WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi; welcome back to Educator.com.
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Today, we are going to talk about sets, elements, and what they mean for numbers.
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To start, we are going to talk about the idea of sets.
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But a lot of courses don't address this stuff directly.
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You might find, in the course that you are taking, that your teacher never talks about this directly.
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But these ideas build the foundation that the rest of math works on.
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So, you can understand all of the concepts that will come later in this course without ever having watched this lesson.
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But watching this first will help you see how it all fits together, which will really help your understanding, which will make it that much easier on you.
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Also, if you want to go on and take later, more advanced math, like calculus or even much more advanced college courses
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(like abstract mathematics), this stuff is going to be really useful to have already ingrained in your mind.
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These ideas are great later for you; so if you are going to take advanced math, really, definitely, watch this.
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Get an understanding of what is going on.
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Once again, you don't really have to deeply understand any of these things; we are just going to be touching it on the surface.
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But you want to get a glimpse of this sort of stuff, so that later on, you can really understand what is going on.
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And also, it is just going to make things a lot smoother, especially when we are talking about functions.
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So, you can get an abstract idea of how a function works, which will help you understand what is going on.
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All right, let's get started!
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A **set** is a collection of distinct objects; each of the objects inside of a set is called an **element**.
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An example here: we have two sets: {1, 2, 3} is a set--each one of those elements is different from all the other elements.
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1 is not 2 or 3; 2 is not 1 or 3; and 3 is not 1 or 2.
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Similarly, for the set {cat, dog}, we have cat different than dog, and dog different than cat.
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Also, really quickly: the way that we show that we are talking about objects inside of a set is: we have these curly braces.
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And I am not that great at making a curly brace, but it is something like that for the left side, and something like that for the right side.
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You can see a nice typography (typed-out font) brace in my slide here.
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But when I am actually writing it out, I do something like these right here.
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So, we put our elements inside of that, and we separate each of the elements with a comma.
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That is just what is happening on a type point of view.
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If we want to, we can also name these: we can decide, "I will name {1,2,3}; I will name that set A."
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And if I want to, I can also name the set {cat,dog} B, because I might want to be able to talk about this;
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and instead of having to say {1,2,3} every time I want to talk about that set, I can just say A.
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"The set A has such-and-such property," or "the set B, when it interacts with it..."
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That way, I don't have to say {cat,dog}, or if it was an even longer list, like 10 or 50 objects...
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it would start to get really hard, practically impossible, to say.
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So instead, we can just change it to using a single letter, or whatever symbol is convenient for our purposes.
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Furthermore, the order that the elements come in has no effect on the set itself.
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So, the order that the elements appear in doesn't matter; we don't care about the order here.
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A = {1,2,3}, but that is the exact same thing as saying {3,2,1}, and that is the exact same thing as saying {2,1,3),
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or any other way you have of ordering those things.
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The important part is that it has all of those elements; the way that they come in--their places in line--that doesn't matter.
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It is just the group that you are considering, not the specific permutation of the line.
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All right, that is the basic idea of a set.
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If we want to describe a set, there are a bunch of different ways to describe it.
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Here are the three most common ways that you are going to see.
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Directly saying all of the elements: we could go through and, like I was talking about before with the curly braces and the comma,
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we just say each of the elements inside of the set: ice, water, steam.
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Our set has three elements; we have just said each of the three elements; that is the most basic method.
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We just say what is inside of the set.
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Another way is that we can clearly describe all of the members of the set.
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So, we also might describe it without it being inside of the curly braces; but sometimes we will actually leave it inside of the curly braces.
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The point of it is that we are able to say, "Oh, yes, that is everything that makes it up."
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So, we could make a set out of the first 80 elements of the periodic table, so we would know that hydrogen would be in the set;
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helium would be in the set; lithium would be in the set; all sorts of different elements
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are going to be inside of the set, up until the eightieth element.
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The eightieth element would be in it; the eighty-first element would not be in it.
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So, another way of describing it is to just say what is inside of it: here is what makes up my set, and there we go--we have a set.
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The final way that we can do it is: we can describe the quality, or it may be qualities, that each member of the set has in common.
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So, the way that you want to parse this--the way you want to read this--is: "x is saying this here is what our set is made up of."
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Our set is made up of all of the x; and then, you read this vertical bar as saying "such that."
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So, all of the x such that x is the first name of a teacher at Educator.com would be this set.
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Another way of reading that vertical bar is the word "where"--"x where x is the first name of a teacher at Educator.com."
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Anything will do here, so long as it is getting across the idea that this thing here, in the second part, is describing the quality
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required of the thing in the first part; so this part, the second part, describes what happens over here in the first part.
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So, for this set, if it is x such that x is the first name of a teacher at Educator.com, then it is going to be a bunch of first names
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of all of the teachers who teach at Educator.com.
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My name is Vincent; I am teaching at Educator.com (since you are watching this right now).
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So, that means that "Vincent" is inside of this set.
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There are going to be a bunch of other names; if you go and look at all of the teachers, you will see a whole bunch of different first names.
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But we know for sure that Vincent is one of the names inside of the set.
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Great! We can also symbolize things--if an element is contained in the set, and we want to talk about an element
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being in that set, we have a convenient symbol to show it, this symbol right here: "element of," "contained in."
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For example, if A is equal to the set {a,b,c}, then we know that a is contained in A; b is contained in A; c is contained in A,
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because they showed up right here in our description of what the set was.
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So, we know that a is an element in it; and we use this symbol right here to show "element of."
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We can also talk about the idea of subsets (if a set is contained inside of another set).
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If an entire set is contained in another set, then formally (as a formal definition) that means that every element in the first set is contained in the second set.
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So, for every element we name in that first set, it shows up in the second set; that is how we are going to formally define it.
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But you could just think of it as it being inside of the other set.
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We are going to call it a subset, because it is part of the other thing; it is like a sub-part, so we call it a subset.
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The symbol for this is this right here, "subset of."
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So, if X is the set {3}, and Y is the set {1,3}, and Z is the set {1,2,3}, then X is a subset of Y, because 3 shows up inside of Y.
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And then, Y is a subset of Z, because 1 and 3 both show up in Z.
