WEBVTT 00:00:00.000 --> 00:00:03.872 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:03.872 --> 00:00:13.070 align:middle line:90% 00:00:13.070 --> 00:00:17.120 align:middle line:84% CASEY RODRIGUEZ: So last time we spoke about-- 00:00:17.120 --> 00:00:19.250 align:middle line:90% we covered sets and induction. 00:00:19.250 --> 00:00:24.950 align:middle line:84% This time, I want to ask a question about sets, 00:00:24.950 --> 00:00:28.820 align:middle line:84% which turns out is actually quite a deep question. 00:00:28.820 --> 00:00:31.130 align:middle line:84% I mean, I didn't come up with it myself. 00:00:31.130 --> 00:00:37.930 align:middle line:84% This question is at least 150 years old probably. 00:00:37.930 --> 00:00:49.600 align:middle line:84% So the question that I want to ask is if A and B are sets-- 00:00:49.600 --> 00:00:58.440 align:middle line:90% 00:00:58.440 --> 00:01:01.470 align:middle line:84% or let's phrase it this way, at least a little more 00:01:01.470 --> 00:01:04.310 align:middle line:90% efficiently. 00:01:04.310 --> 00:01:19.655 align:middle line:84% When do two sets, A and B, have the same size? 00:01:19.655 --> 00:01:26.060 align:middle line:90% 00:01:26.060 --> 00:01:27.890 align:middle line:84% Now, this is especially interesting 00:01:27.890 --> 00:01:30.920 align:middle line:84% if these two sets are not finite, 00:01:30.920 --> 00:01:33.140 align:middle line:84% meaning there's not five of them, 00:01:33.140 --> 00:01:37.340 align:middle line:84% there's not eight of them, but there's infinitely 00:01:37.340 --> 00:01:39.020 align:middle line:84% many members, whatever that means. 00:01:39.020 --> 00:01:41.520 align:middle line:84% And I'll actually define what that means. 00:01:41.520 --> 00:01:44.330 align:middle line:84% For example, do the natural numbers and the integers 00:01:44.330 --> 00:01:46.370 align:middle line:90% have the same size? 00:01:46.370 --> 00:01:49.310 align:middle line:84% Do the natural numbers and the rational numbers 00:01:49.310 --> 00:01:51.140 align:middle line:90% have the same size? 00:01:51.140 --> 00:01:53.660 align:middle line:84% What about the rational numbers and the real numbers? 00:01:53.660 --> 00:01:55.460 align:middle line:84% Even though we haven't defined that yet, 00:01:55.460 --> 00:02:02.000 align:middle line:84% you can still keep in mind your notion of the real numbers 00:02:02.000 --> 00:02:03.515 align:middle line:90% from calculus. 00:02:03.515 --> 00:02:06.410 align:middle line:90% 00:02:06.410 --> 00:02:10.169 align:middle line:90% And so this question is-- 00:02:10.169 --> 00:02:12.270 align:middle line:90% why is it deep? 00:02:12.270 --> 00:02:17.250 align:middle line:84% Because it depends on this word here, 00:02:17.250 --> 00:02:23.150 align:middle line:84% "size," and what exactly that means. 00:02:23.150 --> 00:02:34.200 align:middle line:84% So I'm not-- so kind of an answer, 00:02:34.200 --> 00:02:35.925 align:middle line:90% this is due to George Cantor. 00:02:35.925 --> 00:02:38.640 align:middle line:90% 00:02:38.640 --> 00:02:40.300 align:middle line:90% He said that two sets-- 00:02:40.300 --> 00:02:43.080 align:middle line:84% so not an answer, but a way of defining size 00:02:43.080 --> 00:02:46.200 align:middle line:84% or a way of saying the two sets have the same size. 00:02:46.200 --> 00:03:03.930 align:middle line:84% He said two sets have the same size when the two sets-- 00:03:03.930 --> 00:03:19.040 align:middle line:84% the elements of the two sets can be paired off, meaning-- 00:03:19.040 --> 00:03:24.170 align:middle line:84% and I will make this much more precise in a minute or so. 00:03:24.170 --> 00:03:26.750 align:middle line:90% What does he mean? 00:03:26.750 --> 00:03:28.050 align:middle line:90% Or what do I mean? 00:03:28.050 --> 00:03:35.510 align:middle line:84% For example, the set a, b, and c and the set 1, 2, and 3. 