For more information, please see full course syllabus of Math Analysis
For more information, please see full course syllabus of Math Analysis
Sets, Elements, & Numbers
- We learn about the idea of a set in this lesson. While many courses won't directly address these concepts, they form the foundation that a lot of math rests upon. While you can understand later concepts without these ideas, knowing them can really help, especially in the long term.
- A set is a collection of distinct (different) objects. We call each object in a set an element.
- We can denote a set with any symbol, but we commonly use capital letters like A.
- We can write out a set in various ways. We often use curly braces { } to denote that a set contains what is inside the braces. Here are some of the ways to write out a set.
- Directly name the elements in the set:
A = { x,y,z} - Clearly describe all the members of the set:
B={the first 10 letters of the alphabet} - Describe the quality (or qualities) each member of the set has in common:
C = { x | x is an English word that rhymes with ` thing′}
- Directly name the elements in the set:
- If an element is contained in a set, we show it with the symbol ∈
For example, if x ∈ A, that means that x is an element contained in the set A. - If an entire set is contained in another set (all the elements of one set are elements of the second set as well), we say it is a subset. If A is a subset of B, we denote this as A ⊂ B. This means for any x ∈ A, we know x ∈ B as well.
- We can consider a set that has no elements at all: the empty set (also called the null set). It is a set with nothing in it. We represent it with ∅.
Since the empty set contains nothing at all, it is a subset of all sets (after all, each of its elements trivially appears in every other set). Thus, for any set A, we have ∅ ⊂ A. - The union of two sets is a set that contains the elements of each. We denote this with ∪.
- The intersection of two sets is a set that contains the elements (and only those) that are in both of them. We denote this with ∩.
- We can think of numbers as being elements from sets. Each set makes up a category of numbers.
- Natural Numbers: The numbers we would count objects in the real world with.
ℕ = { 1, 2, 3, 4, 5, 6, 7, …} - Integers: Expanding on ℕ, we also include 0 and the negatives.
ℤ = { …, −3, −2, −1 , 0, 1, 2, 3, …} - Rational Numbers: We take ℤ and use division to create fractions.
ℚ = ⎧
⎨
⎩m n⎢
⎢m ∈ ℤ, n ∈ ℕ ⎫
⎬
⎭ - Irrational Numbers: There are some numbers that cannot be expressed as a fraction of integers. These numbers are not rational, so we call them irrational. Some examples are π and √{47}.
- Reals: Combining the rational and irrational numbers together into a single set, we get the real numbers. We denote them with ℝ. You have been using them for years, and they are our bread and butter in math. ℝ contains any number you might normally use.
- Natural Numbers: The numbers we would count objects in the real world with.
- We can express intervals of ℝ using interval notation.
- To include the end numbers, we use square brackets: [−1, 3].
- To exclude the end numbers, we use parentheses: ( −1, 3).
- If we want, we can mix these types to include one end but exclude the other end: [−1, 3).
- To talk about one end of the interval going on forever, we use −∞ or ∞ (depending on which direction). We always use parentheses with −∞ and/or ∞ because we can't actually include it in the interval: ∞ isn't actually a number, just the idea of continuing forever: [−1, ∞).
Sets, Elements, & Numbers
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Introduction
- Sets and Elements
- Describing/ Defining Sets
- Directly Say All the Elements
- Clearly Describing All the Members of the Set
- Describing the Quality (or Qualities) Each member Of the Set Has In Common
- Symbols: 'Element of' and 'Subset of'
- Empty Set
- Union and Intersection
- Sets Can Be Weird Stuff
- Can Have Elements in a Set
- We Can Have Infinite Sets
- Example
- Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
- This Set Has Infinitely Many Distinct Elements
- Numbers as Sets
- Natural Numbers ℕ
- Including 0 and the Negatives ℤ
- Rational Numbers ℚ
- Can Express Rational Numbers with Decimal Expansions
- Irrational Numbers
- Real Numbers ℝ: Put the Rational and Irrational Numbers Together
- Interval Notation and the Real Numbers
- Interval Notation: Infinity
- Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
- Always Use Parentheses
- Examples
- Example 1
- Example 2
- Example 3
- Example 4
- Intro 0:00
- Introduction 0:05
- Sets and Elements 1:19
- Set
- Element
- Name a Set
- Order The Elements Appear In Has No Effect on the Set
- Describing/ Defining Sets 3:28
- Directly Say All the Elements
- Clearly Describing All the Members of the Set
- Describing the Quality (or Qualities) Each member Of the Set Has In Common
- Symbols: 'Element of' and 'Subset of' 6:01
- Symbol is ∈
- Subset Symbol is ⊂
- Empty Set 8:07
- Symbol is ∅
- Since It's Empty, It is a Subset of All Sets
- Union and Intersection 9:54
- Union Symbol is ∪
- Intersection Symbol is ∩
- Sets Can Be Weird Stuff 12:26
- Can Have Elements in a Set
- We Can Have Infinite Sets
- Example
- Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
- This Set Has Infinitely Many Distinct Elements
- Numbers as Sets 16:03
- Natural Numbers ℕ
- Including 0 and the Negatives ℤ
- Rational Numbers ℚ
- Can Express Rational Numbers with Decimal Expansions
- Irrational Numbers
- Real Numbers ℝ: Put the Rational and Irrational Numbers Together
- Interval Notation and the Real Numbers 26:45
- Include the End Numbers
- Exclude the End Numbers
- Example
- Interval Notation: Infinity 29:09
- Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
- Always Use Parentheses
- Examples
- Example 1 31:23
- Example 2 35:26
- Example 3 38:02
- Example 4 42:21
Math Analysis Online
Transcription: Sets, Elements, & Numbers
Hi; welcome back to Educator.com.0000
Today, we are going to talk about sets, elements, and what they mean for numbers.0002
To start, we are going to talk about the idea of sets.0007
But a lot of courses don't address this stuff directly.0010
You might find, in the course that you are taking, that your teacher never talks about this directly.0013
But these ideas build the foundation that the rest of math works on.0018
So, you can understand all of the concepts that will come later in this course without ever having watched this lesson.0024
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.0029
Also, if you want to go on and take later, more advanced math, like calculus or even much more advanced college courses0036