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So, we are able to see that that is a subset, because everything in here showed up in the other one.
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Furthermore, we know that this property has to be transitive, because X is contained in Y, and Y is contained in Z;
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then since X already lives inside of Y, it must also be inside of Z.
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If we were to see it as sort of a picture, we would see it something like this.
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So, X is contained in Y, is contained in Z; since Z has Y, it must also have X, so we have a transitive property--X is contained inside of Z, as well.
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Great; we can also talk about a set that has no elements at all, the **empty set**.
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And sometimes, it will also be called the null set.
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Either way, it is a set that has nothing in it: it has no elements whatsoever.
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We represent it with this symbol right here, "the empty set" symbol.
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Now, this set is going to be unique, because any set that has no elements inside of it must be the empty set.
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There is only one empty set, because there is only one way to have nothing inside of a set.
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So, the empty set is just nothing at all; there is nothing inside of it--no elements; we have the empty set.
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Since the empty set has nothing inside of it, it must inherently be inside of any other set.
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All of its elements show up in every other set; each of its elements appears in every other set.
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Now, I have the word "trivially" there, because what means is that it is trivial--it is obvious in sort of a silly way.
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Yes, OK, sure, none of them show up...of course nothing shows up, because they don't have any there.
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But that doesn't make it not true; it is trivially true.
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It is kind of an obvious, silly thing, but it is still true; so that means, by our definition of subset, that the empty set is a subset to everything.
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The set A = {walrus} must have the empty set inside of it, because that set has...in a corner...nothing; everything has a little nothing inside of it.
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B, {17,27,47}...the exact same thing: it is also going to have the empty set inside of it.
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A and B don't really have any connection, other than the fact that they both have empty sets inside of them,
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because any set at all, even the empty set itself, is going to contain the empty set,
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because containing yourself is obvious, because it means you already have yourself in there.
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All right, union and intersection: we can create new sets through having our sets interact with each other.
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So, if we have two or more sets, we can have an interaction between those sets and make another set that may or may not be different.
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The **union** of two sets is a set that contains the elements of each.
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We symbolize this with an open cup; that gives us our union symbol.
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The **intersection** of two sets is a set that contains the elements, and only those elements, that are in both sets.
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So, if an elements shows up in both of the sets, it is going to be symbolized with the intersection symbol, sort of like a cup pointing down.
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A cup pointing up--we are filling it up with a bunch of things; a cup pointing down--it is cutting things off.
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We could also see this as a Venn diagram: here we have all of the stuff in set A; here we have all of the stuff in set B.
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What they cover together--what they both cover, here--is A intersect B; the stuff that is in A and in B is A intersect B.
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The stuff that is in everything is going to be A union B; we can see this with the idea of a Venn diagram, as well.
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Union adds everything from all of our sets, and makes a big set out of everything that we have.
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And A intersect B is going to make a smaller set (generally) that is going to see where you cut into each other--
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where you have the exact same thing--and that is all we have left.
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Example using actual things: if A is equal to {cat,mouse}, and B is equal to {cat,dog}, then A union B is...
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cat shows up; mouse shows up; and then, we go over to B, and cat...cat already showed up, so it is not that interesting
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to put it in again; we can't have copies show up in our set, because everything has to be unique;
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but dog hasn't shown up before, so we get dog in there.
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Now, for A intersect B, we ask, "Well, what is the thing that shows up in both of them?"
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Cat, cat...yes, cat showed up in both of them, so it gets to go here.
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But mouse doesn't show up over there; dog doesn't show up in A; so it doesn't show up either.
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You have to be in both of the two sets; intersection is if you were in both of them--you get to go on to the intersection.
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If you are only in one of them, that is not good enough.
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But union is where you only have to be in one of them, and you automatically make it in.
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You can be in both of them, and that is great; you still get in that way, as well.
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Sets can be weird stuff: we have talked about fairly simple stuff so far, that has been finite--just a couple of elements at a time.
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And there have been some numbers; there have been some words; but we haven't encountered anything that crazy.
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Now, the sets you are going to see for math, at least for the next couple of years, are going to generally just be sets of numbers.
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But, as we have seen, we can also contain a lot of different ideas.
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We don't just have to be stuck with numbers; we can also have elements other than numbers,
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like words, or maybe even symbols or faces; we could have a bunch of different things inside of our set.
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The important thing is that they are distinct objects.
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We have also only talked about the idea of finite sets; a **finite set** means that it has a limited number of elements--it doesn't just keep going forever.
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But we can also have an **infinite set**; that is going to be a set where the elements just keep going forever.
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So, an infinite set means the elements keep going forever--they never stop; there is an unlimited number of elements.
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So, how can we see an infinite set?
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Well, let's just start counting and never stop: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17...
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There is no reason I have to stop; I am going to stop, because I am mortal and I am not going to be able to count forever.
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But we get the idea that, even though I can't count forever,
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even though there is no way to literally write an infinite number of things, it still exists as an idea.
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And so, as an idea, it is a perfectly fine set.
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All of the numbers--all of the counting numbers, just listed out forever and ever and ever and ever--that gives us a set.
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It is an infinite set, because it has an unlimited number of elements; but it is a perfectly reasonable set.
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We can make even weirder, more interesting, stranger infinite sets if we want.
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Consider the set where we take a word, and then we repeat it an ever-increasing number of times:
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word, and then wordword, and then wordwordword, and then...etc., etc., etc.
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So, each time we do this, we will make a new "word"; it is not really a word in English, but it is a word in our sense of making up a new thing.
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And if we keep doing this forever, we are going to have an infinite number of words.
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For example, what if we took the word cat?
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Then we would have cat, catcat, catcatcat, catcatcatcat, catcatcatcatcat (I think I said cat 5 times)...
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and we could keep going and going and going and going; this set has infinitely many distinct elements.
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No matter what number you say, there is an element in the set that is going to have that word, "cat," repeated that many times.
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If you say 72, inside of this set right here, there is somewhere (not on this slide, but somewhere)--
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if you just keep going, you are going to be able to imagine the idea of "cat" being repeated 72 times in a row.
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So, we are creating new elements out of doing this; we build a set out of this idea.
