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.