 Hi everyone, I'm Denchie, and in today's video we're going to be talking about sets. Now, sets are the basic building block of mathematics. You might think that's arithmetic or numbers, probably because that was the first stuff you were exposed to as a child learning math at school. However, when you really think about it deeply, sets make up almost everything in math. And if that sounds confusing, then just follow along this video and you might begin to understand a few things about sets. So let's begin with the definition of a set. Now, just thinking about the word set in English, you normally think of it as like a group of objects. So let's write that down. Group of objects. And that's a pretty good definition for now. We can say something like all fruit is a set. So that would be stuff like, you know, apples. That's an apple, definitely. An orange, a lemon, and, you know, a pear. That's a pear, apparently. And every other type of fruit. And this would be a finite set because there's a finite variety of fruit in the world. As you can see, I'm not using numbers. I'm not using weird letters or fancy symbols. What I am doing, however, is using these curly brackets. This is what we call notation in mathematics. It's basically a way of writing different concepts in math in a way that human beings can read them. Now, I don't expect you or anyone else watching this series to memorize the entirety of notation, although you should become somewhat familiar with it because I'm going to be writing a lot of it in this video. However, just understand that the underlying concepts under all of this are actually quite simple. Or even a better word we could use is they are natural. Nature has got a lot to do with math. Nature is intimately related to math. And I'm not talking about the nature like, you know, leaves and animals, the stuff you would study in biology. That stuff is always changing to some extent. But nature in math really isn't. Nature in math has got to do with the universe and the fundamental laws that govern it and values and relations and things like that. Basically the rules of logic in the world. And that's what math explores. Math is the study of natural logic and the way we write it is called notation. Now that we're done with all of this introductory stuff, let's take a good look at sets. So the first thing we're going to do is we're going to create a new set. And this is going to be a finite set. And the items in the set are going to be letters. So, you know, A, B, C, D, whatever. Don't worry about me using letters specifically. These could be anything. This could be fruit, oranges, whatever. Just make sure you understand that A is not B, which is not C, which is not D, and stuff like that. They aren't the same. They're like different objects. And that's going to become important when we take a look at comparing the values in sets or the objects in sets. So first of all, let's create a new set. I'm going to call it capital letter A. You might notice a lot of the time when we're talking about sets, we use capital or big letters. That's just because sets are big, I guess. I don't know. They're just like that. And A is going to be equal to, we're going to put curly brackets again. We're going to put the objects inside of this set, the contents of the set. And we're going to say, OK, A will contain A, B, C, D, E, and F. So these letters are part of the set A. If I told you pick a random item from set A, you would say, oh, C or E or A or F or A again, stuff like that. And what we can say is that any of these letters, let's say E, for example, is a member with this fancy-looking E symbol over here. That's a notation thing, so pay attention to that. E is a member of A. And we can also say that, let's say, G is not a member for the cross through it of A, because it's not contained within A. There's no G here. So this is an easy way for us to understand whether items are in or not in a set. And an easy way to denote all the items in a set is by writing them between two curly brackets with an equal sign and then the name of the set, in this case, A. However, let's start learning how to deal with multiple sets. Let's make a new set. The set is going to be called B. And B is going to contain the letters G, H, I, J, K, and L. So now we can say that while E is not in B, G is in B. Or more specifically, G is a member of B. That's the fancy math word for it. Here what we've created is two sets which are completely different. If you look at all the items in this set and all the items in this set, they are all completely different items. There's not one item which is in both set, not one object, I guess. And this is important because now what we can do is we can draw a diagram of these two sets. Now remember when I said that notation of math is not about the fundamental natural thing we're studying, but it's just a way for humans to understand it? Well, there are multiple ways to denote sets that may make them easier to understand in certain situations. One of these is called a Venn diagram, or a V-E-N-N diagram. And a Venn diagram is pretty easy to draw. All we have to do is draw both sets. And in this case, we have A. And the way we denote the items in this set is just by putting a shape, like a circle. And in here, we're going to put the items A, B, C, D, E, and F. And then we're going to draw B over here. We're going to do another circle. And it's going to contain G, H, I, J, K, and L. So these are two sets. And as you can see, they happily live apart from each other with no intersection as you often do see in a Venn diagram. In this Venn diagram, we're just drawing two sets because these are two different sets. But now let's take a look at what happens when we have two sets where there are items which are present in both sets. So let's