(like abstract mathematics), this stuff is going to be really useful to have already ingrained in your mind.0043
These ideas are great later for you; so if you are going to take advanced math, really, definitely, watch this.0049
Get an understanding of what is going on.0056
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.0057
But you want to get a glimpse of this sort of stuff, so that later on, you can really understand what is going on.0062
And also, it is just going to make things a lot smoother, especially when we are talking about functions.0068
So, you can get an abstract idea of how a function works, which will help you understand what is going on.0072
All right, let's get started!0077
A set is a collection of distinct objects; each of the objects inside of a set is called an element.0079
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.0086
1 is not 2 or 3; 2 is not 1 or 3; and 3 is not 1 or 2.0095
Similarly, for the set {cat, dog}, we have cat different than dog, and dog different than cat.0100
Also, really quickly: the way that we show that we are talking about objects inside of a set is: we have these curly braces.0107
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.0115
You can see a nice typography (typed-out font) brace in my slide here.0122
But when I am actually writing it out, I do something like these right here.0127
So, we put our elements inside of that, and we separate each of the elements with a comma.0131
That is just what is happening on a type point of view.0135
If we want to, we can also name these: we can decide, "I will name {1,2,3}; I will name that set A."0140
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;0146
and instead of having to say {1,2,3} every time I want to talk about that set, I can just say A.0152
"The set A has such-and-such property," or "the set B, when it interacts with it..."0156
That way, I don't have to say {cat,dog}, or if it was an even longer list, like 10 or 50 objects...0160
it would start to get really hard, practically impossible, to say.0166
So instead, we can just change it to using a single letter, or whatever symbol is convenient for our purposes.0168
Furthermore, the order that the elements come in has no effect on the set itself.0175
So, the order that the elements appear in doesn't matter; we don't care about the order here.0180
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),0185
or any other way you have of ordering those things.0191
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.0194
It is just the group that you are considering, not the specific permutation of the line.0201
All right, that is the basic idea of a set.0206
If we want to describe a set, there are a bunch of different ways to describe it.0209
Here are the three most common ways that you are going to see.0213
Directly saying all of the elements: we could go through and, like I was talking about before with the curly braces and the comma,0216
we just say each of the elements inside of the set: ice, water, steam.0222
Our set has three elements; we have just said each of the three elements; that is the most basic method.0227
We just say what is inside of the set.0232
Another way is that we can clearly describe all of the members of the set.0234
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.0238
The point of it is that we are able to say, "Oh, yes, that is everything that makes it up."0243
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;0247
helium would be in the set; lithium would be in the set; all sorts of different elements0253
are going to be inside of the set, up until the eightieth element.0258
The eightieth element would be in it; the eighty-first element would not be in it.0261
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.0265
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.0272
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."0279
Our set is made up of all of the x; and then, you read this vertical bar as saying "such that."0288
So, all of the x such that x is the first name of a teacher at Educator.com would be this set.0301
Another way of reading that vertical bar is the word "where"--"x where x is the first name of a teacher at Educator.com."0309
Anything will do here, so long as it is getting across the idea that this thing here, in the second part, is describing the quality0319
required of the thing in the first part; so this part, the second part, describes what happens over here in the first part.0328
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 names0334
of all of the teachers who teach at Educator.com.0341
My name is Vincent; I am teaching at Educator.com (since you are watching this right now).0345
So, that means that "Vincent" is inside of this set.0349
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.0352
But we know for sure that Vincent is one of the names inside of the set.0356
Great! We can also symbolize things--if an element is contained in the set, and we want to talk about an element0360
being in that set, we have a convenient symbol to show it, this symbol right here: "element of," "contained in."0367
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,0373
because they showed up right here in our description of what the set was.0385
So, we know that a is an element in it; and we use this symbol right here to show "element of."0388
We can also talk about the idea of subsets (if a set is contained inside of another set).0395
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.0400
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.0410
But you could just think of it as it being inside of the other set.0416
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.0420
The symbol for this is this right here, "subset of."0427
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.0431
And then, Y is a subset of Z, because 1 and 3 both show up in Z.0444
So, we are able to see that that is a subset, because everything in here showed up in the other one.0453
Furthermore, we know that this property has to be transitive, because X is contained in Y, and Y is contained in Z;0459
then since X already lives inside of Y, it must also be inside of Z.0465
If we were to see it as sort of a picture, we would see it something like this.0469
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.0475
Great; we can also talk about a set that has no elements at all, the empty set.0486
And sometimes, it will also be called the null set.0492
Either way, it is a set that has nothing in it: it has no elements whatsoever.0495
We represent it with this symbol right here, "the empty set" symbol.0500
