 Welcome back to our lecture series Math 3120, transition to advanced mathematics for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. In lecture three, we're going to talk about the idea of subsets for which this is a continuation of our conversation about sets that we introduced in lecture two. As the name suggests, a subset is going to be a set, but it has this prefix sub attached to it. That is to say sub, meaning underneath, there are going to be sets inside of sets. Consider two sets, A and B. If A and B, before we define what a subset is, let's first define what set equality means. If A and B are two sets, we say that A and B are equal to each other if they have exactly the same elements. A and B are equal if they have all the same elements. We saw this in lecture two. We had an example like 1, 3, 2, 2 is equal to the set 1, 3, 2, something like this. Let's call this first set A, let's call this second set B. These two sets are actually equal to each other, A equals B, and that's because when it comes to A, you have 1, 2, 1, 2, and 3 belong to A, and then you also have that 1, 2, 3 belong to B, and nothing else belongs to them. These two sets contain the exact same elements, so we say they're equal as sets. Repetition doesn't matter because containment is not affected by repetition. Same thing by order. If I list the elements in a different order, so if I write something like the set 1, 2, 3, and this is equal to the set C, well, C still contains the elements 1, 2, 3, and it doesn't contain anything else, so C is likewise equal to these sets, because the order doesn't matter because the quality is determined by containment. If two sets contain exactly the same elements, then we say they're equal to it. Now, much like real numbers, we can talk about when two real numbers are equal. We can also talk about when two numbers are comparable to each other, so we can say something like two is less than or equal to four. There's this symbol that has to do with one number being bigger than another number. We can do a similar thing for sets as well, and that's what leads to this notion of a subset. If every element of a is an element of b, then we say that a is a subset of b, and this will be denoted by a symbol that looks a whole lot like less than or equal to. Our symbol for set containment, that is, we have a subset here, so we read this as like a is a subset of b, a is a subset of b. We often say a subset because there are often multiple subsets for a fixed set b here, so a is a subset. This is supposed to be similar to something like two is less than or equal to four. These symbols look identical, except for less than or equal to, we'll have this pointed while this one's more rounded, and we'll see this pattern again in the future as well. So let's do an example here. Let's take a to be the set one and two, and b to be the set one, two, and three. Then the first set a is a subset of the second set one, two, three, because every element of a belongs to b. One belongs to a, one also belongs to b. Two belongs to a, and two also belongs to b, and since one and two are the only elements inside of the set, that makes it a subset. Now, if we had something like the following, the set one, two, four is not a subset of the set one, two, three. Now, sure, the set on the left does contain one and two, so does the second set, but because the second, the first set has a four and a four is not in one, two, three, it's not a subset. If the set contains anything that's not in the other set, then it's not a subset. So that's an important example to keep in mind there. And so the idea of a subset is we're trying to think of it's like it's a set inside of a larger set. But the way that subsets are defined, you have to also accept that the definition does allow for a set to be a subset of itself, because after all, every element of a is an element of a, right, that's clear. And so by this definition, a is a subset of itself. This is often referred to as the improper subset. Now, it might seem weird to include it, but there are reasons too. And so we actually do accept that a set is a subset of itself, okay? That's an important thing. We'll get to some reasons why that is in this lecture. On the other hand though, if a subset is not the whole set, then we call it a proper subset. That is, if a is a subset of b, but a is not equal to b, then we call this a proper subset. Some people will denote a proper subset using this symbol, a subset b, but there's no line on the bottom. And the idea there is a, you have a subset of b, and you're supposed to consider this with something like two is less than four. That is where we drop the line because equality is not required anymore. In this case, it's not even allowed. But I have to warn you that in the literature, some people use this symbol to mean exactly the same thing as this symbol right here. That is they use this subset symbol to mean it's a proper subset or not, okay? So you have to be very cautious about it. As such, some people will use the symbol a, you have subset, but then they put a little slash to the bottom line, which says it's a subset but not equal. It gets a little bit more cumbersome. Be cautious of these things. In our class, if ever you see this symbol, it'll mean not equal to, but honestly, because all of the confusion, I'll probably just spell it out. a is a subset of b, but a is not equal to b. That's one of the best ways to avoid confusion is just to be explicit. So I want us to consider some examples here. So consider the following three sets. So a is gonna be a set of all even numbers. b is gonna be the set one, two, excuse me, two, four, six. And c is gonna be the set two, three, four, six. So I want you to note that b is a subset of a. And this is because, cause here, two, four, and six all belong to a. Every element of b belongs to a. Therefore, b is a subset of a. On the other hand, c is not a subset of a. And the reason for that is because c contains three, three is in c, but three is not inside of a. So since c contains an element not contained inside of a, that makes