start with a new set. We're going to call it C. And C is going to contain the items G, H, I, J, K, and L. OK. And then make a new set called D. And D is going to contain A, B, G, I, L, and F. All right. As you can see, we've created two new sets. And just by looking at them, you can notice that there are items in both of these sets that are present. So I is present in both of them, L is present in both of them, and G is present in both of them. In this case, our Venn diagram will change. So we're going to write over here C and put a little circle. But because we know both sets contain items which are in both sets, so like L and I and G, we're going to draw the D set overlapping the C set. So these two groups of objects can be visualized as overlapping each other because their content, in a way, overlaps when you think about it. It's like these two things are laying on top of each other, but only partially because they're not the same set. So in this case, we can start writing the stuff which overlaps in the middle. In this case, it's G, I, and L, so we're going to put G, I, and L. And the items which are exclusive to C are H, I, J, and K. And the items exclusive to D are A, B, and F. Now let's take this a step further by looking at the notation to define the situation over here. Now we can say that C obviously is not equal to D. But we can also say that there is a new set formed over here. In fact, there are three new sets formed over here. And looking at this Venn diagram, I think you can guess what they are. There's a set of everything in C that isn't in D. There's a set of everything in D that isn't in C. There's a set of everything in both sets. And there's a set of everything that is intersected over here. So in this specific case, we can see that G, I, and L are the intersecting items, so the items that are formed by the intersection of C and D. And we can denote that as C intersection with this funky little symbol, D. Once again, don't feel too pressured to learn all the little symbols and the names. Just make sure you understand that what I'm talking about here is the items which are present both in C and both in D, which in this case would be equal to G, I, and L. So remember, C is a group of objects. D is a group of objects. But by using both of these sets together, we can form a new group of objects, which is the group of objects which both of them contain. That's a valid group of objects. Now let's take a look at something else. What is C union D? C union D is every single item in the set. So that would be I, H, J, K, G, I, L, A, B, and F. So that's every single item in the set. So the union means you unite the two sets and have everything in it. So this set is the biggest set we can get out of these two sets. It's got the most items in it. In fact, the size of a set, the amount of items, number of items, is what we call the cardinality of a set. Cardinality. It's a funny, funny, funny word. Don't worry about that too much. Just remember that what we're doing over here is we're taking these two sets and we're forming the union of them. And it forms this big set over here, which contains all the items in them. Also, one important thing to note. This union set doesn't contain duplicates of the items. We're not going here and listing these off and then listing these off. We're canceling out the items which are the same. And we're only including the unique, unique items. Because in this case, the sets are dealing with objects which exist in the universe, I guess. You could say that the objects themselves are unique and the sets are just descriptions of groups of them. It's like if you were dealing in real life, if multiple people own the same house or the same object, these aren't new declarations. These aren't new objects being created each time. This is just C saying that it owns these and D saying that it owns these. And both of them can own the same letter at once. If you want to think of it with that analogy, just remember they have to be unique. You can't have I and I or H and H or L and L or something like that. You can't have multiple items in the sets. Anyways, now that we've taken a look at the basics of uniting and intersecting sets, let's take a look at a few other advanced set concepts. So one thing you might start thinking about is what happens if I have a set? Let's pick a new letter, like E, for example. And the set is basically gonna contain, you know, A, B, C, D, E, F, and that's it. And then we make a new set called F. And F contains B, C, F, and E. Notice something about these sets. What's very unique over here is that the items in F are all contained within E. And guess what? Our Venn diagram is gonna change again. We're going to draw a new Venn diagram over here. We're gonna draw the set of E. And then we're gonna draw the set of F. And as you can see, the set of F is nestled. It's inside E. There is a funky way to denote this, and that would be to say that F is a proper subset of E. Don't worry too much about the symbol. The symbol basically means that F contains items. All of its items are within E. And the line means that the two sets are different. So F, it has all the items as E, but that doesn't mean that F is equal to E. That's why there's a cross. It's like that not equal sign. F is not equals to E, but everything in F is in E. So F is a proper subset of E. That's the funky way of saying it, the fancy way. So now let's write all of our items in here. We're gonna put A over here, B over here, C, F, and E over here. And we're gonna put D over here. And I think that's it, yeah. So as you can see, what we've done over here is created a set called F, which is a proper subset of E. And we've drawn this in our Venn diagram as a circle inside of the circle. And when you think about it, this makes sense