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.