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And each one of these is distinct from the others: "cat" is not the same as "catcat," which is not the same as "catcatcat."
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So, each one of these elements is distinct from the others, and there is an unlimited number of them; we have an infinite set.
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We can make really interesting, weird things in set theory--it is really, really cool stuff: we have just scratched the surface of how cool this stuff can get.
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I love set theory personally; but it is something you will have to study in college if you are really, really interested in it.
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So, I just want to finish by saying that sets can be strange and beautiful things, and that there is a whole bunch of stuff out there.
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Now, let's start talking about how all of this set theory stuff applies to what we are going to be seeing,
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in the near future in Precalculus, and then hopefully one day in calculus.
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We can talk about numbers as sets: we understand the notion of "set" now, and that is great;
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so we can now look at sets that make up numbers that we are going to use in math.
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We have already seen one of the most essential sets: it was our first example of an infinite set, the natural numbers:
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N = {1,2,3,4,5,6,7...}; this is just starting at 1 and counting on forever.
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This is our first, most basic infinite set, in many ways.
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We get this idea of counting and never stopping from the age of 3 on, if not even earlier.
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We are getting this idea where you start counting, and you just never stop.
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And of course, as a child, you realize, "Eventually I have to stop--I will say I will count to 100, and then I will not count any further."
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But you could just keep going; and that is the idea of the natural numbers--
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you just keep going forever and ever and ever, and you have an infinite number of elements.
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One thing to note is that some teachers will define the natural numbers as starting with the 0.
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So, you might, instead, have N be {0,1,2,3,4...}; so it is the exact same thing on the latter part of it.
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The tail of it is going to look the same, but you might start with 0; you might start with 1.
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I prefer the version without 0, that starts with 1; but some teachers make a distinction,
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and will call the one starting with 1 the counting numbers, and the one starting with 0 the natural numbers.
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The point is to just pay attention to what your teacher is teaching you, if you are taking an outside class.
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And make sure you are using their definition, so that you get everything right on the homework,
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and that you understand what they are trying to teach you.
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There is nothing better or less good about one or the other; it is just sort of a taste thing.
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And I happen to prefer the version without 0.
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Also, I just really quickly want to talk about this symbol that we use: that is N in blackboard bold,
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which is to say what we would write out if we were writing the symbol by hand.
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However, it is kind of hard to make that symbol by hand, since it is such a fancy typography symbol.
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Instead, if you were writing this out by hand, the symbol that you write is like this.
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You start out with an N, and then you just drop another line down here; and that is seen as ℕ if you want to write it by hand.
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Probably, you are not going to have to write this stuff by hand for at least a while, and maybe not ever.
00:18:10.300 --> 00:18:14.000
But I want you to know, in case you were interested in doing that.
00:18:14.000 --> 00:18:15.900
So, let's keep expanding on these ideas.
00:18:15.900 --> 00:18:20.700
We can take the natural numbers and say, "Well, we have positive numbers; but we also have negative numbers."
00:18:20.700 --> 00:18:24.600
So, let's count, not just forward, but let's count backward, as well.
00:18:24.600 --> 00:18:27.500
We will hit 0, and then we will just drive on into the negatives.
00:18:27.500 --> 00:18:34.300
This gives us the integers: we start at 0, and then we go forward for positive, but we also go backward for negative.
00:18:34.300 --> 00:18:42.100
0 forward is 1, 2, 3, 4, 5, 6... 0 backward is -1, -2, -3, -4, -5, -6, -7, -8.
00:18:42.100 --> 00:18:45.600
We are going off in both directions, and that gives us the integers.
00:18:45.600 --> 00:18:51.600
We use this symbol here with the special Z...if you want to make this ℤ, you start off with just a fairly normal Z;
00:18:51.600 --> 00:18:59.300
and then you drop down this extra diagonal and join it with your Z, and you have the symbol for the integers in handwritten form.
00:18:59.300 --> 00:19:05.100
Now, I think you are wondering, "Wait, that made sense with the natural numbers; that was N; but why Z?"
00:19:05.100 --> 00:19:11.000
Z comes from German; I believe *zollen* is the word for integer...no, *zollen* is just numbers;
00:19:11.000 --> 00:19:15.800
and so, *zollen* is numbers, and because German mathematicians were doing this work
00:19:15.800 --> 00:19:20.200
around the same time that English-speaking mathematicians were, and it was all being codified into symbols,
00:19:20.200 --> 00:19:27.600
we ended up using Z for the German version of numbers, because those mathematicians did a lot of great work when setting up set theory.
00:19:27.600 --> 00:19:32.500
All right, the next idea: we can add yet another layer of depth by including the idea of division.
00:19:32.500 --> 00:19:38.400
So, we have 1, 2, 3...-1, -2, -3...these are great, and they get us a good idea of what is in the real world.
00:19:38.400 --> 00:19:46.000
But what if I want to talk about wanting half of a pie, or if I want to talk about..."He got one and a half dollars,"
00:19:46.000 --> 00:19:49.600
or something where I want to break a number into pieces?
00:19:49.600 --> 00:19:51.800
Now, we have to be able to talk about fractions.
00:19:51.800 --> 00:19:54.300
To do that, we use the rational numbers.
00:19:54.300 --> 00:20:01.000
So, here we have that interesting format where we have the middle bar meaning "where," "such that," something like that.
00:20:01.000 --> 00:20:08.800
So, what this means is that we have m/n, where m comes from the integers (m is one of these integers);
00:20:08.800 --> 00:20:14.400
it can be a negative number; it can be a positive number; but it is going to be a whole number.
00:20:14.400 --> 00:20:20.500
And n has to be contained in the natural numbers, which is good, because we certainly don't want to be able to divide by 0,
00:20:20.500 --> 00:20:25.100
and because of my definition of the natural numbers, we are not allowed to have 0 in the naturals.
00:20:25.100 --> 00:20:28.000
That means we can't divide by 0, so we are safe there.
00:20:28.000 --> 00:20:34.400
This gives us the ability to have any number up top, divided by any whole number that is positive on the bottom,
00:20:34.400 --> 00:20:36.800
which lets us make any fraction that we want to.