Now, this set is going to be unique, because any set that has no elements inside of it must be the empty set.0504
There is only one empty set, because there is only one way to have nothing inside of a set.0513
So, the empty set is just nothing at all; there is nothing inside of it--no elements; we have the empty set.0517
Since the empty set has nothing inside of it, it must inherently be inside of any other set.0525
All of its elements show up in every other set; each of its elements appears in every other set.0531
Now, I have the word "trivially" there, because what means is that it is trivial--it is obvious in sort of a silly way.0538
Yes, OK, sure, none of them show up...of course nothing shows up, because they don't have any there.0546
But that doesn't make it not true; it is trivially true.0552
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.0555
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.0564
B, {17,27,47}...the exact same thing: it is also going to have the empty set inside of it.0575
A and B don't really have any connection, other than the fact that they both have empty sets inside of them,0580
because any set at all, even the empty set itself, is going to contain the empty set,0585
because containing yourself is obvious, because it means you already have yourself in there.0590
All right, union and intersection: we can create new sets through having our sets interact with each other.0595
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.0601
The union of two sets is a set that contains the elements of each.0608
We symbolize this with an open cup; that gives us our union symbol.0612
The intersection of two sets is a set that contains the elements, and only those elements, that are in both sets.0618
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.0625
A cup pointing up--we are filling it up with a bunch of things; a cup pointing down--it is cutting things off.0632
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.0637
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.0649
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.0660
Union adds everything from all of our sets, and makes a big set out of everything that we have.0668
And A intersect B is going to make a smaller set (generally) that is going to see where you cut into each other--0673
where you have the exact same thing--and that is all we have left.0680
Example using actual things: if A is equal to {cat,mouse}, and B is equal to {cat,dog}, then A union B is...0686
cat shows up; mouse shows up; and then, we go over to B, and cat...cat already showed up, so it is not that interesting0693
to put it in again; we can't have copies show up in our set, because everything has to be unique;0702
but dog hasn't shown up before, so we get dog in there.0706
Now, for A intersect B, we ask, "Well, what is the thing that shows up in both of them?"0710
Cat, cat...yes, cat showed up in both of them, so it gets to go here.0714
But mouse doesn't show up over there; dog doesn't show up in A; so it doesn't show up either.0720
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.0728
If you are only in one of them, that is not good enough.0735
But union is where you only have to be in one of them, and you automatically make it in.0738
You can be in both of them, and that is great; you still get in that way, as well.0741
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.0746
And there have been some numbers; there have been some words; but we haven't encountered anything that crazy.0754
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.0759
But, as we have seen, we can also contain a lot of different ideas.0766
We don't just have to be stuck with numbers; we can also have elements other than numbers,0769
like words, or maybe even symbols or faces; we could have a bunch of different things inside of our set.0773
The important thing is that they are distinct objects.0778
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.0781
But we can also have an infinite set; that is going to be a set where the elements just keep going forever.0789
So, an infinite set means the elements keep going forever--they never stop; there is an unlimited number of elements.0794
So, how can we see an infinite set?0801
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...0803
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.0815
But we get the idea that, even though I can't count forever,0822
even though there is no way to literally write an infinite number of things, it still exists as an idea.0825
And so, as an idea, it is a perfectly fine set.0832
All of the numbers--all of the counting numbers, just listed out forever and ever and ever and ever--that gives us a set.0836
It is an infinite set, because it has an unlimited number of elements; but it is a perfectly reasonable set.0842
We can make even weirder, more interesting, stranger infinite sets if we want.0848
Consider the set where we take a word, and then we repeat it an ever-increasing number of times:0854
word, and then wordword, and then wordwordword, and then...etc., etc., etc.0860
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.0866
And if we keep doing this forever, we are going to have an infinite number of words.0874
For example, what if we took the word cat?0877
Then we would have cat, catcat, catcatcat, catcatcatcat, catcatcatcatcat (I think I said cat 5 times)...0880
and we could keep going and going and going and going; this set has infinitely many distinct elements.0889
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.0895
If you say 72, inside of this set right here, there is somewhere (not on this slide, but somewhere)--0901
if you just keep going, you are going to be able to imagine the idea of "cat" being repeated 72 times in a row.0909
So, we are creating new elements out of doing this; we build a set out of this idea.0917
And each one of these is distinct from the others: "cat" is not the same as "catcat," which is not the same as "catcatcat."0923
So, each one of these elements is distinct from the others, and there is an unlimited number of them; we have an infinite set.0929
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.0935
I love set theory personally; but it is something you will have to study in college if you are really, really interested in it.0943
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.0949
Now, let's start talking about how all of this set theory stuff applies to what we are going to be seeing,0955
in the near future in Precalculus, and then hopefully one day in calculus.0959
We can talk about numbers as sets: we understand the notion of "set" now, and that is great;0963
