This video is about notation.
I'm going to describe the most common pieces of notation that we use to talk about sets.
Um... For our purposes, we're gonna think
of a set as a collection of things, erm, normally numbers.
We put an asterisk here, because if you ask a logician they will tell you that
this is a terrible way to think of sets and it causes trouble. And they're right.
But for our purposes, ahh... it's fine; it works, and it's simple, so we're gonna leave it at that.
So we think of a set as a collection of things,
and those things are the elements of the set.
Um... Here's an example. I'm gonna describe one set.
When I write it this way, I have given a name to this set.
Here, this "A" is the name of the set.
And then, I am using curly braces here to describe the set. In between the curly braces, I describe it.
In this case for example, this is a set that has three elements,
and the elements are the number four, the number five, and the number six.
There are a few common symbols that we use to describe the common operations and statements
we make about sets. So, let's introduce them.
First, these two symbols are used to say that something is an element of a set,
or that something is not an element of a set. For example, I am gonna be talking about these three sets:
A, B, and C. And then I could say that 1 is an element of A, but 1 is not an element of B.
Okay. So those two symbols describe the relation between an element and a set.
This symbol is different. This symbol describes the relation between two different sets.
And it means "is a subset of." So instead of saying when an element is in a set, we say:
"I'm comparing two sets and I want to see if one is contained in the other."
For example, using this case, I can say that C is a subset of A. So it's contained in A.
And unfortunately here I have to pause and give a warning.
Because this piece of notation comes with a little bit of trouble: it's not entirely universal. Here's why:
I've been using this symbol, this symbol here, to mean that something is a subset of, and may be equal.
Hence the little second bar there, that simulates an equals sign.
If I ever want to say that something is a subset of and I want to specify "but they are not equal," then I write in this.
So, this symbol means the smaller subset is strictly smaller or equal, and this means it is strictly smaller.
But unfortunately, some books do things differently. Some books use this symbol
to mean "is a subset of, and may be equal." And then if they want to specify that it may
NOT be equal then they use a different symbol, for example, this one. Um... But this is a problem,
because then this can mean two different things depending on who is writing it. Um...
My personal preference is to always use either this symbol, or this symbol and then
there is no ambiguity possible.
Let's go back to all the pieces of notation.
[Paper rustling]
Um... These two symbols are used to describe operations between sets.
The first one means "union of sets."
For example if I write "A union B," it means I am gonna construct a new set,
and the elements of this new set are anything that was already either in A, or in B, or possibly in both.
So for example in this case, I will take all the elements in A or B or both, and that would be this set.
And this symbol means "intersection of sets." So it's also used to construct a new set, given two.
In this case I am gonna take all the elements which are in BOTH the sets A and B.
So in this case, that would be a set with one single element, which is 2.
Finally, this symbol is used to mean the empty set, and that's simply the set with no elements.
So another way to describe it could be "the empty set is this set. And there's nothing in it."
Finally, there are a few sets of numbers which are important, because they appear a lot.
And it's convenient to have names for them, so we can refer to them. So let's learn about those.
Um... These are the major four sets of numbers we'll use.
There's a fifth one, which is the complex numbers. We won't be talking about those.
We have special names for them and those are universal; everybody agrees on these names.
They are uppercase letters with an extra line in them (double-struck).
This "N" represents the natural numbers.
Those are the "counting numbers," which is the first set you learn in kindergarten.
This comes with a caveat: I started here at 0, uh, but this is also a matter of convention.
Some books start at 0; some books start at 1. There are good arguments to do both of them;
we haven't agreed on it.
The second symbol, Z, represents the integer numbers. Those are (include) both positive and negative.
"Q" represents the rationals. Those are the fractions; anything you can write as a quotient of integers.
And finally, R stands for the real numbers, ah,
those are anything that has a decimal expansion.
So, notice that, ah, these are subsets of each other. Each one is bigger than the next.