because everything in here is in here. It's like having a big set like E and having a little set F resting on top of it. They're not intersecting because intersecting would imply that F has items to it, which are unique, which is not the case over here. So whenever you see sets like this, make it a fun exercise to draw little Venn diagrams to understand what's actually going on. Also, don't be confused by this. F, this is just the circle, it's not an item within E. Although F is a subset of E, F is not an element, which is like, it's not an object that's part of E, it's just a set. Anyways, here's a fun little primer question. Does the set of all sets contain itself? I don't know, do you have an answer to that? If you have an answer to that, comment below. But yeah, does the set of all sets contain it? If you made a set and it had all the sets in it, would it contain itself? Another fun thing you can do with sets just as a little exercise is you can take something called the Cartesian product. So if you take E and you write a little cross next, this is not multiplication, this is just a cross. It means Cartesian product. And you cross it with F, you end up getting a new set, which is a set of sets. And I'm gonna put the other sets inside of the set inside parentheses and not curly brackets just to make it a little easier to understand. Basically, when you take a Cartesian product, you're like taking all the combinations of the sets. So that means that the items in the set, the sets in this set are going to be things like AB, AC, AF, and like AE. So there's a little set that contains A and B and that set is inside of F and there's another little set and so on. And the idea here is I'm taking all of the items in E, like A, and I'm saying, okay, what are the combinations it can have with F? So AB, AC, AF, AE, and then I'm starting with B again, B, B, BC, B, F, BE, and then I'm starting with C, C, B, C, E, C, F, C, E, then D, B, and so on. And you can create something called the Cartesian product. That's just something you might be asked on a test if you're watching this video for school or something, just a fun little thing you can do with sets. Now that we've learned about union, intersection, Cartesian products, and subsets and stuff, let's learn all about number sets, sets of numbers, which is kind of the main part of this video and the shortest but most important part that you can understand after learning all of that stuff that you just learned about with sets. When you're dealing with numbers in real life, there's a set of numbers which we're pretty much always thinking about in our brains whenever we see objects, at least if you think about math daily for whatever reason. And that would be natural numbers. Natural numbers are the only numbers that we can see and properly touch. Now what I mean by that is, let's say I have one pen. Let's say I have zero pens. Let's say I have two pens, I don't have two pens, but if I did, I would show you two pens. Three pens, four pens, five pens, so on. As you can probably guess, this set, which we're denoting with this big fancy-looking N over here, is gonna contain all of the numbers, zero, one, two, three, four, five, and so on, all of these counting numbers. So think of these as counting numbers. And one fun thing we can do with the natural numbers is put them on a number line. So here's a line, and we're gonna put all of the basic little numbers we have here on it. One, two, three, four, five, six. Oh no, we ran out of space. Well, we're always going to run out of space because this is an infinite set. It never ends. It goes on to infinity. That's the worst infinity symbol ever, but you get the idea. You could technically go on forever. For any item in the natural numbers, you could find it plus one. So think of a natural number right now, quick. 487, well, I caught you one better. 488, it's a bigger number. So you can keep going forever with this pattern of adding one, you can always add one. And so it's an infinite set. Now, when you think about this number line over here, we can start doing some basic arithmetic operations. I'm not gonna focus too much on arithmetic today, but this will be very useful for anybody who's already quite familiar with addition and subtraction and you probably are if you're watching this video. So let's think about some basic concepts. Let's think what happens when I add, I don't know, three to two. So two over here, that represents two objects and three over here, that represents three objects. And if you want, you can count this and be like one, two, three, four, five. Okay, the answer is five. Or you could use the number line, which would mean going from two and then going to three, four, and five. I've done three hops. One hop, two hop, three hop. And now I've gotten to my answer. So by using the number line, we can go through and figure out the answer to basic arithmetic questions like two plus three, which as we can see is equal to five. All right, now we can also do the opposite. Let's say we want to go three and want to find out what happens when we go back two steps on a number line, or we go back by a negative amount. So three minus two. We use the dash symbol to show when we're going backwards on the number line. So we would start from three and we go back two times, one, two. That means we get to one. So the answer would be one. I've done two hops, but I've done them backwards because we're subtracting two. We're subtracting two, so we're going two hops backwards from three. Now let me ask you a tricky question. What if I asked you to do five minus seven? Now, let's try doing that on a number line. Let's go seven hops backwards like we were doing two hops backwards at three. One, two, three, four, five. But oh no, we've encountered a bit of a problem. We