so we can now look at sets that make up numbers that we are going to use in math.0968
We have already seen one of the most essential sets: it was our first example of an infinite set, the natural numbers:0972
N = {1,2,3,4,5,6,7...}; this is just starting at 1 and counting on forever.0978
This is our first, most basic infinite set, in many ways.0985
We get this idea of counting and never stopping from the age of 3 on, if not even earlier.0990
We are getting this idea where you start counting, and you just never stop.0996
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."0999
But you could just keep going; and that is the idea of the natural numbers--1006
you just keep going forever and ever and ever, and you have an infinite number of elements.1009
One thing to note is that some teachers will define the natural numbers as starting with the 0.1015
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.1020
The tail of it is going to look the same, but you might start with 0; you might start with 1.1026
I prefer the version without 0, that starts with 1; but some teachers make a distinction,1030
and will call the one starting with 1 the counting numbers, and the one starting with 0 the natural numbers.1036
The point is to just pay attention to what your teacher is teaching you, if you are taking an outside class.1040
And make sure you are using their definition, so that you get everything right on the homework,1044
and that you understand what they are trying to teach you.1047
There is nothing better or less good about one or the other; it is just sort of a taste thing.1050
And I happen to prefer the version without 0.1055
Also, I just really quickly want to talk about this symbol that we use: that is N in blackboard bold,1058
which is to say what we would write out if we were writing the symbol by hand.1064
However, it is kind of hard to make that symbol by hand, since it is such a fancy typography symbol.1068
Instead, if you were writing this out by hand, the symbol that you write is like this.1073
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.1078
Probably, you are not going to have to write this stuff by hand for at least a while, and maybe not ever.1086
But I want you to know, in case you were interested in doing that.1090
So, let's keep expanding on these ideas.1094
We can take the natural numbers and say, "Well, we have positive numbers; but we also have negative numbers."1096
So, let's count, not just forward, but let's count backward, as well.1101
We will hit 0, and then we will just drive on into the negatives.1104
This gives us the integers: we start at 0, and then we go forward for positive, but we also go backward for negative.1107
0 forward is 1, 2, 3, 4, 5, 6... 0 backward is -1, -2, -3, -4, -5, -6, -7, -8.1114
We are going off in both directions, and that gives us the integers.1122
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;1125
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.1131
Now, I think you are wondering, "Wait, that made sense with the natural numbers; that was N; but why Z?"1139
Z comes from German; I believe zollen is the word for integer...no, zollen is just numbers;1145
and so, zollen is numbers, and because German mathematicians were doing this work1151
around the same time that English-speaking mathematicians were, and it was all being codified into symbols,1156
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.1160
All right, the next idea: we can add yet another layer of depth by including the idea of division.1167
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.1172
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,"1178
or something where I want to break a number into pieces?1186
Now, we have to be able to talk about fractions.1189
To do that, we use the rational numbers.1192
So, here we have that interesting format where we have the middle bar meaning "where," "such that," something like that.1194
So, what this means is that we have m/n, where m comes from the integers (m is one of these integers);1201
it can be a negative number; it can be a positive number; but it is going to be a whole number.1209
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 0,1214
and because of my definition of the natural numbers, we are not allowed to have 0 in the naturals.1220
That means we can't divide by 0, so we are safe there.1225
This gives us the ability to have any number up top, divided by any whole number that is positive on the bottom,1228
which lets us make any fraction that we want to.1234
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.1237
If we want to talk about the fraction -52/101, well, we can turn that into being equivalent to -52/101;1247
and so, we have -52...well, that is an integer; and 101 is a natural number; so right there, we have the natural numbers.1260
We are able to build any fraction that we are used to seeing in normal circumstances.1270
Any sort of normal fraction that we would talk about, we can make now with the rational numbers.1274
This gives us a lot of ability to make numbers.1280
We can get pretty much anywhere we want to be by using the rational numbers.1282
Also, you might wonder why it is Q; really quickly, if we wanted to write this by hand, you make a Q first;1288
and then, you drop a vertical line like that; so you have ℚ; that gives us our blackboard bold once again,1294
which is just to say something we can write by hand that makes it other than just writing the letter Q.1301
So, that lets us talk about that set of all the rational numbers.1306
And why do we use the letter Q? Because a fraction is connected with the idea of quotients.1309
So, as opposed to using F (which we kind of use for functions a lot, as we will talk about later),1314
we use Q to talk about quotients; so that is where we get the letter Q from.1319
All right, onward: we can also talk about rational numbers as a decimal expansion.1324
We have this idea of expanding a rational number into a decimal version; there is nothing wrong with decimal versions.1330
And we can have pretty much any number turn into a decimal version of itself.1336
So, the decimal expansion of every rational number (you probably learned this in grade school)...1340
every rational is either going to terminate (which means it ends), or it continues with repeating digits.1344
For our first example, something terminating, we have 0.09375; that is what we get from 3/32.1351
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.1357
So, we just cut it off, and it terminates--it stops at a certain point.1364
