Universal set and absolute complement

What I want to do
in this video is

introduce the idea
of a universal set,

or the universe that we care
about, and also the idea

of a complement, or an
absolute complement.

If we're for doing
it as a Venn diagram,

the universe is usually
depicted as some type

of a rectangle right over here.

And it itself is a set.

And it usually is denoted
with the capital U--

U for universe-- not to be
confused with the union set

notation.

And you could say
that the universe

is all possible things
that could be in a set,

including farm animals
and kitchen utensils

and emotions and
types of Italian food

or even types of food.

But then that just
becomes somewhat crazy,

because you're thinking
of all possible things.

Normally when people talk
about a universal set,

they're talking about a universe
of things that they care about.

So the set of all people or
the set of all real numbers

or the set of all countries,
whatever the discussion

is being focused on.

But we'll talk about in
abstract terms right now.

Now, let's say you have a subset
of that universal set, set A.

And so set A literally
contains everything

that I have just shaded in.

What we're going
to talk about now

is the idea of a complement, or
the absolute complement of A.

And the way you could
think about this

is this is the set of all
things in the universe that

aren't in A. And
we've already looked

at ways of expressing this.

The set of all things in the
universe that aren't in A,

we could also write as
a universal set minus A.

Once again, this is a
capital U. This is not

the union symbol
right over here.

Or we could literally
write this as U,

and then we write that little
slash-looking thing, U slash A.

So how do we represent
that in the Venn diagram?

Well, it would be all the stuff
in U that is not in A. One way

to think about it, you
could think about it

as the relative complement
of A that is in U.

But when you're taking
the relative complement

of something that is
in the universal set,

you're really talking about
the absolute complement.

Or when people just talk
about the complement,

that's what they're saying.

What's the set of all
the things in my universe

that are not in A.

Now, let's make things a
little bit more concrete

by talking about
sets of numbers.

Once again, our sets-- we could
have been talking about sets

of TV personalities
or sets of animals

or whatever it might be.

But numbers are a nice,
simple thing to deal with.

And let's say that
our universe that we

care about right over here
is the set of integers.

So our universe is
the set of integers.

So I'll just write
U-- capital U--

is equal to the set of integers.

And this is a little
bit of an aside,

but the notation for the set of
integers tends to be a bold Z.

And it's Z for Zahlen, from
German, for apparently integer.

And the bold is
this kind of weird

looking- they call
it blackboard bold.

And it's what mathematicians
use for different types of sets

of numbers.

In fact, I'll do a little
aside here to do that.

So for example, they'll
write R like this

for the set of real numbers.

They'll write a Q in that
blackboard bold font,

so it looks something like this.

They'll write the Q; it might
look something like this.

This would be the set
of rational numbers.

And you might say,
why Q for a rational?

Well, there's a
couple of reasons.

One, the R is already taken up.

And Q for quotient.

A rational number
can be expressed

as a quotient of two integers.

And we just saw you
have your Z for Zahlen,

or integers, the
set of all integers.

So our universal
set-- the universe

that we care about
right now-- is integers.

And let's define a subset of it.

Let's call that
subset-- I don't know.

Let me use a letter that I
haven't been using a lot.

Let's call it C,
the set C. Let's say

it's equal to negative
5, 0, and positive 7.

And I'm obviously not
drawing it to scale.

The set of all
integers is infinite,

while the set C is a finite set.

But I'll just kind
of just to draw

it, that's our set
C right over there.

And let's think about
what is a member of C,

and what is not a member of
C. So we know that negative 5

is a member of our set C.

This little symbol right
here, this denotes membership.

It looks a lot like the
Greek letter epsilon,

but it is not the
Greek letter epsilon.

This just literally means
membership of a set.

We know that 0 is a
member of our set.

We know that 7 is a
member of our set.

Now, we also know
some other things.

We know that the number negative
8 is not a member of our set.

We know that the number 53
is not a member of our set.

And 53 is sitting
someplace out here.

We know the number 42 is
not a member of our set.

42 might be sitting
someplace out there.

Now let's think
about C complement,

or the complement
of C. C complement,

which is the same thing as
our universe minus C, which

is the same thing
as universe, or you

could say the relative
complement of C

in our universe.

These are all
equivalent notation.

What is this, first of
all, in our Venn diagram?

Well, it's all
this stuff outside

of our set C right over here.

And now, all of a sudden,
we know that negative 5

is a member of C, so it can't
be a member of C complement.

So negative 5 is not a
member of C complement.

0 is not a member
of C complement.

0 sits in C, not
in C complement.

I could say 53-- 53 is a
member of C complement.

It's outside of C. It's in the
universe, but outside of C. 42

is a member of C complement.

So anyway, hopefully that helps
clear things up a little bit.