it not a subset. And so that's what it means to be a subset. You're a set inside of a set. Every element of the subset belongs to the other set, the larger set. Like I mentioned before, every set is a subset of itself. So a is a subset of a. It's also important to notice that given any set a whatsoever, the empty set is actually a subset of a. Now that one might be a little bit confusing or difficult to believe at first because to be a subset, every element of this set has to belong to this set, but the empty set doesn't contain any element. So is it true that every element of the empty set belongs to a? This is an example of what we refer to something as being vacuously true. We'll talk some more about that in the future, but let me rephrase the definition of being a subset to clarify here. In fact, let's actually work with not being a subset. A set is not a subset if there's something in this set that's not in this set, okay? Now when you look at the empty set, there's nothing in there at all. So there's nothing in the empty set that's not in a. So that means that the empty set is not, not a subset. Oh boy, a double negative. But just much like in English, we can resolve this, right? The double negative actually cancel each other out. If you're not not a subset, you're actually a subset. And so the empty set is in fact a subset of every set. This is important to mention because I wanna do an example right now where we're gonna write down all the subsets of the set ABC, it contains these elements A, B and C. And so for the sake of example, I'm gonna refer to this set as capital A here, okay? Now as I list down the subsets, this will include the empty set, which is sometimes called the trivial subset. It'll include the improper subset, which is everything. It'll also include all of the non-trivial proper subsets for which at Apeori, I might not know what those are. Now whenever you're asked to write down something or to list it or to enumerate, that is you're trying to describe all the elements inside of a set. And it can be difficult to write all of these if the set starts to get big. So if you're trying to write them all down, list them, you might need to use some type of strategy to make sure you get everything and don't miss anything. Now for this example, as there's only three elements in the set A, what I'm gonna do is actually write down all of the subsets based upon their size, that is their supposed cardinality. So let's write down all the subsets that have cardinality zero. There's only one set that has cardinality zero, that's the empty set. But like we mentioned above, every set contains the empty set as a subset. So I do have to include it. What about sets of cardinality one? This would be a subset that contains only one element. Well, if it contains an element, it's gotta be one of the three elements inside of A. So you're gonna have the set that contains just little A, just little B, or just little C. It turns out those are the only ones there are. And a set that contains only one element is commonly referred to as a singleton because it contains a single element there. What about sets that contain exactly two elements? Well, they have to be two elements from A here. So it could be A and B. It could be A and C. It could also be perhaps B and C, like so. And so these sets of cardinality two, you can think of as pairs. Now, later on in this lecture series, we'll talk about an ordered pair, which is a related, but a different thing. Because with a set, it doesn't matter the order you list the elements. And then lastly, there is a set of cardinality three, that is the whole set itself, A, B, and C. And so this gives us all of the sets. And if we count them, one, two, three, four, five, six, seven, eight, there are eight subsets in total. So remember this number. We're gonna come back and think about it in just a second. So there's eight subsets total. Now, when I presented this problem here on the screen here, I worked through it. I make it look kind of easy, very systematic. If you were asked to do this exact same task without me doing it first, you might struggle with it. Maybe you'll knock it out of the park. That would be great, but you might struggle with it. And if you struggle, there's probably two reasons why. One might be is that you've included too many things in your list. Perhaps, for example, you list B, A as a different set. When all reality, B, A is the same thing as A, B. The order in which you list an element does not matter. And so if you're thinking of these as ordered pairs as opposed to just a pair, then you might erroneously add the same set twice. You've double counted it or over counted it. You'd have to watch out for that. It's also possible that maybe you put in the set A, A, B, thinking this is a subset of cardinality three because it contains three things, but as the two things A and A are actually the same thing, this is a set of cardinality two. This is actually this set right here. This is same instance where you might count the same element more than once and get an over count. You have to watch out for that. Okay, another problem that people sometimes have is as you're listing it, you might have forgotten one. Just, you know, on good faith, you tried to list everything but by just a mere accident of a clerical error, you might have missed something. And so maybe BC was forgotten from the list. And now if you forget elements, this might undercount. Now, if you're undercounting because you forgot something but over counting because you counted the same thing twice, then your number might not be anywhere close to what the correct number was. In this case, it was eight, all right? Or maybe you magically got eight, but for the wrong reason, that's no good either. So we have to be more strategic in our counting. For this example, there's only three elements. It really wasn't the hard to do by listing them by order. But what if the order, what if the cardinality of the set A was actually 100? That would be a lot harder to do, right? Sure, the single tens, there's gonna be 100 of those. There's