's a comparison there. Or we could say like negative five is less than or equal to seven. We can compare real numbers based upon their size, whatever that means, right? But sets, we can do a similar thing as well. We can actually define a comparison between sets that very much acts like the less than or equal to symbol we've seen already. And this is going to come into play as we talk about subsets. So imagine we have two sets A and B. We say that two sets A and B are equal to each other if A and B have exactly the same elements as each other. So if every element of A is in B and if every element of B is in A, then we say the two sets are going to be equal. This very simple but fundamental property is going to be very useful as we start talking sets, which we'll do later in this lecture. So I'm going to come back to this one in just a moment here. We say that if every element is an element of B, then we say that A is a subset of B and that's denoted in the following way. A is a subset of B and so our symbol right here, our symbol right here, this symbol represents the subset symbol that if you take a set on the left, A is a subset of B meaning that everything inside of A is also inside of B. Just as a little bit of late-tech experience, in case you do want to tech these things up on your own, this symbol right here, typically you're going to use the symbol backslash subset, EQ. The EQ at the end puts an equal sign at the bottom of these things. If you omitted the EQ, you just do backslash subset, you're just going to symbol that looks like this because this kind of looks like a less than or equal to or just a less than sign, except it doesn't come to a point. It's rounded when you write it. If you just do backslash subset, then you don't have the equal sign on the bottom, which you might not want for reasons that'll be clear in a little bit. I should also mention that if you want to say an element like X is inside of a big set X right here, this symbol right here in late-tech is backslash N. Pretty simple. This is some notation you're going to want to use as you describe sets. Also with these curly braces, you can't just write curly braces in late-tech because this is a controlled symbol. You're going to have to do backslash curly brace, backslash curly brace, if you want to get these symbols for late-tech. And likewise for the empty set, which will be coming up in just a moment. If you want to describe the empty set, you'll just do backslash empty set. Many symbols in late-tech are quite intuitive, and oftentimes you could probably guess what the symbol's name is going to be. But getting back to the lecture at hand, right? If we have a subset, an example of that would be something like the following. The set 1, 2 is a subset of 1, 2, 3. Because if you take every element on the left-hand side, 1 is a member of the set, and 2 is likewise a member of the set. So in fact, we have an example of a subset right here. A subset, as the name kind of suggests here, sub meaning underneath, like subterranean is that area that's underneath the earth. A subset here means it's a set inside of another set. And so 1, 2, 1, 2 is a set, subset of 1, 2, 3. In fact, this is what we call a proper subset, because 1, 2 is not the set itself. A proper subset would be a subset, which is not equal. The thing is the two sets are not equal, because the second set has the element 3, but the first set doesn't have 3 in it anywhere. There's no 3 to be found. So the two sets are not equal, but one is a subset of the other. And so we call this a proper subset. The reason why we have to talk about a proper subset is because it's possible to have an improper subset. It's possible, for example, for a set to be a subset of itself. That's actually always the case. Like if you take the set 1, 2, 3, it's a subset of itself, 1, 2, 3. Because if you look at the definition, 1 is in both of the sets. 2 is in both of the sets. And 3 is in both of the sets. So in fact, the set 1, 2, 3 is a subset of itself, because every element in A belongs to A. And this is actually something that's true for every set. Every set is a subset of itself. And we often call this the improper subset, which is why we have to sometimes talk about proper subsets, if we want to make sure we're only talking about those sets that are strictly smaller than it. I should also mention that the empty set is actually a subset of every set. And this is true because the empty set is vacuously a subset. Because the definition says that every element of the set on the left belongs to the set on the right. Now, can we find an element in the empty set that's not contained inside set A? Think about it for a second. Is there an element of the empty set that doesn't belong to set A? Well, the empty set contains nothing. Therefore, there's no counter example. There is no element of the empty set that doesn't belong to A, so therefore it's a subset. This can be a little bit tricky for students when they're first exposed to logic, because this right here is when we say something is vacuously true. We have this conditional for which the hypothesis is not satisfied. And so it doesn't matter what the conclusion is, the conditional statement is considered a true statement. I'll give you a counter example to the empty set not being a subset. So there's always a subset. And so given any set under the sun, any set, it doesn't matter. You always have the following statements, always, always, always. Given any set A, A will be a subset of itself. And the empty set will be a subset of A. That's always true. Now, what subsets live in the middle? Well, that depends a whole lot on A. And so we get these, we often look for these proper non-trivial subsets. The improper subset, of course, referring to the whole set itself. The trivial subset being the empty set, right? So are there non-trivial proper subsets? That depends on the set itself. So consider the following three sets. Let A be the set of all even numbers. We'll say even natural numbers there. And so B will be the set 2, 4, 6. And then C is the set 2, 3, 4, 6. Notice that B is a subset of A because 2, 4, and 6 are all even numbers. It's a proper subset because there are certainly even numbers which do not belong to 2, 4, 6, like 8 or 0. So there's not equal. And so because, for example, 8 is an A, but it's not a B, we have that A is not a subset of B. So they're not equal as sets. And so what we often can do is we can denote this using B subset A, but notice there's no equal sign underneath it. Because much like if we say 2 is strictly less than 3, by omitting the equal sign there, we're saying this is a strict inequality. 2 is strictly less than 3. So in our context, that's what we mean right here, that B is strictly less than A. Now I'm going to warn you here that much like when we define the natural numbers, there are other mathematicians who take a very different convention here. When they write this symbol right here without the equal sign on that, that actually means the subset symbol we have. So they use this symbol and this symbol interchangeably. And so that's sort of a different convention that some people take. And so even though we will use it for proper containment, some authors, some papers, some textbooks would use this symbol to mean subset. It could be improper subset. Now to help distinguish between that, some people use the symbol subset and then there's a line underneath it, but there's a slash through the line to indicate that it's not equal, but it's a subset. That's easy to do in Leytec. It's subset and then it's NEQ for not equal to. And so again, some people use that symbol. We won't be using it that much, but honestly, if I was writing a research paper and I want to indicate that this is a proper subset, I'd probably use this subset NEQ just to clear. Just because again, some authors use that interchangeably. How do we live in here? Some other things to mention. B is a subset of C. One or two, four, six does belong to the set two, three, four, six. On the other hand, C is not a subset of A because three belongs to C, but three does not belong to A. Something that belongs to C that doesn't belong to A. Therefore, it's not a subset. Everything that belongs to C must be part of A if it is in fact a subset. And so this is how one actually shows equality of sets. If A is a subset of B and B is a subset of A, then that actually implies that A equals B. Now, this principle right here, it seems basic, but it's actually a very important property. A very important proof technique, one might say, about sets. And we'll see that at the end of this lecture here, that if you want to show that two sets are equal, one of the best ways to do it is to show that the two sets are subsets of one another. That is, taking an arbitrary element of A, you show it belongs to B, and take an arbitrary element of B, you show it belongs to A. That then would show that the two sets, so this is our first proof technique. And most of the proofs you would write for this lecture in the homework is going to follow that template. Show the two sets are subsets of each other, and that forces equality of the two sets. Thank you.