can't really go back. There's a limit over here. There's zero over here and the natural numbers doesn't contain anything before zero if there even is something. But what does before zero even mean? So what are we gonna do? So there is a simple solution to this and it's called making it up. Now obviously mathematicians didn't make up everything. These are natural concepts which are very useful in real life. However, when I extend the number line this way and start adding negative numbers, negative one, negative two, negative three, negative four and so on. Well, what does that mean? What does it mean to have a negative one? Obviously I can have one pen. I can have zero pens. I could have two pens if I had them. But can I have negative one of a pen? No, that's impossible. It's impossible to have negative of a pen. However, now that we've done this and I've kind of playfully put it over here, put the negative numbers to the left side so it makes it easier for us to understand how this links into subtraction. But if we go from five and try to hop seven times backwards. So let's do the exercise we were doing again where in subtraction we hop to the left seven times. One, two, three, four, five, six, seven. You can see that our extended number system allows us to get a new answer which is negative two. So the answer to this question would be negative two. Now for a very, very long time, mathematicians actually did not believe in this. They did not believe in the idea of negative numbers. So what they would have done here is not write negative two. They would have rewritten this in a different way. They would have said five plus two is equal to seven. And when you think about it really hard in your brain, these two statements are kind of the same thing. They mean the same thing about these numbers. Because when you think about it, if two things added together make a thing, then surely one of those things taken away from that should make a similar thing, shouldn't it? Now we're gonna look into a birth to take a lot better in the next episode where we're actually looking at addition, subtraction, multiplication and so on. But for now I want you to understand that when I say that we're going to extend the natural number system to include something called the integers, which we denote with this Z over here, then it's not just an arbitrary decision. It's actually fundamental to our natural understanding of math because it makes questions like this answerable. You can find the answer now because we've created, we've made up a new number system. So what are the integers? Let's draw another number line and take an example. So let's say we have another number line with all the numbers on it. So you know zero at the center, then one, then two, then three, four, five, and of course this would go on to infinity. And now this newly invented negative number system we've just been talking about, negative one, negative two, negative three, negative four, negative five. Just knowing this number line, I'm sure you could already tell that the natural numbers belong over here. These are the natural numbers. So all the numbers, zero, one, two, three and so on, on this side of the number line. However, the integers, which I was talking about with that Z symbol from before, are the entire number line, at least the one we're looking at now. So Z over here would make up all of these numbers we're looking at here and all of their extensions towards infinity and negative infinity. So negative five, negative six, negative seven, and five, six and seven, so on. All of these are part of the integer set or the Z set. When you think about it now, if you're thinking about these sets, very, very hard and think back to the Venn diagrams we were looking at earlier, linking back all the stuff which we've been talking about in this lesson, we can now draw a Venn diagram of these two number systems, the natural counting numbers, the ones which we can count and hold in our hands and the integer numbers. And we can say the following statement with a Venn diagram. We're gonna say that the natural numbers are a proper subset of the integer numbers. So like we were doing before, with the sets that we were looking at before, we're one of the set, all of its items contain the other set's items. In this case, we're looking at how Z or Z contains all the items in N because Z is a set of all integers, so every single whole number and N is a set of all the counting numbers. So N, if we write it in set notation, is a perfect subset of Z. And so we've linked back everything we've done in the lesson so far. Now number systems expand far, far, farther than this. There's things like the rational numbers with this little Q symbol. There's things like the complex numbers with this little C symbol. There's even quaternions and lots of other fancy numbers. And of course, the infamous real numbers, which are every single possible number, which you can think of negative and positive that does not go into the complex plane. But obviously you won't have to worry about that for quite a while in this math series, but we will take a look at these different sets of numbers after we've taken a good look at arithmetic. Because arithmetic is when math starts getting especially useful, but also is when a lot of people start struggling with math because maybe they're trying to do things in their head or maybe they just struggled to do calculations or maybe they just struggled to understand the dynamics of negative numbers and positive numbers, how they interact with each other and multiplication. And then there's fractions when we start looking at rational numbers, which are these ones. But for now, ignoring all the confusion that's to come, I hope you enjoyed this fun math video. I've been Denchie and goodbye.