If, on the other hand, it continues with repeating digits, then that means there is some block of digits that will keep repeating forever.1368
So, with 77/270, we get .2851851851...we realize that 851851851...point 2 happens first, and then our repeating block shows up: 851851851.1376
And it is just going to march out forever and ever and ever.1391
So, if we have a rational number, it is going to do one of these two things.1395
It either terminates (it ends), or it repeats.1398
Every rational number, anything that can be expressed as an integer divided by an integer, by whole numbers over whole numbers,1401
with maybe a positive or a negative sign--that is going to have either the decimal ending or the decimal going forever, but repeating.1408
Why is this important? This idea of the rational numbers is really great, but there are still some numbers we can't express.1417
So, you might remember that decimal expansions of all the rationals either terminate, or they go into repetition.1423
There is at least one number you have heard of by now that keeps changing: pi.1431
You have learned about the number π for probably quite a few years now.1437
And you know that it just keeps shifting around: 3.1415...and you can memorize a bunch of digits, if you want.1440
But it is never going to just lock down and turn into something where you are done memorizing it.1447
There are always going to be infinitely many more digits to remember.1452
So, π never stops--it never repeats; it is not a rational number.1455
You have probably also heard that √2 is also not a rational number.1462
These turn out to be true; we can't express them as rational numbers--we can't express them as a fraction of integers.1467
The decimal expansion of an irrational number, unlike a rational, never stops, and it always keeps changing.1474
They are these sort of shifting, mixed-up numbers that just always keep doing interesting things.1480
They keep us working hard, unlike the rational numbers.1486
So, if we want to really be able to describe everything that is out there--all of the numbers we might encounter--1489
we need to be able to talk about the irrationals, in addition to the rationals.1493
Also, why do we call them irrationals?1497
It is nothing because they are crazy and they are something weird; it is because they are just not rational--they are irrational.1499
Irrational numbers...it is just because they are not rational, not because there is anything wrong with them,1506
but just because they are not that set that we call the rationals; that is it.1510
So, if we want to put the rational and irrational numbers together to get something1516
where we can really have all the numbers we work with, we have a great set.1519
That will give us the real numbers: we put them together, and we get the real numbers.1524
These are our bread and butter in mathematics.1529
You are going to be using them for years; you have been using them for pretty much everything you have ever done,1532
unless you have worked on the complex numbers for a little while.1536
And even if you did work on the complex numbers before, it was still using real numbers as part of those complex numbers.1539
The only thing was that i, and it still had a real number right next to it.1545
So, real numbers make up a huge portion of mathematics.1549
And unless you go for a whole bunch more math in college (which I would recommend--I really like math),1553
you are not going to end up seeing, probably, anything other than the real numbers,1559
until you get to some really abstract, interesting math.1563
But it is going to take a while before you see anything other than the reals.1565
They are great things to get at home with, and settle down with, and get a good understanding of.1569
And the purpose of all these set concepts, beforehand, is to be able to get a sense of how this work--1573
"Where do the reals live when we are not moving them around and working with them and doing things with them?"1578
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.1584
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.1592
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).1598
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,1606
"Well, I just want to talk about this one chunk," we can use interval notation.1610
For example, we might want to talk about everything from -1 to 3.1615
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 3.1618
So, we use interval notation; if we want to include the end numbers (-1 and 3), we use square brackets.1625
So, square brackets here give us inclusion; they keep those endpoints in it.1633
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.1640
If we want to exclude them (we want everything in between them, but we don't want the end things),1653
then we exclude them by using parentheses; parentheses give us exclusion.1659
That gets us -1 to 3, but without actually having -1 and 3.1665
So, -1 does not show up; 3 does not show up.1671
We use, if we want to symbolize it in a graphical manner (as a picture), open circles like this right here to show exclusion.1677
We use filled-in dots to show inclusion.1685
Exclusion is with parentheses, a curve, empty circle; and inclusion is with a filled-in dot or a nice square, solid bracket.1688
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.1697
It is just a question of whether or not we are going to include the ends of the interval.1704
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].1710
So, that is going to be all of the real numbers between 4 and 7, of course;1722
but it will keep the number 7 (because we have the square bracket);1726
but it is going to not include 4 (because we have the parenthesis).1729
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.1734
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.1741
If we want to talk about the idea of infinity, then we can talk about going on forever.1750
So, the symbol for infinity--that nice infinity sign--gives us a nice, convenient way to talk about going on forever.1754
So, if we want to talk about the interval going forever in one direction or the other, we will use -∞ or positive ∞.1763
And keep in mind: when there is no symbol in front of it, we just assume that it is positive.1769
So, negative infinity has the negative sign; positive infinity doesn't have anything.1773
If you absolutely had to symbolize that it was the positive version, you could put a little plus sign in front of it.1777
So, that will show us which direction we are going to go forever.1782
Depending on the direction that we want to talk about going forever, we will choose the appropriate infinity, negative or positive.1785
