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