1
00:00:00,854 --> 00:00:01,710
PROFESSOR: OK.

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00:00:01,710 --> 00:00:04,760
So we come to the idea
of countable sets, which

3
00:00:04,760 --> 00:00:07,130
are the most familiar
kind of infinite sets.

4
00:00:07,130 --> 00:00:12,130
And a countable set is one where
you can list the elements-- a0,

5
00:00:12,130 --> 00:00:13,910
a1, a2 and so on.

6
00:00:13,910 --> 00:00:17,430
So there's a list of
all of the elements of A

7
00:00:17,430 --> 00:00:20,920
in which every element in
A appears at some point.

8
00:00:20,920 --> 00:00:23,835
You can count up to
any given element of A,

9
00:00:23,835 --> 00:00:26,900
and every element of A you
will eventually get to,

10
00:00:26,900 --> 00:00:28,590
you'll be able to
count up to it.

11
00:00:28,590 --> 00:00:31,470
So it's just a
matter of listing it.

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00:00:31,470 --> 00:00:33,980
And the technical definition
of "A is countable"

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00:00:33,980 --> 00:00:36,650
is if there's a
bijection between A

14
00:00:36,650 --> 00:00:37,910
and the non-negative integers.

15
00:00:37,910 --> 00:00:40,100
Because this listing,
in effect, really

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00:00:40,100 --> 00:00:44,490
is a mapping from the
non-negative integers to A. 0

17
00:00:44,490 --> 00:00:48,600
is a0, 1 maps to
a1, 2 maps to a2,

18
00:00:48,600 --> 00:00:51,830
and implicitly there's a
bijection being indicated here.

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00:00:51,830 --> 00:00:55,060
That's assuming that all of the
[? a's ?] are distinct for it

20
00:00:55,060 --> 00:00:56,310
to be a bijection.

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00:00:56,310 --> 00:00:58,420
So we also have,
as a special case,

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00:00:58,420 --> 00:01:01,000
the finite sets are also
considered to be countable.

23
00:01:01,000 --> 00:01:04,365
So really, if n is
a bijection to A,

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00:01:04,365 --> 00:01:06,550
then A is called
countably infinite.

25
00:01:06,550 --> 00:01:08,730
The other possibility
is that A is finite.

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00:01:08,730 --> 00:01:12,959
And the two together, I
just say A is countable.

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00:01:12,959 --> 00:01:14,500
So what we've just
figured out, then,

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00:01:14,500 --> 00:01:18,010
from the previous examples,
is that the positive integers

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00:01:18,010 --> 00:01:19,040
are countable.

30
00:01:19,040 --> 00:01:20,650
And all the integers
are countable,

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00:01:20,650 --> 00:01:23,527
because in both cases
we exhibited bijections

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00:01:23,527 --> 00:01:24,735
to the non-negative integers.

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00:01:27,850 --> 00:01:30,140
Another important
and not very hard

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00:01:30,140 --> 00:01:32,580
example is the set of
finite binary words.

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00:01:32,580 --> 00:01:35,520
So we use this notation,
"0, 1 star," meaning

36
00:01:35,520 --> 00:01:38,190
all the finite-- star means
all the finite sequences

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00:01:38,190 --> 00:01:39,940
of these elements, 0 and 1.

38
00:01:39,940 --> 00:01:42,910
So this is just the
finite binary words.

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00:01:42,910 --> 00:01:43,870
How are they countable?

40
00:01:43,870 --> 00:01:46,890
Well, I need a way to be able to
list them in some orderly way.

41
00:01:46,890 --> 00:01:48,750
Well, let's just
do it by length.

42
00:01:48,750 --> 00:01:51,770
Let's begin by listing
the empty word, or string,

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00:01:51,770 --> 00:01:52,840
of length zero.

44
00:01:52,840 --> 00:01:55,930
And then I'm going to list
all the one-bit strings,

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00:01:55,930 --> 00:01:58,162
the strings of length one.

46
00:01:58,162 --> 00:01:59,310
And there are two of those.

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00:01:59,310 --> 00:02:03,252
So let the second element,
the next element of the list

48
00:02:03,252 --> 00:02:05,710
after the empty string, be 0,
and then let the next element

49
00:02:05,710 --> 00:02:07,150
after that be 1.

50
00:02:07,150 --> 00:02:09,130
Then let's list all
the length two strings.

51
00:02:09,130 --> 00:02:12,090
Well, there's four length
two binary strings.

52
00:02:12,090 --> 00:02:15,290
And let's just list them
in some sensible order--

53
00:02:15,290 --> 00:02:17,420
say, by their binary
representation.

54
00:02:17,420 --> 00:02:20,410
And then keep going.

