WEBVTT
00:00:00.680 --> 00:00:04.170
Probability models often
involve infinite sample
00:00:04.170 --> 00:00:08.780
spaces, that is,
infinite sets.
00:00:08.780 --> 00:00:12.570
But not all sets are
of the same kind.
00:00:12.570 --> 00:00:17.530
Some sets are discrete and we
call them countable, and some
00:00:17.530 --> 00:00:20.960
are continuous and we call
them uncountable.
00:00:20.960 --> 00:00:23.470
But what exactly is the
difference between these two
00:00:23.470 --> 00:00:24.860
types of sets?
00:00:24.860 --> 00:00:27.760
How can we define
it precisely?
00:00:27.760 --> 00:00:31.320
Well, let us start by first
giving a definition of what it
00:00:31.320 --> 00:00:33.920
means to have a countable set.
00:00:33.920 --> 00:00:37.530
A set will be called countable
if its elements can be put
00:00:37.530 --> 00:00:41.450
into a 1-to-1 correspondence
with the positive integers.
00:00:41.450 --> 00:00:45.280
This means that we look at the
elements of that set, and we
00:00:45.280 --> 00:00:47.990
take one element-- we call
it the first element.
00:00:47.990 --> 00:00:50.790
We take another element--
we call it the second.
00:00:50.790 --> 00:00:54.090
Another, we call the third
element, and so on.
00:00:54.090 --> 00:00:58.980
And this way we will eventually
exhaust all of the
00:00:58.980 --> 00:01:02.920
elements of the set, so that
each one of those elements
00:01:02.920 --> 00:01:06.840
corresponds to a particular
positive integer, namely the
00:01:06.840 --> 00:01:09.140
index that appears underneath.
00:01:09.140 --> 00:01:13.110
More formally, what's happening
is that we take
00:01:13.110 --> 00:01:18.810
elements of that set that are
arranged in a sequence.
00:01:18.810 --> 00:01:24.420
We look at the set, which is the
entire range of values of
00:01:24.420 --> 00:01:28.580
that sequence, and we want that
sequence to exhaust the
00:01:28.580 --> 00:01:31.470
entire set omega.
00:01:31.470 --> 00:01:37.250
Or in other words, in simpler
terms, we want to be able to
00:01:37.250 --> 00:01:43.360
arrange all of the elements
of omega in a sequence.
00:01:43.360 --> 00:01:46.800
So what are some examples
of countable sets?
00:01:46.800 --> 00:01:50.330
In a trivial sense, the positive
integers themselves
00:01:50.330 --> 00:01:54.960
are countable, because we can
arrange them in a sequence.
00:01:54.960 --> 00:01:58.110
This is almost tautological,
by the definition.
00:01:58.110 --> 00:02:00.820
For a more interesting example,
let's look at the set
00:02:00.820 --> 00:02:02.370
of all integers.
00:02:02.370 --> 00:02:04.610
Can we arrange them
in a sequence?
00:02:04.610 --> 00:02:09.669
Yes, we can, and we can do it
in this manner, where we
00:02:09.669 --> 00:02:13.330
alternate between positive
and negative numbers.
00:02:13.330 --> 00:02:16.880
And this way, we're going to
cover all of the integers, and
00:02:16.880 --> 00:02:20.190
we have arranged them
in a sequence.
00:02:20.190 --> 00:02:24.870
How about the set of all pairs
of positive integers?
00:02:24.870 --> 00:02:27.280
This is less clear.
00:02:27.280 --> 00:02:28.480
Let us look at this picture.
00:02:28.480 --> 00:02:32.560
This is the set of all pairs of
positive integers, which we
00:02:32.560 --> 00:02:35.590
understand to continue
indefinitely.
00:02:35.590 --> 00:02:39.450
Can we arrange this sets
in a sequence?
00:02:39.450 --> 00:02:40.870
It turns out that we can.
00:02:40.870 --> 00:02:45.310
And we can do it by tracing
a path of this kind.
00:02:50.130 --> 00:02:52.990
So you can probably get
the sense of how
00:02:52.990 --> 00:02:54.820
this path is going.
00:02:54.820 --> 00:03:00.090
And by continuing this way, over
and over, we're going to
00:03:00.090 --> 00:03:05.370
cover the entire set of all
pairs of positive integers.
00:03:05.370 --> 00:03:07.840
So we have managed to arrange
them in a sequence.
00:03:07.840 --> 00:03:11.870
So the set of all such pairs
is indeed a countable set.
00:03:11.870 --> 00:03:15.160
And the same argument can be
extended to argue for the set
00:03:15.160 --> 00:03:19.816
of all triples of positive
integers, or the set of all
00:03:19.816 --> 00:03:23.350
quadruples of positive
integers, and so on.