Now, keep in mind: you are always going to use parentheses with negative infinity or infinity.1791
Why is it that we always use parentheses when we are talking about them in interval?1795
It is because we can't actually include infinity: infinity isn't a number.1798
Infinity is just the idea of continuing forever.1803
So, since infinity is an idea of just keeping going, it is not an actual place; so we can't end on it.1806
To have a square bracket implies that we end on it, and it is there.1812
The parenthesis, on the other hand, will just show the idea of keeping going, keeping reaching towards it.1815
You will never actually reach it, but the interval will just keep going towards that notion of infinity.1821
So, for example, we could have -∞ to 2, with a square bracket on the 2.1827
That is going to be all numbers less than or equal to--everything starting at negative infinity, and working all the way up until 2.1831
And we will actually get to 2, and we will achieve 2.1838
(3,∞) is going to be all of the numbers greater than 3, but we won't include 3,1841
because we don't have a bracket on it; we have a parenthesis on the 3.1847
So, it is going to be everything from 3, but not actually including 3.1850
So, we will get really, really, really close to 3, but we will never actually touch it; we will never actually achieve 3.1854
And finally, if we want to just talk about the entire real line, that is the same thing as saying -∞ to positive ∞,1860
because that is everything that the real numbers have.1866
Start all the way from the very beginning; reach all the way to the beginning, and reach all the way to the end.1868
Just keep reaching forever and ever; go all the way to negative infinity; go all the way to positive infinity.1874
That is going to be the same thing as just saying "all the real numbers at once."1879
All right, let's do some examples.1883
We have the set X = {a,b,c}, the set Y = {b,c,d}, and the set Z = {c,d,e}.1885
Let's figure out a couple of different ways to talk about unions and intersections.1892
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.1896
And then, we add "union Z" on that; it is going to be in addition to all of the units with Z.1906
So, it is going to be all of the elements that show up in all of them: a shows up; b shows up;1910
c shows up; well, b already showed up; c already showed up; but d is new.1916
c already showed up; d already showed up; but e is new.1921
So, it is going to be {a,b,c,d,e}: there we go.1924
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?1932
What are the elements that are in each and every one of them?1940
Well, a does not show up in Z, nor does it show up in Y.1943
b does not show up in Z; it does show up in Y, but it has to show up in all three of them.1949
c does show up in Y and does show up in Z, so c is in.1954
And since everything else must not show up in X, it must be that the only thing inside of it is c.1957
We can also break this down into two pieces: we can say, "Well, what is X ∩ Y, first?"1963
X ∩ Y would be b and c, because those are the elements X and Y share in common.1967
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.1974
If they are all unions, and they are all intersections, it doesn't really matter the order that we choose--1985
which ones to intersect, which ones to "union" first...it is going to be a question of how they all interact.1990
What if we put all the elements in all of them together, or what element is inside of every single one of these sets?1996
So, it doesn't matter about the order; it doesn't matter about how we approach doing it.2003
But it does sometimes matter, if we talk about intersection and union working together.2006
So, for example, if we had (X ∩ Y), and then union Z, well, we have parentheses around it.2012
While we haven't explicitly reminded you of the order of operations, I am sure you remember to do things inside of parentheses first.2019
So, if X ∩ Y is inside of parentheses, then we have to do it first.2025
So, X ∩ Y gives {b,c}; and now we are going to do union Z.2028
Z is going to be c, d, and e; so that gives us a total of {b,c,d,e} in our set.2035
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?2044
Now, we need to start by asking, "Well, what is the intersection of Y and Z?"2054
Well, c and d show up in both of them; e does not show up; b does not show up in both of them.2059
So, c and d make up the intersection of Y and Z.2066
So, X ∪ {c,d} is going to be a and b (because they are new), and c and d (were already there).2070
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}.2079
Notice: these two things are not the same--there is not an equivalence between those two sets; they are not equal sets.2088
They aren't the same set, because how we approach putting these things together matters.2096
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 3.2101
It matters how we put these together, because we have different things going on.2109
It is not just multiplication; in a way, it is multiplication and addition--it matters the order that we do it in.2113
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.2118
The next example: we have ℕ, ℤ, ℚ, and ℝ; we have all of those big number sets that we talked about before.2126
Which one of them will be subsets to the others? How will the subsets work?2133
Well, first, let's start with reminding ourselves about what these are.2136
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.2138
The integers are going to be going off in the negative direction and the positive direction.2151
We have ... up until...and then we meet up...and then we just keep going that way.2156
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.2164
So, that is going to give us the rationals.2173
And the reals are just all numbers--what we are used to as thinking of all the possible numbers--all numbers are the reals.2176
Well, with that in mind, it is pretty easy to see that the natural numbers...2184
Well, since the integers...not equal...subset is what I meant to write...2188
Since the natural numbers are {1,2,3,4...}--they are all the positive integers--they must show up in the integers,2194
because the integers are the positive integers, and the negative integers, and 0.2201
So, ℕ is a subset of ℤ.2205
Now, ℤ shows up in the rationals; how is that possible?2209
Well, if you give me any integer number, I can very easily make a rational number out of it.2212
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.2218
You give me any integers (like -572), and I just put it over 1, and once again, we are back inside of the rationals.2232
So, whatever integer you give me, pretty clearly, has a rational version, as well.2239