00:20:36.800 --> 00:20:47.500
You give me any fraction (like, say, 47/9), and look: we have 47 (which belongs to the integers); 9 belongs to the natural numbers.
00:20:47.500 --> 00:21:00.100
If we want to talk about the fraction -52/101, well, we can turn that into being equivalent to -52/101;
00:21:00.100 --> 00:21:09.700
and so, we have -52...well, that is an integer; and 101 is a natural number; so right there, we have the natural numbers.
00:21:09.700 --> 00:21:14.600
We are able to build any fraction that we are used to seeing in normal circumstances.
00:21:14.600 --> 00:21:20.000
Any sort of normal fraction that we would talk about, we can make now with the rational numbers.
00:21:20.000 --> 00:21:22.300
This gives us a lot of ability to make numbers.
00:21:22.300 --> 00:21:28.300
We can get pretty much anywhere we want to be by using the rational numbers.
00:21:28.300 --> 00:21:34.500
Also, you might wonder why it is Q; really quickly, if we wanted to write this by hand, you make a Q first;
00:21:34.500 --> 00:21:41.000
and then, you drop a vertical line like that; so you have ℚ; that gives us our blackboard bold once again,
00:21:41.000 --> 00:21:46.200
which is just to say something we can write by hand that makes it other than just writing the letter Q.
00:21:46.200 --> 00:21:49.200
So, that lets us talk about that set of all the rational numbers.
00:21:49.200 --> 00:21:54.400
And why do we use the letter Q? Because a fraction is connected with the idea of quotients.
00:21:54.400 --> 00:21:59.100
So, as opposed to using F (which we kind of use for functions a lot, as we will talk about later),
00:21:59.100 --> 00:22:03.800
we use Q to talk about quotients; so that is where we get the letter Q from.
00:22:03.800 --> 00:22:10.500
All right, onward: we can also talk about rational numbers as a decimal expansion.
00:22:10.500 --> 00:22:16.000
We have this idea of expanding a rational number into a decimal version; there is nothing wrong with decimal versions.
00:22:16.000 --> 00:22:20.100
And we can have pretty much any number turn into a decimal version of itself.
00:22:20.100 --> 00:22:24.100
So, the decimal expansion of every rational number (you probably learned this in grade school)...
00:22:24.100 --> 00:22:31.100
every rational is either going to terminate (which means it ends), or it continues with repeating digits.
00:22:31.100 --> 00:22:37.400
For our first example, something terminating, we have 0.09375; that is what we get from 3/32.
00:22:37.400 --> 00:22:44.200
And see how it just ends right here: if we were to keep going, it would be 00000...we would just have 0's forever.
00:22:44.200 --> 00:22:48.300
So, we just cut it off, and it terminates--it stops at a certain point.
00:22:48.300 --> 00:22:56.100
If, on the other hand, it continues with repeating digits, then that means there is some block of digits that will keep repeating forever.
00:22:56.100 --> 00:23:11.500
So, with 77/270, we get .2851851851...we realize that 851851851...point 2 happens first, and then our repeating block shows up: 851851851.
00:23:11.500 --> 00:23:15.000
And it is just going to march out forever and ever and ever.
00:23:15.000 --> 00:23:18.000
So, if we have a rational number, it is going to do one of these two things.
00:23:18.000 --> 00:23:21.200
It either terminates (it ends), or it repeats.
00:23:21.200 --> 00:23:27.900
Every rational number, anything that can be expressed as an integer divided by an integer, by whole numbers over whole numbers,
00:23:27.900 --> 00:23:36.700
with maybe a positive or a negative sign--that is going to have either the decimal ending or the decimal going forever, but repeating.
00:23:36.700 --> 00:23:43.500
Why is this important? This idea of the rational numbers is really great, but there are still some numbers we can't express.
00:23:43.500 --> 00:23:51.600
So, you might remember that decimal expansions of all the rationals either terminate, or they go into repetition.
00:23:51.600 --> 00:23:57.100
There is at least one number you have heard of by now that keeps changing: π.
00:23:57.100 --> 00:24:00.500
You have learned about the number π for probably quite a few years now.
00:24:00.500 --> 00:24:07.400
And you know that it just keeps shifting around: 3.1415...and you can memorize a bunch of digits, if you want.
00:24:07.400 --> 00:24:12.000
But it is never going to just lock down and turn into something where you are done memorizing it.
00:24:12.000 --> 00:24:15.100
There are always going to be infinitely many more digits to remember.
00:24:15.100 --> 00:24:22.500
So, π never stops--it never repeats; it is not a rational number.
00:24:22.500 --> 00:24:26.700
You have probably also heard that √2 is also not a rational number.
00:24:26.700 --> 00:24:33.800
These turn out to be true; we can't express them as rational numbers--we can't express them as a fraction of integers.
00:24:33.800 --> 00:24:40.000
The decimal expansion of an irrational number, unlike a rational, never stops, and it always keeps changing.
00:24:40.000 --> 00:24:45.800
They are these sort of shifting, mixed-up numbers that just always keep doing interesting things.
00:24:45.800 --> 00:24:48.900
They keep us working hard, unlike the rational numbers.
00:24:48.900 --> 00:24:53.600
So, if we want to really be able to describe everything that is out there--all of the numbers we might encounter--
00:24:53.600 --> 00:24:57.000
we need to be able to talk about the irrationals, in addition to the rationals.
00:24:57.000 --> 00:24:59.400
Also, why do we call them irrationals?
00:24:59.400 --> 00:25:06.200
It is nothing because they are crazy and they are something weird; it is because they are just not rational--they are irrational.
00:25:06.200 --> 00:25:10.400
Irrational numbers...it is just because they are not rational, not because there is anything wrong with them,
00:25:10.400 --> 00:25:16.200
but just because they are not that set that we call the rationals; that is it.
00:25:16.200 --> 00:25:19.300
So, if we want to put the rational and irrational numbers together to get something
00:25:19.300 --> 00:25:24.200
where we can really have all the numbers we work with, we have a great set.
00:25:24.200 --> 00:25:29.000
That will give us the **real numbers**: we put them together, and we get the real numbers.
00:25:29.000 --> 00:25:31.800
These are our bread and butter in mathematics.