one empty set, but what about the pairs? That might be a little bit harder to calculate if you haven't considered that problem before. But what about the triples? What about all the sets of cardinality four, five, six, 56, right? That would get a lot more difficult to list them all if you had to do it. Like maybe you could write a computer program, but what does the program do? There has to be a strategy that can be implemented to guarantee that everything is there. And so in order to do that, we're gonna play a game right now. We're gonna play a game called build the subset. Build a subset. Now, this is kind of like the store that sometimes appears in a mall called build a bear. And build a bear like a child goes inside of the store and they get options on how to build their own teddy bear. They can pick the color of the teddy bear. Maybe they want a black bear, maybe they want a brown bear, maybe they want a white bear. They get it to decide, does the bear wear a shirt? And what color is that shirt? Does it wear a dress? What color is the dress? Does it wear a hat? What does the hat look like? And so they dressed the bear to be what they want it to be, hence build a bear, right? It's a simple idea there. We could build subsets in a similar manner, but instead of deciding like the color of the fur, the hat, the shirt, our decisions are things like the following. Does A belong to the set? Does B belong to the set? Or does C belong to the set? We have to make those three decisions. And each and every situation, there are exactly two options. The answers to the question one, two and three is either yes or no. Now, the answer to number one could be yes, but the second one could be no. They don't have to be the same answer, but each of these questions has two options, yes or no. And so we have to decide that. So if we're building a set, we're like, okay, I want it to have A, I don't want it to have B, I do want it to have C. We've just been constructed the set A, C. All right, does that make sense? And so what I can then do is visualize this. Imagine we have like a conveyor belt that's building all of our subsets here. And so the objects will then be categorized, kind of like a coin sorting machine of some kind. We're gonna categorize them based upon a path that'll take different decisions will leave us on different roads. Choose your own adventure. That's the analogy we should be using right now. So the very first decision we have to make is the first question, does A belong to the set? And there are two options here. There's the option that A does belong, and there's the option that A doesn't belong. That is an option. And then we have to make the second one. So we made our choice. Now the second choice has to be, does B belong to it or not? Now, much like we build a bear, when we decide whether B belongs to the subset or not, our choice for A doesn't actually affect our choice for B. There's so-called independent choices. I can choose either of the two options regardless of which path I'm on. B could belong to A or maybe not. And then it branches off over here as well. Maybe B belongs to A or maybe it doesn't. And then regardless of those choices, we have a third decision to make. Does C belong to the set or not? Well, for those you have two options. You have C belongs to A or C doesn't belong to A. Those are your options. And for each of these, they branch off in the similar manner. They branch off. I'm not gonna write down all the possibilities here. Just assume that the right always means that the element belongs and the left always means the element doesn't belong like so. And so I now claim that this binary tree that I've written on the screen right here, I'm gonna zoom out a little bit. This binary tree then represents all of the subsets. Like let's take the subset we were considering a moment ago, A comma C. So A comma C, it contains A. It doesn't contain B, but it does contain C. So you have an A, you don't have a B, you have a C. So at the end, we've now constructed the set A comma C. Now, conversely, let's say we wanna construct the set B. How do you construct that? Well, this singleton doesn't contain A. It does contain B, but it doesn't contain C. So you just get the singleton B down here. And every set can be constructed by following some path on this binary tree. Take for example, the empty set. The empty set says you don't have A, you don't have B, you don't have C. You always choose the left. And so the empty sets over here. But conversely, if you wanna find the set A itself, it contains everything, always choose the right. When there's a choice placed before, you always choose the right. You always choose A, B, and C. And so you get the set A, B, C at the end right here. And every path then constructs a unique subset here. Notice how many leaves are at the bottom of this tree. I know it's weird, because in biology, leaves are usually at the top of the tree, but they're at the bottom for us right now. You get one, two, three, four, five, six, seven, eight. Oh, there are eight leaves on this binary tree, and each leaf corresponds to a path on the tree. So if we count the paths, that actually counts the subsets. And that's where this eight came from. Okay, so notice what we've done here is we've used a diagram that helps us calculate all the subsets. So if we had a computer program, we could write it to follow all these paths and now count all the subsets and build all the subsets. Like so, now again, if we come back to what if our set has cardinality 100? Are we gonna draw this tree? Oh, that'd be pretty big, but we could still predict how big it is. Because after all, there's eight at the bottom. That eight is not a coincidence. Notice that the first decision, there was two options. The second decision, there was two options. The third decision, there was two options at each branching, okay? So regardless of which direction you go, there was two options for the first choice. And then base, so you have these two branches. And then each of those nodes branches off in two directions, thus giving us four. And