We can keep going and now include the reals; we can talk about the reals.2244
And the reals are going to have everything, because we define the reals as having all of the rationals and all of the irrationals.2247
So, the rationals fit inside of the reals, as well; so we have subsets going up:2253
ℕ is a subset of ℤ, is a subset of ℚ, is a subset of ℝ.2257
That also means that, because this is transitive, ℕ is also a subset of ℚ, and ℕ is also a subset of ℝ.2261
ℤ is also a subset of ℝ, as well; and those are all of the relations that we can get out of this.2270
ℕ is a subset, and ℤ is a subset, and ℚ is a subset, inside of ℝ.2277
The third example: if we let A be the set of all titles of all published written works;2282
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.2288
Now, we start with...there are not just a couple of answers to this; there is not just one finished answer.2296
There are many more answers than I am aware of.2301
But I can give you some examples, and talk about how to think about this.2304
Let's also just rephrase this, so we have another way of thinking about it.2308
A is the same thing as talking about...A is every title of books and magazines and poems...2311
it is everything that is a written piece of work that has been published, that we could have actually2328
gone to a store and bought, or found in a published book; A is every title of books, etc., etc., etc.--2332
everything written, that is published--that is what A makes up.2338
Now, B is everything (from the way we are writing this) that is three words long.2342
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.2361
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."2373
So, A ∩ B is just titles that are three words long.2386
To be able to answer this question, we just need to figure out what are some titles that are three words.2397
So, we start thinking, and here are some of the ones that I thought of.2404
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.2408
Romeo and Juliet: there is a title that is three words long, written by Shakespeare, and it is a published piece of work.2414
We have all been able to find a copy of Romeo and Juliet if we have been looking for it.2424
So, Romeo and Juliet is one.2427
What about another one--how about Things Fall Apart by Chinua Achebe?2429
Or we could also talk about something by Kurt Vonnegut: Kurt Vonnegut wrote Breakfast of Champions.2438
So, Breakfast of Champions is another example of something where we have a phrase that is 3 words long,2446
and is the title of something that is a written work.2462
We could also talk about To the Lighthouse by Virginia Woolfe; To the Lighthouse is another example.2466
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.2475
And I don't know them; but it is going to be anything that is written and has three words in it...2482
3 words...not just in it, but 3 words for the title--precisely 3 words.2488
As much as I would like to be able to say Cannery Row, or Of Mice and Men, or 1984,2493
I can't talk about those, because they are not precisely 3 words long.2502
There are a lot of books out there that aren't 3 words long in the title.2507
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).2512
But any phrase that is three words long would be in B, and any title would be in A.2523
But what we are looking for is the intersection of A and B--titles that are three words.2528
Romeo and Juliet, Things Fall Apart, Breakfast of Champions,2533
To the Lighthouse: these are all some examples from various different authors.2536
The final example, Example 4: List all of the subsets of {x,y,z}.2542
The very first subset that we have to remember is the empty set: the empty set shows up as a subset for everything.2546
The empty set is our very first subset.2553
The next one--well, let's look at all of the subsets that have one element inside of them.2556
{x} (oops, I made a really bad bracket there) is going to be a set, just on its own; and that is a subset.2561
Another one would be {y}; that is another subset.2570
Another one would be {z}; those are all of the sets that are one element long, and are subsets of {x,y,z}.2574
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."2582
What about {x,z}? And then, finally, there is {y,z}.2593
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}2600
that have 2 elements precisely in them: x and y, x and z, y and z.2606
You could rearrange them in different orders, but remember, since it is a set we are talking about, order is not important.2612
It doesn't matter the order that it shows up in--just that it did show up at all.2617
Those are all of the sets that are going to be two elements long, and are subsets of {x,y,z}.2621
And then, finally, we have {x,y,z} itself; it is a subset of itself, because remember, by the formal definition2625
of being a subset, it just means that all of the elements inside of your set show up in the other set.2632
And every element {x,y,z} shows up inside of {x,y,z}; it makes sense; so every set is a subset of itself.2638
It is kind of obvious, and not that really interesting; but it is another trivial assertion.2646
It is interesting to think about, but not something that really gains us a lot of knowledge of any specific thing.2651
But it is still an interesting idea, and might have other connections later on, if we think about it a lot.2656
All right, so that gives us a total of 8 subsets; and those are all of them.2661
All right, I hope you enjoyed this; I hope you learned something about sets.2667
Like I said before, we are not going to really focus on the ideas that we had here.2670
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.2674
Virtually all of modern mathematics is built upon the idea of set theory.2679
It can be explained through the idea of set theory.2683
So, I just wanted you to get some exposure to this foundation, so that later things we talk about,2685
like when we talk about functions and a whole bunch of things, in fact, we have some idea of being able2689
to refer back to these sets, pulling things out from sets, going to other sets.2693
There is really cool stuff here; set theory is really fascinating; I totally recommend studying it sometime, if you get the chance.2697
I am glad that you managed to get here, and that you have some idea of how sets work.2704
And we will see you in the next lesson--goodbye!2707
Talk to you later at Educator.com!2709
2 answers
Last reply by: Professor Selhorst-Jones
Mon Jul 4, 2016 12:39 AM
Post by Javed Ghanniaiman on July 2, 2016
Hi Professor,
I have a question about one of the practice questions.