55
00:02:20,410 --> 00:02:23,560
List all the length
three binary strings--

56
00:02:23,560 --> 00:02:24,560
there's eight of those.

57
00:02:24,560 --> 00:02:28,850
And finally, keep going up
until you get to the length n

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00:02:28,850 --> 00:02:31,190
binary strings, of which
there are 2 to the n.

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00:02:31,190 --> 00:02:35,120
And this is a description of a
way to list, one after another,

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00:02:35,120 --> 00:02:39,080
all of the finite binary words,
or finite binary strings.

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00:02:39,080 --> 00:02:42,710
And that listing is
implicitly a description

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00:02:42,710 --> 00:02:46,320
of a bijection from the
non-negative integers

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00:02:46,320 --> 00:02:49,830
n to the nth element
in my listing.

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00:02:49,830 --> 00:02:53,620
And that's a bijection, so the
binary words are countable.

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00:02:57,515 --> 00:02:59,370
Another example
of a countable set

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00:02:59,370 --> 00:03:03,150
is the pairs of
non-negative integers.

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00:03:03,150 --> 00:03:05,420
So how can-- now I've got
the non-negative integers.

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00:03:05,420 --> 00:03:08,477
I've got to find a bijection of
pairs of non-negative integers.

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00:03:08,477 --> 00:03:09,560
How am I going to do that?

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00:03:09,560 --> 00:03:12,440
Well, it's the same idea as
we used with binary strings.

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00:03:12,440 --> 00:03:13,940
There's a bunch of
ways to prove it,

72
00:03:13,940 --> 00:03:17,650
but let's just propagate
the binary string idea.

73
00:03:17,650 --> 00:03:22,850
Let's start listing the pairs
of non-negative integers.

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00:03:22,850 --> 00:03:28,380
And after 0, 0, I'm going to
list two pairs-- 0, 1 and 1, 0.

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00:03:28,380 --> 00:03:32,306
And after them, I'm going to
list three pairs-- 0, 2, 2, 0,

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00:03:32,306 --> 00:03:33,870
and 1, 1.

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00:03:33,870 --> 00:03:37,050
And after them, 0,
3, 3, 0, 1, 2, 2, 1.

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00:03:37,050 --> 00:03:38,690
And if you can see
what I'm doing,

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00:03:38,690 --> 00:03:43,790
I'm basically listing the
pairs in the order of the sum

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00:03:43,790 --> 00:03:44,940
of their coordinates.

81
00:03:44,940 --> 00:03:49,060
So the nth block of pairs
that I'm going to list

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00:03:49,060 --> 00:03:54,680
will be the pairs the sum of
whose two coordinates is n.

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00:03:54,680 --> 00:03:56,210
There'll be n plus one of those.

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00:03:56,210 --> 00:03:57,460
And I keep going in this way.

85
00:03:57,460 --> 00:04:00,120
This is a nice orderly
description of--

86
00:04:00,120 --> 00:04:01,780
or a description of
a nice orderly way

87
00:04:01,780 --> 00:04:04,580
to list all of the pairs
of non-negative integers.

88
00:04:04,580 --> 00:04:07,720
Within a block, invent
some alphabetical rule

89
00:04:07,720 --> 00:04:10,210
for listing the pairs.

90
00:04:10,210 --> 00:04:14,060
So I'm going to--
I've hinted at a rule

91
00:04:14,060 --> 00:04:17,375
here for listing the finite
set of pairs whose sum is n,

92
00:04:17,375 --> 00:04:19,940
and you can invent--
any one will do.

93
00:04:19,940 --> 00:04:23,390
So that tells us that
we have a bijection

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00:04:23,390 --> 00:04:26,500
between the
non-negative integers

95
00:04:26,500 --> 00:04:29,180
and the pairs of
non-negative integers.

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00:04:29,180 --> 00:04:32,420
So that's another
important bijection.

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00:04:32,420 --> 00:04:34,420
Now, when you're trying
to prove countability,

98
00:04:34,420 --> 00:04:36,640
it's very useful to have
the following lemma, which

99
00:04:36,640 --> 00:04:38,181
gives an alternative
characterization

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00:04:38,181 --> 00:04:41,750
of countability-- namely,
a set A is countable if

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00:04:41,750 --> 00:04:45,520
and only if you can
list A allowing repeats.

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00:04:45,520 --> 00:04:47,050
Remember, our
original definition

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00:04:47,050 --> 00:04:50,930
is that you can list A without
repeats if it's infinite,

104
00:04:50,930 --> 00:04:52,290
or else it's finite.

105
00:04:52,290 --> 00:04:55,340
So that was-- the bijection
between the non-negative

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00:04:55,340 --> 00:04:57,060
integers and A, in
effect, is saying

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00:04:57,060 --> 00:04:59,510
that that's a listing of
all of an infinite set A

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00:04:59,510 --> 00:05:04,440
with no repeats, because it's
a bijection we're mapping.