00:25:31.800 --> 00:25:36.400
You are going to be using them for years; you have been using them for pretty much everything you have ever done,
00:25:36.400 --> 00:25:38.900
unless you have worked on the complex numbers for a little while.
00:25:38.900 --> 00:25:45.100
And even if you did work on the complex numbers before, it was still using real numbers as part of those complex numbers.
00:25:45.100 --> 00:25:48.700
The only thing was that i, and it still had a real number right next to it.
00:25:48.700 --> 00:25:53.400
So, real numbers make up a huge portion of mathematics.
00:25:53.400 --> 00:25:59.400
And unless you go for a whole bunch more math in college (which I would recommend--I really like math),
00:25:59.400 --> 00:26:02.800
you are not going to end up seeing, probably, anything other than the real numbers,
00:26:02.800 --> 00:26:05.500
until you get to some really abstract, interesting math.
00:26:05.500 --> 00:26:08.700
But it is going to take a while before you see anything other than the reals.
00:26:08.700 --> 00:26:13.600
They are great things to get at home with, and settle down with, and get a good understanding of.
00:26:13.600 --> 00:26:18.400
And the purpose of all these set concepts, beforehand, is to be able to get a sense of how this work--
00:26:18.400 --> 00:26:24.100
"Where do the reals live when we are not moving them around and working with them and doing things with them?"
00:26:24.100 --> 00:26:32.100
We express them...if we want to be able to talk about them with this nice, simple symbol, we use ℝ, in this blackboard bold font.
00:26:32.100 --> 00:26:38.200
If we want to be able to write this by hand, we make a normal R, and then we throw down this extra vertical line right here.
00:26:38.200 --> 00:26:46.200
And that is the symbol for the real numbers (and R stands for real numbers; it makes a lot of sense, unlike some of the other ones).
00:26:46.200 --> 00:26:50.600
If we want to talk about an interval of the real numbers, if we want to go into that home of real numbers and say,
00:26:50.600 --> 00:26:55.100
"Well, I just want to talk about this one chunk," we can use interval notation.
00:26:55.100 --> 00:26:58.600
For example, we might want to talk about everything from -1 to 3.
00:26:58.600 --> 00:27:04.900
We don't want to talk about 100; we don't want to talk about negative one billion; we just want to talk about everything from -1 to 3.
00:27:04.900 --> 00:27:13.500
So, we use interval notation; if we want to include the end numbers (-1 and 3), we use square brackets.
00:27:13.500 --> 00:27:20.600
So, square brackets here give us inclusion; they keep those endpoints in it.
00:27:20.600 --> 00:27:33.000
We go from -1 up until 3, and those points will be there; they are actually going to be part of our interval: -1 and 3 show up.
00:27:33.000 --> 00:27:39.000
If we want to exclude them (we want everything in between them, but we don't want the end things),
00:27:39.000 --> 00:27:44.800
then we exclude them by using parentheses; parentheses give us exclusion.
00:27:44.800 --> 00:27:51.000
That gets us -1 to 3, but without actually having -1 and 3.
00:27:51.000 --> 00:27:56.900
So, -1 does not show up; 3 does not show up.
00:27:56.900 --> 00:28:04.900
We use, if we want to symbolize it in a graphical manner (as a picture), open circles like this right here to show exclusion.
00:28:04.900 --> 00:28:08.100
We use filled-in dots to show inclusion.
00:28:08.100 --> 00:28:17.500
Exclusion is with parentheses, a curve, empty circle; and inclusion is with a filled-in dot or a nice square, solid bracket.
00:28:17.500 --> 00:28:24.500
But in either case (-1 to 3 with square brackets or -1 to 3 with parentheses), we are going to always include everything between those.
00:28:24.500 --> 00:28:30.300
It is just a question of whether or not we are going to include the ends of the interval.
00:28:30.300 --> 00:28:42.200
If we want to talk about 4 to 7, but we want to not include 4, and we want to include 7, we have (4,7].
00:28:42.200 --> 00:28:46.200
So, that is going to be all of the real numbers between 4 and 7, of course;
00:28:46.200 --> 00:28:49.400
but it will keep the number 7 (because we have the square bracket);
00:28:49.400 --> 00:28:53.700
but it is going to not include 4 (because we have the parenthesis).
00:28:53.700 --> 00:29:01.000
So, the parenthesis next to the 4 will exclude it--will keep it out; but the square bracket next to the 7 will keep it in.
00:29:01.000 --> 00:29:10.200
So, we can talk about intervals where one end gets left out, and one end gets kept in, by mixing up how we use this interval notation.
00:29:10.200 --> 00:29:14.100
If we want to talk about the idea of infinity, then we can talk about going on forever.
00:29:14.100 --> 00:29:22.900
So, the symbol for infinity--that nice infinity sign--gives us a nice, convenient way to talk about going on forever.
00:29:22.900 --> 00:29:29.500
So, if we want to talk about the interval going forever in one direction or the other, we will use -∞ or positive ∞.
00:29:29.500 --> 00:29:33.100
And keep in mind: when there is no symbol in front of it, we just assume that it is positive.
00:29:33.100 --> 00:29:37.100
So, negative infinity has the negative sign; positive infinity doesn't have anything.
00:29:37.100 --> 00:29:42.400
If you absolutely had to symbolize that it was the positive version, you could put a little plus sign in front of it.
00:29:42.400 --> 00:29:45.500
So, that will show us which direction we are going to go forever.
00:29:45.500 --> 00:29:50.700
Depending on the direction that we want to talk about going forever, we will choose the appropriate infinity, negative or positive.
00:29:50.700 --> 00:29:54.800
Now, keep in mind: you are always going to use parentheses with negative infinity or infinity.
00:29:54.800 --> 00:29:58.300
Why is it that we always use parentheses when we are talking about them in interval?
00:29:58.300 --> 00:30:03.400
It is because we can't actually include infinity: infinity isn't a number.
00:30:03.400 --> 00:30:06.400
Infinity is just the idea of continuing forever.
00:30:06.400 --> 00:30:12.000
So, since infinity is an idea of just keeping going, it is not an actual place; so we can't end on it.