The question is, if A = (2, 13) and B = [11, 19], then what is A ?B (in interval notation)?
My answer was A ?B = [11, 12]. But the answer that they gave was A ?B = [11, 13)
My understanding is that 13 isn't included in set A, therefore it cannot intersect with B, right?
2 answers
Last reply by: Professor Selhorst-Jones
Mon Nov 16, 2015 1:28 PM
Post by Kitt Parker on November 16, 2015
Had a question about one of the practice questions. It stumped me a bit.
M = (-47, n] N = [-108,3)
What is M U N (Interval Notation)?
I knew this had to either be [-108,3) or [-108,n]. It seems that we are allowed to assume that the value of n is greater than 3. Is this because n cannot without further elaboration be defined as a number we are aware. For example we know that-47 exists in both sets. Seems odd because when just shown set M it seems that n could be any number, maybe -2 which would have changed the official answer.
1 answer
Last reply by: Micheal Bingham
Mon Jul 27, 2015 9:49 AM
Post by Duy Nguyen on July 3, 2015
Network error! [Error #2032] I encountered this problem. Please help me fix me. Thank you
1 answer
Last reply by: Professor Selhorst-Jones
Thu Jan 8, 2015 1:08 PM
Post by Ana Chu on January 7, 2015
For 3:20, can you write it like this:
{x | x + b = 20}
1 answer
Last reply by: Professor Selhorst-Jones
Tue Jan 6, 2015 12:08 PM
Post by enya zh on December 29, 2014
What is the use of the null set when there is nothing in it? What does it represent?
1 answer
Last reply by: Professor Selhorst-Jones
Wed Apr 23, 2014 10:06 AM
Post by Sameh Mahmoud on April 22, 2014
Hi Mr.Vincent,
Thanks for the Lecture, Q is Set which contain the Integer fractions , what about the Non Integer Fraction like 3.5/1.5 , in which set we can find?
Thanks
1 answer
Last reply by: Professor Selhorst-Jones
Mon Oct 7, 2013 3:58 PM
Post by guled habib on October 7, 2013
Hello Professor Vincent,
Thank you for your breathtaking lesson about sets, elements, and numbers. I really like your work and keep it up.
The formula for rational numbers set Q does it work when M and N are both negative numbers. I think they are opposing the set N thus making M divided N where both are negative numbers not a rational number.
1 answer
Last reply by: Professor Selhorst-Jones
Fri Oct 4, 2013 6:10 PM
Post by Mahmoud Osman on October 4, 2013
just want to thank You , You save my life <3
1 answer
Last reply by: Professor Selhorst-Jones
Sun Sep 22, 2013 1:09 PM
Post by enya zh on September 22, 2013
In example one didn't you forget the nothing set for X intersection Y intersection Z?
1 answer
Last reply by: Professor Selhorst-Jones
Sun Aug 18, 2013 11:44 AM
Post by Abhijith Nair on August 18, 2013
I like the way you teach.
1 answer
Last reply by: Professor Selhorst-Jones
Sun Aug 18, 2013 11:41 AM
Post by Bree Thomas on August 18, 2013
When you were talking about unions. The definition states that two steps is a set that contains the elements of each. Isn't that the same as a subset?
2 answers
Last reply by: antoni szeglowski
Fri Apr 11, 2014 11:53 PM
Post by Kiyoshi Smith on August 8, 2013
Hi Vincent,
I'm really enjoying your pre calculus course. Do you think you could do a course on mathematical proofs? I'm currently reading "How To Prove It" by Daniel Velleman and I thought it would be great if there were lectures about this subject. I've looked around and I can't find any.
Would you consider teaching a course on proofs?
Thanks,
Kiyoshi
1 answer
Last reply by: Professor Selhorst-Jones
Sun Jul 28, 2013 8:16 PM
Post by Chudamuni Dahal on July 26, 2013
Hi professor, is there any book recommendation for this course?
3 answers
Last reply by: Professor Selhorst-Jones
Tue Jan 6, 2015 12:04 PM
Post by StudentAN on June 29, 2013
i like
1 answer
Last reply by: Professor Selhorst-Jones
Sun Jun 16, 2013 11:18 AM
Post by Montgomery Childs on June 15, 2013
Excellent!
3 answers
Last reply by: Professor Selhorst-Jones
Thu Jun 13, 2013 8:26 PM
Post by Magesh Prasanna on June 4, 2013
Sir please give some practical applications of set theory.
I enjoyed the way you explain the concepts from skin to core.