00:03:35.510 --> 00:03:38.960 align:middle line:84% So this one, this set, consists of three letters-- a, b, and c. 00:03:38.960 --> 00:03:40.970 align:middle line:84% This set consists of three integers-- 00:03:40.970 --> 00:03:42.020 align:middle line:90% 1, 2, and 3. 00:03:42.020 --> 00:03:44.900 align:middle line:84% They have the same size because I can essentially 00:03:44.900 --> 00:03:45.665 align:middle line:90% pair them off. 00:03:45.665 --> 00:03:49.590 align:middle line:90% 00:03:49.590 --> 00:03:54.660 align:middle line:84% So a and 1 go together, b and 2 go together, 00:03:54.660 --> 00:03:59.760 align:middle line:84% and c and 3 go together, meaning each member of this set 00:03:59.760 --> 00:04:05.200 align:middle line:84% gets paired off with an element of this set. 00:04:05.200 --> 00:04:07.930 align:middle line:84% And every element of this set gets paired off 00:04:07.930 --> 00:04:09.550 align:middle line:84% with a unique element from this set. 00:04:09.550 --> 00:04:17.570 align:middle line:90% 00:04:17.570 --> 00:04:25.860 align:middle line:84% Now, to make-- and so this goes under the name 00:04:25.860 --> 00:04:35.640 align:middle line:84% of the theory of cardinality or cardinal numbers, if you like. 00:04:35.640 --> 00:04:38.440 align:middle line:84% But we won't go too deeply into it. 00:04:38.440 --> 00:04:40.620 align:middle line:90% So the way you make-- 00:04:40.620 --> 00:04:44.640 align:middle line:84% or the way one makes this pairing off business 00:04:44.640 --> 00:04:48.065 align:middle line:84% more precise is using the language of functions. 00:04:48.065 --> 00:04:50.190 align:middle line:84% And you've been dealing with functions since you've 00:04:50.190 --> 00:04:53.400 align:middle line:90% been doing calculus. 00:04:53.400 --> 00:05:00.690 align:middle line:84% So I'm just going to quickly reintroduce or review 00:05:00.690 --> 00:05:04.770 align:middle line:84% some terminology that goes along with functions that 00:05:04.770 --> 00:05:09.990 align:middle line:84% will be the precise meaning of this pairing off business 00:05:09.990 --> 00:05:11.940 align:middle line:90% that I'm writing here. 00:05:11.940 --> 00:05:14.650 align:middle line:84% But just with an eye towards the future, 00:05:14.650 --> 00:05:19.560 align:middle line:84% this is why I'm reviewing some of this terminology related 00:05:19.560 --> 00:05:20.670 align:middle line:90% to functions. 00:05:20.670 --> 00:05:25.370 align:middle line:90% 00:05:25.370 --> 00:05:28.800 align:middle line:84% So a brief review of some terminology for functions-- 00:05:28.800 --> 00:05:39.350 align:middle line:84% so let me just recall that if A and B are sets, 00:05:39.350 --> 00:05:47.920 align:middle line:90% then function f A B-- 00:05:47.920 --> 00:05:52.540 align:middle line:84% so a function's usually written as f colon A arrow to B, 00:05:52.540 --> 00:05:57.160 align:middle line:84% meaning it takes elements from A into elements from B. 00:05:57.160 --> 00:06:06.765 align:middle line:90% This is a mapping that-- 00:06:06.765 --> 00:06:10.140 align:middle line:90% 00:06:10.140 --> 00:06:11.910 align:middle line:90% or if you like, an assignment. 00:06:11.910 --> 00:06:16.410 align:middle line:90% 00:06:16.410 --> 00:06:26.570 align:middle line:84% It assigns to each x in A a unique element which 00:06:26.570 --> 00:06:30.560 align:middle line:90% we denote f of x in B. 00:06:30.560 --> 00:06:35.850 align:middle line:84% So one single input, x, gives me a single output. 00:06:35.850 --> 00:06:38.750 align:middle line:84% And the input does not give me three outputs. 00:06:38.750 --> 00:06:43.220 align:middle line:84% Now, this is what one would call a naive definition probably, 00:06:43.220 --> 00:06:46.610 align:middle line:84% not necessarily completely unambiguous. 