00:30:12.000 --> 00:30:15.400
To have a square bracket implies that we end on it, and it is there.
00:30:15.400 --> 00:30:21.600
The parenthesis, on the other hand, will just show the idea of keeping going, keeping reaching towards it.
00:30:21.600 --> 00:30:27.200
You will never actually reach it, but the interval will just keep going towards that notion of infinity.
00:30:27.200 --> 00:30:31.500
So, for example, we could have -∞ to 2, with a square bracket on the 2.
00:30:31.500 --> 00:30:38.300
That is going to be all numbers less than or equal to--everything starting at negative infinity, and working all the way up until 2.
00:30:38.300 --> 00:30:41.400
And we will actually get to 2, and we will achieve 2.
00:30:41.400 --> 00:30:47.600
(3,∞) is going to be all of the numbers greater than 3, but we won't include 3,
00:30:47.600 --> 00:30:50.400
because we don't have a bracket on it; we have a parenthesis on the 3.
00:30:50.400 --> 00:30:54.600
So, it is going to be everything from 3, but not actually including 3.
00:30:54.600 --> 00:30:59.700
So, we will get really, really, really close to 3, but we will never actually touch it; we will never actually achieve 3.
00:30:59.700 --> 00:31:05.900
And finally, if we want to just talk about the entire real line, that is the same thing as saying -∞ to positive ∞,
00:31:05.900 --> 00:31:08.300
because that is everything that the real numbers have.
00:31:08.300 --> 00:31:14.400
Start all the way from the very beginning; reach all the way to the beginning, and reach all the way to the end.
00:31:14.400 --> 00:31:19.200
Just keep reaching forever and ever; go all the way to negative infinity; go all the way to positive infinity.
00:31:19.200 --> 00:31:23.400
That is going to be the same thing as just saying "all the real numbers at once."
00:31:23.400 --> 00:31:25.500
All right, let's do some examples.
00:31:25.500 --> 00:31:32.400
We have the set X = {a,b,c}, the set Y = {b,c,d}, and the set Z = {c,d,e}.
00:31:32.400 --> 00:31:36.000
Let's figure out a couple of different ways to talk about unions and intersections.
00:31:36.000 --> 00:31:46.300
First, X ∪ Y ∪Z: that is going to be equal to...X ∪ Y is going to be all of the elements included in X and Y.
00:31:46.300 --> 00:31:50.500
And then, we add "union Z" on that; it is going to be in addition to all of the units with Z.
00:31:50.500 --> 00:31:55.700
So, it is going to be all of the elements that show up in all of them: a shows up; b shows up;
00:31:55.700 --> 00:32:01.200
c shows up; well, b already showed up; c already showed up; but d is new.
00:32:01.200 --> 00:32:04.100
c already showed up; d already showed up; but e is new.
00:32:04.100 --> 00:32:11.800
So, it is going to be {a,b,c,d,e}: there we go.
00:32:11.800 --> 00:32:19.700
If we want to talk about X ∩ Y ∩ Z, then that is going to be...what is the only place that they all have in common?
00:32:19.700 --> 00:32:22.700
What are the elements that are in each and every one of them?
00:32:22.700 --> 00:32:28.900
Well, a does not show up in Z, nor does it show up in Y.
00:32:28.900 --> 00:32:33.800
b does not show up in Z; it does show up in Y, but it has to show up in all three of them.
00:32:33.800 --> 00:32:37.500
c does show up in Y and does show up in Z, so c is in.
00:32:37.500 --> 00:32:42.700
And since everything else must not show up in X, it must be that the only thing inside of it is c.
00:32:42.700 --> 00:32:47.600
We can also break this down into two pieces: we can say, "Well, what is X ∩ Y, first?"
00:32:47.600 --> 00:32:54.400
X ∩ Y would be b and c, because those are the elements X and Y share in common.
00:32:54.400 --> 00:33:05.600
And then, we intersect that with Z, as well; the only thing that {b,c} shares with Z is the c right here, so we get {c} as our answer to all of them intersecting.
00:33:05.600 --> 00:33:10.400
If they are all unions, and they are all intersections, it doesn't really matter the order that we choose--
00:33:10.400 --> 00:33:15.800
which ones to intersect, which ones to "union" first...it is going to be a question of how they all interact.
00:33:15.800 --> 00:33:22.800
What if we put all the elements in all of them together, or what element is inside of every single one of these sets?
00:33:22.800 --> 00:33:26.600
So, it doesn't matter about the order; it doesn't matter about how we approach doing it.
00:33:26.600 --> 00:33:32.000
But it does sometimes matter, if we talk about intersection and union working together.
00:33:32.000 --> 00:33:39.100
So, for example, if we had (X ∩ Y), and then union Z, well, we have parentheses around it.
00:33:39.100 --> 00:33:44.700
While we haven't explicitly reminded you of the order of operations, I am sure you remember to do things inside of parentheses first.
00:33:44.700 --> 00:33:47.800
So, if X ∩ Y is inside of parentheses, then we have to do it first.
00:33:47.800 --> 00:33:55.400
So, X ∩ Y gives {b,c}; and now we are going to do union Z.
00:33:55.400 --> 00:34:03.800
Z is going to be c, d, and e; so that gives us a total of {b,c,d,e} in our set.
00:34:03.800 --> 00:34:14.400
So, {b,c,d,e}: but compare--what if we did it a different way--if we had X being "unioned" with the intersection of Y and Z?
00:34:14.400 --> 00:34:18.900
Now, we need to start by asking, "Well, what is the intersection of Y and Z?"
00:34:18.900 --> 00:34:25.900
Well, c and d show up in both of them; e does not show up; b does not show up in both of them.
00:34:25.900 --> 00:34:30.200
So, c and d make up the intersection of Y and Z.
00:34:30.200 --> 00:34:39.600
So, X ∪ {c,d} is going to be a and b (because they are new), and c and d (were already there).
00:34:39.600 --> 00:34:48.000
So, {a,b,c,d) is (X ∪ Y) ∩ Z; but we get a different one if we do (X ∩ Y) ∪ Z: we get {b,c,d,e}.
00:34:48.000 --> 00:34:56.300
Notice: these two things are not the same--there is not an equivalence between those two sets; they are not equal sets.