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00:05:04,440 --> 00:05:05,880
If an element
appeared twice, we'd

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00:05:05,880 --> 00:05:07,627
have two different
non-negative integers

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00:05:07,627 --> 00:05:09,960
mapping to it, which would
break the bijection property,

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00:05:09,960 --> 00:05:11,930
the injection property.

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00:05:11,930 --> 00:05:14,370
And so suppose we allow repeats.

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00:05:14,370 --> 00:05:18,020
And the claim is that that's
fine, because you can fix that.

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00:05:18,020 --> 00:05:20,810
So the lemma says
that, if there's

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00:05:20,810 --> 00:05:24,770
a surjective function from the
non-negative integers to A,

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00:05:24,770 --> 00:05:26,480
then A is countable.

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00:05:26,480 --> 00:05:29,180
Well, let's just check
quickly in one direction.

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00:05:29,180 --> 00:05:33,470
If A is finite, then there's
clearly a surjective function

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00:05:33,470 --> 00:05:36,270
from the non-negative
integers to A.

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00:05:36,270 --> 00:05:38,390
There's lots of extra
non-negative integers

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00:05:38,390 --> 00:05:39,720
you don't need.

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00:05:39,720 --> 00:05:42,450
If it's a finite set,
like 10 elements in A,

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00:05:42,450 --> 00:05:44,500
map 0 through 9 to
those 10 elements,

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00:05:44,500 --> 00:05:47,530
and map every other
non-negative integer, say,

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00:05:47,530 --> 00:05:52,650
to 10th element, or
last element, of A.

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00:05:52,650 --> 00:05:55,270
So there's certainly a
surjection if A is finite.

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00:05:55,270 --> 00:05:58,250
Now, suppose that A is infinite,
and I have a surjection

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00:05:58,250 --> 00:06:00,600
from the non-negative
integers to A.

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00:06:00,600 --> 00:06:03,272
So I'm listing A with repeats.

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00:06:03,272 --> 00:06:05,530
And I'm supposed to
have a bijection if it

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00:06:05,530 --> 00:06:06,789
matches the other definition.

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00:06:06,789 --> 00:06:07,580
How do you do that?

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00:06:07,580 --> 00:06:09,550
Well, if you're a
computer scientist,

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00:06:09,550 --> 00:06:12,310
you know how to change
a sequence with repeats

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00:06:12,310 --> 00:06:13,980
into a sequence without repeats.

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00:06:13,980 --> 00:06:15,780
You just filter
it for duplicates,

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00:06:15,780 --> 00:06:17,370
going from left to right.

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00:06:17,370 --> 00:06:20,030
Take this infinite
sequence of elements of A

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00:06:20,030 --> 00:06:23,100
in which there are repeats, and
keep only the first occurrence

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00:06:23,100 --> 00:06:23,920
of each element.

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00:06:23,920 --> 00:06:26,880
That will define a bijection
with the non-negative integers

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00:06:26,880 --> 00:06:29,320
if a is infinite.

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00:06:29,320 --> 00:06:32,750
And that's how we prove this
lemma, which I'm just going

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00:06:32,750 --> 00:06:34,460
to settle for talking through.

146
00:06:34,460 --> 00:06:36,420
So now we have
another convenient way

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00:06:36,420 --> 00:06:39,590
to show that a set is
countable, just by describing,

148
00:06:39,590 --> 00:06:41,690
not a bijection,
but a surjection

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00:06:41,690 --> 00:06:44,260
between the non-negative
integers in A. Surjections

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00:06:44,260 --> 00:06:47,140
are often easier to describe
than bijections, which

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00:06:47,140 --> 00:06:48,960
is why this is a useful lemma.

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00:06:48,960 --> 00:06:53,230
A corollary of this
is that, if I'm

153
00:06:53,230 --> 00:06:55,280
trying to show that
a set A is countable,

154
00:06:55,280 --> 00:06:58,190
all that I really need to
do is find some other set

155
00:06:58,190 --> 00:06:59,860
that I know to be
countable and describe

156
00:06:59,860 --> 00:07:03,150
a surjection from
that other set C to A.

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00:07:03,150 --> 00:07:07,602
Because I know that
if C is countable,

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00:07:07,602 --> 00:07:09,810
then there'll be a bijection
between the non-negative

159
00:07:09,810 --> 00:07:13,520
integers and C. And
since when you combine

160
00:07:13,520 --> 00:07:16,700
a bijection with a surjection,
you wind up with a surjection,

161
00:07:16,700 --> 00:07:19,520
that will implicitly
define a surjection

162
00:07:19,520 --> 00:07:23,890
from the non-negative integers
to A, which by the lemma

163
00:07:23,890 --> 00:07:25,450
tells me that A is countable.

