Probability models often
involve infinite sample
spaces, that is,
infinite sets.
But not all sets are
of the same kind.
Some sets are discrete and we
call them countable, and some
are continuous and we call
them uncountable.
But what exactly is the
difference between these two
types of sets?
How can we define
it precisely?
Well, let us start by first
giving a definition of what it
means to have a countable set.
A set will be called countable
if its elements can be put
into a 1-to-1 correspondence
with the positive integers.
This means that we look at the
elements of that set, and we
take one element-- we call
it the first element.
We take another element--
we call it the second.
Another, we call the third
element, and so on.
And this way we will eventually
exhaust all of the
elements of the set, so that
each one of those elements
corresponds to a particular
positive integer, namely the
index that appears underneath.
More formally, what's happening
is that we take
elements of that set that are
arranged in a sequence.
We look at the set, which is the
entire range of values of
that sequence, and we want that
sequence to exhaust the
entire set omega.
Or in other words, in simpler
terms, we want to be able to
arrange all of the elements
of omega in a sequence.
So what are some examples
of countable sets?
In a trivial sense, the positive
integers themselves
are countable, because we can
arrange them in a sequence.
This is almost tautological,
by the definition.
For a more interesting example,
let's look at the set
of all integers.
Can we arrange them
in a sequence?
Yes, we can, and we can do it
in this manner, where we
alternate between positive
and negative numbers.
And this way, we're going to
cover all of the integers, and
we have arranged them
in a sequence.
How about the set of all pairs
of positive integers?
This is less clear.
Let us look at this picture.
This is the set of all pairs of
positive integers, which we
understand to continue
indefinitely.
Can we arrange this sets
in a sequence?
It turns out that we can.
And we can do it by tracing
a path of this kind.
So you can probably get
the sense of how
this path is going.
And by continuing this way, over
and over, we're going to
cover the entire set of all
pairs of positive integers.
So we have managed to arrange
them in a sequence.
So the set of all such pairs
is indeed a countable set.
And the same argument can be
extended to argue for the set
of all triples of positive
integers, or the set of all
quadruples of positive
integers, and so on.
This is actually not just a
trivial mathematical point
that we discuss for some curious
reason, but it is
because we will often
have sample spaces
that are of this kind.
And it's important to know
that they're countable.
Now for a more subtle example.
Let us look at all rational
numbers within the range
between 0 and 1.
What do we mean by
rational numbers?
We mean those numbers that
can be expressed as a
ratio of two integers.
It turns out that we can arrange
them in a sequence,
and we can do it as follows.
Let us first look at rational
numbers that have a
denominator term of 2.
Then, look at the rational
numbers that have a
denominator term of 3.
Then, look at the rational
numbers, always within this
range of interest, that have
a denominator of 4.
And then we continue
similarly--
rational numbers that have a
denominator of 5, and so on.
This way, we're going
to exhaust all of
the rational numbers.
Actually, this number here
already appeared there.
It's the same number.
So we do not need to include
this in a sequence, but that's
not an issue.
Whenever we see a rational
number that has already been
encountered before,
we just delete it.
In the end, we end up with a
sequence that goes over all of
the possible rational numbers.
And so we conclude that the set
of all rational numbers is
itself a countable set.
So what kind of set would
be uncountable?
An uncountable set, by
definition, is a set that is
not countable.
And there are examples of
uncountable sets, most
prominent, continuous subsets
of the real line.
Whenever we have an interval,
the unit interval, or any
other interval that has positive
length, that interval
is an uncountable set.
And the same is true if, instead
of an interval, we
look at the entire real line,
or we look at the
two-dimensional plane,
or three-dimensional
space, and so on.
So all the usual sets that we
think of as continuous sets
turn out to be uncountable.
How do we know that they
are uncountable?
There is actually a brilliant
argument that establishes that
the unit interval
is uncountable.
And then the argument is easily
extended to other
cases, like the reals
and the plane.
We do not need to know how this
argument goes, for the
purposes of this course.
But just because it is so
beautiful, we will actually be
presenting it to you.