00:34:56.300 --> 00:35:01.200
They aren't the same set, because how we approach putting these things together matters.
00:35:01.200 --> 00:35:08.800
It is not like 3 times 4 times 5, which is the exact same thing as 4 times 3 times 5, which is the exact same thing as 5 times 4 times 3.
00:35:08.800 --> 00:35:13.200
It matters how we put these together, because we have different things going on.
00:35:13.200 --> 00:35:18.400
It is not just multiplication; in a way, it is multiplication and addition--it matters the order that we do it in.
00:35:18.400 --> 00:35:25.800
So, intersection and union--we can't just do it in any order; we have to pay attention to the order that it has been put together in.
00:35:25.800 --> 00:35:32.700
The next example: we have ℕ, ℤ, ℚ, and ℝ; we have all of those big number sets that we talked about before.
00:35:32.700 --> 00:35:35.800
Which one of them will be subsets to the others? How will the subsets work?
00:35:35.800 --> 00:35:38.600
Well, first, let's start with reminding ourselves about what these are.
00:35:38.600 --> 00:35:50.900
ℕ is everything from 0...oops, not from 0--I don't believe in that one!...I said that one wrong: 1, 2, 3, 4...just keep going forever.
00:35:50.900 --> 00:35:56.400
The integers are going to be going off in the negative direction and the positive direction.
00:35:56.400 --> 00:36:03.800
We have ... up until...and then we meet up...and then we just keep going that way.
00:36:03.800 --> 00:36:13.400
And if we talk about the rationals, that is the way of saying all integer fractions--fractions made up with integers on the top and bottom.
00:36:13.400 --> 00:36:16.000
So, that is going to give us the rationals.
00:36:16.000 --> 00:36:24.100
And the reals are just all numbers--what we are used to as thinking of all the possible numbers--all numbers are the reals.
00:36:24.100 --> 00:36:28.300
Well, with that in mind, it is pretty easy to see that the natural numbers...
00:36:28.300 --> 00:36:33.700
Well, since the integers...not equal...subset is what I meant to write...
00:36:33.700 --> 00:36:41.200
Since the natural numbers are {1,2,3,4...}--they are all the positive integers--they must show up in the integers,
00:36:41.200 --> 00:36:45.000
because the integers are the positive integers, and the negative integers, and 0.
00:36:45.000 --> 00:36:48.800
So, ℕ is a subset of ℤ.
00:36:48.800 --> 00:36:52.600
Now, ℤ shows up in the rationals; how is that possible?
00:36:52.600 --> 00:36:57.900
Well, if you give me any integer number, I can very easily make a rational number out of it.
00:36:57.900 --> 00:37:11.900
If you give me -5, well, -5/1 is the same thing as -5; and -5/1 is very clearly contained inside of the rationals: -5/1 is very clearly an element of the rationals.
00:37:11.900 --> 00:37:19.500
You give me any integers (like -572), and I just put it over 1, and once again, we are back inside of the rationals.
00:37:19.500 --> 00:37:23.900
So, whatever integer you give me, pretty clearly, has a rational version, as well.
00:37:23.900 --> 00:37:27.600
We can keep going and now include the reals; we can talk about the reals.
00:37:27.600 --> 00:37:33.200
And the reals are going to have everything, because we define the reals as having all of the rationals and all of the irrationals.
00:37:33.200 --> 00:37:37.300
So, the rationals fit inside of the reals, as well; so we have subsets going up:
00:37:37.300 --> 00:37:41.500
ℕ is a subset of ℤ, is a subset of ℚ, is a subset of ℝ.
00:37:41.500 --> 00:37:50.300
That also means that, because this is transitive, ℕ is also a subset of ℚ, and ℕ is also a subset of ℝ.
00:37:50.300 --> 00:37:56.900
ℤ is also a subset of ℝ, as well; and those are all of the relations that we can get out of this.
00:37:56.900 --> 00:38:02.600
ℕ is a subset, and ℤ is a subset, and ℚ is a subset, inside of ℝ.
00:38:02.600 --> 00:38:08.500
The third example: if we let A be the set of all titles of all published written works;
00:38:08.500 --> 00:38:16.500
and B is all of the phrases that are precisely three words long; let's talk about what would be some elements inside of A ∩ B.
00:38:16.500 --> 00:38:20.700
Now, we start with...there are not just a couple of answers to this; there is not just one finished answer.
00:38:20.700 --> 00:38:24.000
There are many more answers than I am aware of.
00:38:24.000 --> 00:38:28.500
But I can give you some examples, and talk about how to think about this.
00:38:28.500 --> 00:38:31.000
Let's also just rephrase this, so we have another way of thinking about it.
00:38:31.000 --> 00:38:48.100
A is the same thing as talking about...A is every title of books and magazines and poems...
00:38:48.100 --> 00:38:52.100
it is everything that is a written piece of work that has been published, that we could have actually
00:38:52.100 --> 00:38:58.400
gone to a store and bought, or found in a published book; A is every title of books, etc., etc., etc.--
00:38:58.400 --> 00:39:02.000
everything written, that is published--that is what A makes up.
00:39:02.000 --> 00:39:20.900
Now, B is everything (from the way we are writing this) that is three words long.
00:39:20.900 --> 00:39:32.800
So, what we are looking for: if we want to find the intersection of A and B, then A ∩ B is going to be things that are in both.
00:39:32.800 --> 00:39:45.900
So, if you are in both, then to be inside of A ∩ B...that is the same thing as saying "titles that are three words long."
00:39:45.900 --> 00:39:56.700
So, A ∩ B is just titles that are three words long.
00:39:56.700 --> 00:40:04.000
To be able to answer this question, we just need to figure out what are some titles that are three words.
00:40:04.000 --> 00:40:07.700
So, we start thinking, and here are some of the ones that I thought of.
00:40:07.700 --> 00:40:14.300
We could say *Romeo and Juliet*, right? Almost everyone is going to know *Romeo and Juliet*, so that is a good one to start with.
00:40:14.300 --> 00:40:24.000
*Romeo and Juliet*: there is a title that is three words long, written by Shakespeare, and it is a published piece of work.