00:03:23.350 --> 00:03:28.550
This is actually not just a
trivial mathematical point
00:03:28.550 --> 00:03:32.270
that we discuss for some curious
reason, but it is
00:03:32.270 --> 00:03:35.280
because we will often
have sample spaces
00:03:35.280 --> 00:03:36.960
that are of this kind.
00:03:36.960 --> 00:03:41.490
And it's important to know
that they're countable.
00:03:41.490 --> 00:03:44.079
Now for a more subtle example.
00:03:44.079 --> 00:03:48.070
Let us look at all rational
numbers within the range
00:03:48.070 --> 00:03:50.640
between 0 and 1.
00:03:50.640 --> 00:03:53.140
What do we mean by
rational numbers?
00:03:53.140 --> 00:03:56.340
We mean those numbers that
can be expressed as a
00:03:56.340 --> 00:03:59.050
ratio of two integers.
00:03:59.050 --> 00:04:02.240
It turns out that we can arrange
them in a sequence,
00:04:02.240 --> 00:04:04.540
and we can do it as follows.
00:04:04.540 --> 00:04:07.050
Let us first look at rational
numbers that have a
00:04:07.050 --> 00:04:09.450
denominator term of 2.
00:04:09.450 --> 00:04:12.120
Then, look at the rational
numbers that have a
00:04:12.120 --> 00:04:16.140
denominator term of 3.
00:04:16.140 --> 00:04:19.500
Then, look at the rational
numbers, always within this
00:04:19.500 --> 00:04:23.230
range of interest, that have
a denominator of 4.
00:04:26.270 --> 00:04:28.780
And then we continue
similarly--
00:04:28.780 --> 00:04:33.630
rational numbers that have a
denominator of 5, and so on.
00:04:33.630 --> 00:04:35.909
This way, we're going
to exhaust all of
00:04:35.909 --> 00:04:37.710
the rational numbers.
00:04:37.710 --> 00:04:42.650
Actually, this number here
already appeared there.
00:04:42.650 --> 00:04:44.030
It's the same number.
00:04:44.030 --> 00:04:47.720
So we do not need to include
this in a sequence, but that's
00:04:47.720 --> 00:04:49.020
not an issue.
00:04:49.020 --> 00:04:52.920
Whenever we see a rational
number that has already been
00:04:52.920 --> 00:04:56.060
encountered before,
we just delete it.
00:04:56.060 --> 00:05:00.480
In the end, we end up with a
sequence that goes over all of
00:05:00.480 --> 00:05:02.090
the possible rational numbers.
00:05:02.090 --> 00:05:05.310
And so we conclude that the set
of all rational numbers is
00:05:05.310 --> 00:05:07.500
itself a countable set.
00:05:07.500 --> 00:05:11.350
So what kind of set would
be uncountable?
00:05:11.350 --> 00:05:14.240
An uncountable set, by
definition, is a set that is
00:05:14.240 --> 00:05:15.570
not countable.
00:05:15.570 --> 00:05:19.480
And there are examples of
uncountable sets, most
00:05:19.480 --> 00:05:24.560
prominent, continuous subsets
of the real line.
00:05:24.560 --> 00:05:28.110
Whenever we have an interval,
the unit interval, or any
00:05:28.110 --> 00:05:31.710
other interval that has positive
length, that interval
00:05:31.710 --> 00:05:34.290
is an uncountable set.
00:05:34.290 --> 00:05:37.370
And the same is true if, instead
of an interval, we
00:05:37.370 --> 00:05:40.490
look at the entire real line,
or we look at the
00:05:40.490 --> 00:05:43.330
two-dimensional plane,
or three-dimensional
00:05:43.330 --> 00:05:44.760
space, and so on.
00:05:44.760 --> 00:05:48.960
So all the usual sets that we
think of as continuous sets
00:05:48.960 --> 00:05:52.040
turn out to be uncountable.
00:05:52.040 --> 00:05:55.040
How do we know that they
are uncountable?
00:05:55.040 --> 00:05:58.640
There is actually a brilliant
argument that establishes that
00:05:58.640 --> 00:06:01.650
the unit interval
is uncountable.
00:06:01.650 --> 00:06:04.360
And then the argument is easily
extended to other
00:06:04.360 --> 00:06:07.150
cases, like the reals
and the plane.
00:06:07.150 --> 00:06:10.240
We do not need to know how this
argument goes, for the
00:06:10.240 --> 00:06:11.820
purposes of this course.
00:06:11.820 --> 00:06:15.930
But just because it is so
beautiful, we will actually be
00:06:15.930 --> 00:06:17.190
presenting it to you.