Hello, this is section 1.

The topic of the section is

common sets of numbers and

set notation.

So, the goal of this section is

basically just to orient ourselves

and build a common foundation

for learning the language

of mathematics,

including set notation,

set building notation,

and, specifically, therefore,

very common sets of numbers

I want to solve to be aware of

as we move forward in this unit.

The first of these sets of numbers

is written as a capital Z,

or you could say, boldface Z,

and that is concise notation

for the set of all integers.

Now, integer is just synonymous

with a whole number, okay?

So, this capital Z,

stands for the set of all integers.

Now, how big is that set?

Well, that set's an infinite set.

There's an infinite number of integers.

In fact, it's a doubly infinite

array of numbers.

So I'm gonna write another

description of that set

(the set of all integers)

in what's called set notation.

So, for set notation,

you start with a curly brace, okay?

So a left curly brace,

and then at the end of this definition

I have a right curly brace, over here,

and then between them I'm gonna

fill in all this concise description

and all the elements of that set,

separated by commas.

Let me show you how this works.

So we have a dot dot dot.

I'll talk about what that means,

and let's just say

negative three, negative two,

negative one, zero, one,

two, three,

dot dot dot.

I'm gonna append this

with a right curly brace.

Okay.

So there is set notation

for the set of all integers.

What I've done

is that I'm separating

the elements

or, simply, let's call them members

of the set of all integers

by commas,

and on the end here

I'm sort of bookending this description

with what's called ellipsis,

so the dot dot dot just means

we continue on forever

on both directions,

to positive infinity

to the right and to the left

with respect to negative infinity.

That's what I meant by being

a doubly infinite array.

Okay.

Now, I can describe this set.

I'm gonna draw something called

a Venn diagram here.

So let's just say I have this framework.

I have a universe of sets.

So, U just stands for universe.

This is an example of a Venn diagram.

Within that universe, I have this set,

to be except, for sure,

the set of all integers.

So we just label that.

It's kind of a bubble,

and just slap on this Z there,

just to indicate that set of all integers.

Now, the question

I want to ask or pose is:

Are there numbers

outside of that set

that we can imagine?

And the answer is, of course, yes,

and there're lots of numbers

you're probably well familiar with

that aren't contained

in the set of all integers.

I mean, one such example,

would be like the number 1/2, right?

1/2 would be floating out here.

Somewhere.

I'm just gonna use a sort of dot

to indicate

1/2 is out there.

Certainly it's outside of

the set of all integers.

So, on the heels of that,

let's introduce some little bit

more notation.

So, there's set notation.

Now, we'd like to,

when we talk about sets,

be able to coherently describe

what's called a membership

in that set.

So, for instance,

here's the notation as it works.

Let's say, the number 4.

The number 4.

That's a whole number.

That's an integer.

So, I can write this notation.

So, I would read this as follows.

This symbol, if your not familiar,

is epsilon,

and epsilon here,

with relation to sets,

just means "is an element of".

You could read that as

4 is an element of the integers,

or you can say 4 is contained

within the integers.

So, yeah, sure enough.

4 is in the integers.

That's in the bubble.

Somewhere in my Venn diagram,

but 1/2 is not in there.

So, that's how we denote

membership in a set.

"4 is an element of"

We use the epsilon symbol.

I could similarly write

0 is an element of the integers,

or 8 is an element of the integers,

and so on.

How can I show

something's not an element of it

using this membership notation?

Well, for instance,

we just observed, casually,

"1/2 is not an element of"

So, I'm just gonna say

epsilon with a slash through

is the typical way to notate this,

and 1/2 is not an element of,

remember,

of the integers.

In fact, we can see

almost, really, any fraction

is not gonna be

contained in the integers,

so there's a lot more out there

to sort of discover.

That leads us to our next kind of

natural set of numbers.

Think of Q as standing for quotient.

So, capital Q, is again a

concise notation

standing for what's usually referred

as the set of all rational numbers.

What's a rational number?

By definition, a rational number is

just like a ratio.

It's a fraction.

The set of all rational numbers is

the set of all fractions.

We can write all the

elements in that set

in the following fashion.

I can say Q consists of

all the elements that look like this

"a" divided by "b"

and this vertical line in

set builder notation

is read "such that".

So, the set of all things

that look like this

defining membership here

this criterion

a and b are integers

and if we're totally thorough

we can't divide by zero.

We can't have zero

in the denominator.

So we also have to specify

that b cannot be zero.

So there you have it.

One more comment on the

integers and the rational numbers

is they have a nice relationship

if you think about this.

The integers are entirely

contained inside of the rationals.

Another way to put that is

the integers are

a proper subset of the rationals,

and the way that's written in set notation

is that you would say

the integers are contained

inside of the rationals.

So this notation

is what's called a subset

or you could think of containment.

So, again, we would just say

Z is a subset

is contained within

the rationals.

How do I illustrate that

with a Venn diagram?

Let's just draw another bigger bubble

that entirely contains Z.

Also it's gonna contain this fraction

1/2 for sure.

And there, that outer bubble, is my Q

so Z is entirely contained within it

That helps to sort of

connect these things.

Of course we all know, for instance,

in terms of numbers

1 is less than 2.

We use this kind of relation symbol.

Well, similarly for sets,

we can talk about kind of one set

in the sense of being less

than another one.

So, this relation just means

this set, Z, is contained or smaller

in the way

is contained inside of Q.

Okay.

So what we're gonna do is

build up a full sequence

of the most common sets of numbers

you'll ever encounter

in any applied science field

beginning with Z and Q.

That's kind of our basis.