then four nodes right here at this level. And then each of those nodes branches off into two directions because again, the options are yes and no. So you get like so. And so at the end we end up with eight. And so watch the progression of nodes here. You had one, two nodes, four nodes, eight nodes. These are powers of two. What we're saying in hindsight here is eight turns out to be two cubed. Where did the two come from? Two comes from that in each branch, we have two options, yes or no, in or out. And then the three comes with the number of decisions we have to make, which is the number of elements in play here. So when we have three elements, the number of subsets we get is going to be eight, two cubed. I then claim that the number of subsets of a set of size 100 would be two to the 100th, which is a huge number, but the correct number of subsets. We can actually take the argument we just made and formalize it into an actual proof. It isn't a coincidence. If we have a set that contains in elements the number of subsets is equal to two to the end. It's the exact same argument we saw on the previous screen now generalized. So let's now read the proof together here. To see this, imagine that a set A contains in elements and we can label those elements A1, A2, A3 up to An, just arbitrary labels here. Imagine that we want to make a subset of A and that subset will be called B. B for build the subset. We would have to decide whether A1 belongs to B or not. There's two options, either it does or it doesn't. Then after that's decided, we have to decide whether A2 belongs to B or not. Then we have to decide if A3 belongs to B or not. Then we have to decide if A4 belongs to B or not, etc. Going all the way down to N. Each of these questions has two options. Yes, it's in there or no, it's not. And so imagine we have a complete binary tree of depth N, that's this picture right here, although I use some dot, dot, dots to suggest it could be very large. Each decision on whether AI belongs to B is like a fork in the road. It's a branch on our binary tree right here. If AI belongs to B, then we choose the right edge. If AI doesn't belong to B, we choose the left edge. Then when we are done making all of these decisions, we have created a path from the root of the tree to one of its leaves, like so. As it's a rooted tree, each path is uniquely determined by its final leaf. As a complete binary tree has exactly two N mini leaves, so the reasons we talked about before, and each leaf coincides with a path and each path coincides with a subset, there are exactly two to the N mini subsets of A. Now this example we just did here is our first example of a proof technique referred to as a combinatorial proof. This is something we will study again in the future, don't worry about it right now, but the idea is we counted the same set in two different ways. We were counting the paths on a binary tree. The passive binary tree coincide with the subsets of a set, but they also coincide with the leaves of a set. And so using this, we're able to count the number of subsets, which is gonna be two to the N for a finite set. For an infinite set, clearly it's infinite, so we don't need to say much more about that. Now to close this video, I did wanna introduce a little bit of notation. This is probably something you've seen before, but just for the sake of it, we have to define things in mathematics, so here it is. When we talk about subsets of the real numbers, we often use interval notation to describe them. And so there's an example of an open interval, which is defined typically in the following format. We have a left parenthesis, A, B, right parenthesis. This will consist of all real numbers X, which X is greater than A, but less than B, an open interval. There's similar, there's the idea of a closed interval, which is denoted with a square bracket, left bracket A, B, right bracket. This will consist of all real numbers X, such that X is greater than or equal to A and X is less than or equal to B. So the main difference between, of course, the open and the closed interval is that these numbers, A and B, are called the left and right end points of the left and right end points of the interval, respectively. The closed interval contains the end points, the open interval does not. We'll actually prove that in the very next video. You can also have half open or half closed intervals, things like parenthesis A, B bracket, or you could also have bracket A, B parenthesis. Basically, the bracket always means that you include the end point. The parenthesis means you don't. So if you look at parenthesis A, B bracket, that means that X is greater than A, but not equal to A, and X is less than or equal to B. Then if you take bracket A, B parenthesis, that means that A is less than or equal to X and X is strictly less than B in that situation. So A is greater than or equal to, X is greater than or equal to A, but X is less than B in that situation. All right, so we've probably seen this before. We also allow for the possibility that one of the end points could be plus or minus infinity. So you could have bracket A infinity, which means all numbers, all numbers X that are greater than or equal to A. Parenthesis A comma infinity would be all numbers strictly greater than A. You could have negative infinity B bracket, which would be all numbers less than or equal to B, or you could have negative infinity B parenthesis. So this would be all numbers strictly less than B, or you could have everything, all numbers between negative infinity to infinity, that's the whole set of real numbers. So we don't use this one that often. We'll typically just write it as the set of real numbers. Notice that with these ones, you never put a bracket next to infinity or negative infinity because that notation would suggest that infinity or negative infinity belongs to the real numbers, which it does not. And these terminology is about open and closed. They're given meetings in calculus and in topology, but we'll just use this standard notation if we need to describe intervals of the real line.