164
00:07:25,450 --> 00:07:27,080
So the general way
to prove something

165
00:07:27,080 --> 00:07:28,960
is countable is just
describe a surjection,

166
00:07:28,960 --> 00:07:30,770
from something you
know to be countable,

167
00:07:30,770 --> 00:07:33,350
that hits your target.

168
00:07:33,350 --> 00:07:36,160
And let's look at
an example of that.

169
00:07:36,160 --> 00:07:38,030
I claim that the
rationals are countable,

170
00:07:38,030 --> 00:07:39,488
the rational numbers
are countable.

171
00:07:39,488 --> 00:07:43,620
Well, this is kind of a little
bit more striking at first,

172
00:07:43,620 --> 00:07:45,840
because you can see
how you can count

173
00:07:45,840 --> 00:07:49,590
the non-negative integers,
the positive integers, all

174
00:07:49,590 --> 00:07:51,660
the integers, because
there's a nice sensible way

175
00:07:51,660 --> 00:07:52,940
to have one come after other.

176
00:07:52,940 --> 00:07:55,020
But with the
rationals, it's messy.

177
00:07:55,020 --> 00:07:58,110
In between any two rationals,
there's another rational.

178
00:07:58,110 --> 00:07:59,540
There isn't any first rational.

179
00:07:59,540 --> 00:08:02,350
There isn't any obvious
way to list them all.

180
00:08:02,350 --> 00:08:06,170
But really, if you stop thinking
about the rationals of how they

181
00:08:06,170 --> 00:08:10,900
are laid out on the real line,
but just think of them as pairs

182
00:08:10,900 --> 00:08:14,200
of integers, then it becomes
clear how to list them,

183
00:08:14,200 --> 00:08:17,880
because we already know that the
pairs of non-negative integers

184
00:08:17,880 --> 00:08:18,680
are countable.

185
00:08:18,680 --> 00:08:21,680
So I'm just going to map a pair
of non-negative integers m, n

186
00:08:21,680 --> 00:08:24,860
to the rational
number m divided by n.

187
00:08:24,860 --> 00:08:27,610
Well, n might be
zero, so if n is zero,

188
00:08:27,610 --> 00:08:31,220
just map all of those pairs to
your favorite rational number.

189
00:08:31,220 --> 00:08:33,049
Call it a half.

190
00:08:33,049 --> 00:08:38,200
And that gives us a
nice surjective mapping,

191
00:08:38,200 --> 00:08:39,630
because every
rational number can

192
00:08:39,630 --> 00:08:41,546
be expressed as m
over n-- at least

193
00:08:41,546 --> 00:08:43,130
every non-negative
rational number.

194
00:08:43,130 --> 00:08:45,970
So in effect, what we
have is a surjection

195
00:08:45,970 --> 00:08:47,990
from the pairs of
non-negative integers, which

196
00:08:47,990 --> 00:08:52,340
we know is countable, onto the
non-negative real numbers--

197
00:08:52,340 --> 00:08:55,830
sorry, the non-negative rational
numbers, quotients of integers.

198
00:08:55,830 --> 00:08:57,850
Which means that the
rationals, sure enough,

199
00:08:57,850 --> 00:09:01,170
are countable, even
though they seem to be

200
00:09:01,170 --> 00:09:04,180
spread out all over the line.

201
00:09:04,180 --> 00:09:08,410
So, again, we saw that if
N cross N is countable,

202
00:09:08,410 --> 00:09:10,200
and there's a surj,
described above,

203
00:09:10,200 --> 00:09:13,810
to the non
non-negative rationals,

204
00:09:13,810 --> 00:09:15,580
so they're countable.

205
00:09:15,580 --> 00:09:18,580
Well, just looking
ahead a little bit,

206
00:09:18,580 --> 00:09:20,360
it's going to turn
out that, in contrast

207
00:09:20,360 --> 00:09:24,500
to the rational numbers, the
real numbers are not countable.

208
00:09:24,500 --> 00:09:27,230
And in fact, neither
are the infinite binary

209
00:09:27,230 --> 00:09:30,760
sequences that we
saw-- there was

210
00:09:30,760 --> 00:09:33,900
a bijection between the
infinite binary sequences

211
00:09:33,900 --> 00:09:36,716
and the power set of the
non-negative integers.

212
00:09:36,716 --> 00:09:38,090
And both of these
are going to be

213
00:09:38,090 --> 00:09:40,670
basic examples of
uncountable sets,

214
00:09:40,670 --> 00:09:43,030
so sets that are not
countable, which we will be

215
00:09:43,030 --> 00:09:46,260
examining in the next lecture.