00:40:24.000 --> 00:40:27.200
We have all been able to find a copy of *Romeo and Juliet* if we have been looking for it.
00:40:27.200 --> 00:40:29.300
So, *Romeo and Juliet* is one.
00:40:29.300 --> 00:40:38.100
What about another one--how about *Things Fall Apart* by Chinua Achebe?
00:40:38.100 --> 00:40:46.500
Or we could also talk about something by Kurt Vonnegut: Kurt Vonnegut wrote *Breakfast of Champions*.
00:40:46.500 --> 00:41:02.600
So, *Breakfast of Champions* is another example of something where we have a phrase that is 3 words long,
00:41:02.600 --> 00:41:06.300
and is the title of something that is a written work.
00:41:06.300 --> 00:41:15.600
We could also talk about *To the Lighthouse* by Virginia Woolfe; *To the Lighthouse* is another example.
00:41:15.600 --> 00:41:22.400
There are a whole bunch of examples out there; I can't list all of these, because we would be here for days and days and days and days.
00:41:22.400 --> 00:41:28.400
And I don't know them; but it is going to be anything that is written and has three words in it...
00:41:28.400 --> 00:41:33.400
3 words...not just in it, but 3 words for the title--precisely 3 words.
00:41:33.400 --> 00:41:42.300
As much as I would like to be able to say *Cannery Row*, or *Of Mice and Men*, or *1984*,
00:41:42.300 --> 00:41:47.000
I can't talk about those, because they are not precisely 3 words long.
00:41:47.000 --> 00:41:51.700
There are a lot of books out there that aren't 3 words long in the title.
00:41:51.700 --> 00:42:03.100
And there are lots of phrases that are three words long, like "hot in here" (sorry, I didn't come up with any brilliant phrases in that period of time).
00:42:03.100 --> 00:42:07.800
But any phrase that is three words long would be in B, and any title would be in A.
00:42:07.800 --> 00:42:12.900
But what we are looking for is the intersection of A and B--titles that are three words.
00:42:12.900 --> 00:42:16.100
*Romeo and Juliet*, *Things Fall Apart*, *Breakfast of Champions*,
00:42:16.100 --> 00:42:22.100
*To the Lighthouse*: these are all some examples from various different authors.
00:42:22.100 --> 00:42:26.200
The final example, Example 4: List all of the subsets of {x,y,z}.
00:42:26.200 --> 00:42:33.300
The very first subset that we have to remember is the empty set: the empty set shows up as a subset for everything.
00:42:33.300 --> 00:42:36.000
The empty set is our very first subset.
00:42:36.000 --> 00:42:41.100
The next one--well, let's look at all of the subsets that have one element inside of them.
00:42:41.100 --> 00:42:49.900
{x} (oops, I made a really bad bracket there) is going to be a set, just on its own; and that is a subset.
00:42:49.900 --> 00:42:53.900
Another one would be {y}; that is another subset.
00:42:53.900 --> 00:43:02.200
Another one would be {z}; those are all of the sets that are one element long, and are subsets of {x,y,z}.
00:43:02.200 --> 00:43:13.100
Now, we can go with the two-element ones, and we can say, "All right, well, {x,y}--that is going to be a subset."
00:43:13.100 --> 00:43:20.400
What about {x,z}? And then, finally, there is {y,z}.
00:43:20.400 --> 00:43:26.500
And we think about that for a little while, and we realize that those are all the sets I can possibly make out of {x,y,z}
00:43:26.500 --> 00:43:32.000
that have 2 elements precisely in them: x and y, x and z, y and z.
00:43:32.000 --> 00:43:36.800
You could rearrange them in different orders, but remember, since it is a set we are talking about, order is not important.
00:43:36.800 --> 00:43:40.800
It doesn't matter the order that it shows up in--just that it did show up at all.
00:43:40.800 --> 00:43:45.200
Those are all of the sets that are going to be two elements long, and are subsets of {x,y,z}.
00:43:45.200 --> 00:43:52.200
And then, finally, we have {x,y,z} itself; it is a subset of itself, because remember, by the formal definition
00:43:52.200 --> 00:43:58.200
of being a subset, it just means that all of the elements inside of your set show up in the other set.
00:43:58.200 --> 00:44:06.300
And every element {x,y,z} shows up inside of {x,y,z}; it makes sense; so every set is a subset of itself.
00:44:06.300 --> 00:44:10.800
It is kind of obvious, and not that really interesting; but it is another trivial assertion.
00:44:10.800 --> 00:44:15.700
It is interesting to think about, but not something that really gains us a lot of knowledge of any specific thing.
00:44:15.700 --> 00:44:20.800
But it is still an interesting idea, and might have other connections later on, if we think about it a lot.
00:44:20.800 --> 00:44:26.900
All right, so that gives us a total of 8 subsets; and those are all of them.
00:44:26.900 --> 00:44:30.000
All right, I hope you enjoyed this; I hope you learned something about sets.
00:44:30.000 --> 00:44:34.000
Like I said before, we are not going to really focus on the ideas that we had here.
00:44:34.000 --> 00:44:39.600
But what we just did was built the foundation of pretty much everything else that you are going to end up ever seeing in math.
00:44:39.600 --> 00:44:43.100
Virtually all of modern mathematics is built upon the idea of set theory.
00:44:43.100 --> 00:44:45.000
It can be explained through the idea of set theory.
00:44:45.000 --> 00:44:48.700
So, I just wanted you to get some exposure to this foundation, so that later things we talk about,
00:44:48.700 --> 00:44:53.400
like when we talk about functions and a whole bunch of things, in fact, we have some idea of being able
00:44:53.400 --> 00:44:57.500
to refer back to these sets, pulling things out from sets, going to other sets.
00:44:57.500 --> 00:45:03.800
There is really cool stuff here; set theory is really fascinating; I totally recommend studying it sometime, if you get the chance.
00:45:03.800 --> 00:45:07.600
I am glad that you managed to get here, and that you have some idea of how sets work.
00:45:07.600 --> 00:45:09.400
And we will see you in the next lesson--goodbye!
00:45:09.400 --> 00:45:11.000
Talk to you later at Educator.com!