00:06:46.610 --> 00:06:48.320 align:middle line:90% But for us, it suffices. 00:06:48.320 --> 00:06:51.020 align:middle line:84% In the textbook, you can look up a function 00:06:51.020 --> 00:06:57.870 align:middle line:84% is unambiguously defined or as a subset of the Cartesian product 00:06:57.870 --> 00:07:01.470 align:middle line:84% of A and B, which you would recognize 00:07:01.470 --> 00:07:03.990 align:middle line:84% as essentially describing the graph of the function. 00:07:03.990 --> 00:07:06.840 align:middle line:84% But then you never use that definition of again. 00:07:06.840 --> 00:07:10.560 align:middle line:84% And you essentially use this definition 00:07:10.560 --> 00:07:13.020 align:middle line:84% when you think about functions and when you 00:07:13.020 --> 00:07:14.380 align:middle line:90% prove properties of function. 00:07:14.380 --> 00:07:17.550 align:middle line:84% So this we will just take as our definition. 00:07:17.550 --> 00:07:22.240 align:middle line:84% And it will be unambiguous enough for us. 00:07:22.240 --> 00:07:37.240 align:middle line:84% So let f be a function from A to B. 00:07:37.240 --> 00:07:46.440 align:middle line:84% If C is a subset of A, we define a set-- 00:07:46.440 --> 00:07:49.890 align:middle line:84% so this is f of capital C. So C is not an element of A. 00:07:49.890 --> 00:07:51.020 align:middle line:90% It's a subset of A. 00:07:51.020 --> 00:07:55.155 align:middle line:84% And this is a subset of B. This is-- 00:07:55.155 --> 00:07:58.230 align:middle line:90% 00:07:58.230 --> 00:08:01.500 align:middle line:84% let me write it slightly differently here. 00:08:01.500 --> 00:08:05.580 align:middle line:84% This is a set of all y in B such that there 00:08:05.580 --> 00:08:12.660 align:middle line:84% exists an x in C such that y equals to f of x. 00:08:12.660 --> 00:08:17.580 align:middle line:84% And I could write this a little more efficiently as this 00:08:17.580 --> 00:08:20.190 align:middle line:84% is the set of all elements f of x 00:08:20.190 --> 00:08:30.270 align:middle line:84% as x ranges over the elements of this subset C. 00:08:30.270 --> 00:08:44.570 align:middle line:84% And if D is a subset of B, we define the set f inverse of D. 00:08:44.570 --> 00:08:48.140 align:middle line:84% This is the set of guys that get mapped into D. 00:08:48.140 --> 00:08:53.120 align:middle line:84% So I should say this is the inverse image of D, 00:08:53.120 --> 00:08:57.890 align:middle line:84% not the inverse of this function f. 00:08:57.890 --> 00:09:03.080 align:middle line:84% So the inverse, whatever that means for now, 00:09:03.080 --> 00:09:04.920 align:middle line:84% does not necessarily always exist. 00:09:04.920 --> 00:09:08.690 align:middle line:84% So this inverse image of the set D always exists. 00:09:08.690 --> 00:09:19.670 align:middle line:84% This is the set of all elements in A such that f of x is in D. 00:09:19.670 --> 00:09:31.400 align:middle line:90% So for example, let me-- 00:09:31.400 --> 00:09:38.760 align:middle line:90% 00:09:38.760 --> 00:09:48.090 align:middle line:90% so 1, 2, 3, 4 and a, b, c, d. 00:09:48.090 --> 00:09:55.110 align:middle line:84% And let's suppose 1 goes to a, 2 goes to a, 3 goes 00:09:55.110 --> 00:09:57.810 align:middle line:90% to c, 4 goes to d. 00:09:57.810 --> 00:10:02.160 align:middle line:90% 00:10:02.160 --> 00:10:08.040 align:middle line:84% Then f of the set 1, 2, this is where-- 00:10:08.040 --> 00:10:10.270 align:middle line:90% what is the set that gets-- 00:10:10.270 --> 00:10:12.570 align:middle line:90% so this is the subset of-- 00:10:12.570 --> 00:10:20.220 align:middle line:84% so this you should think of as B, a, b, and c, and D. And then 00:10:20.220 --> 00:10:23.610 align:middle line:90% 1, 2, and 3, this is A. 00:10:23.610 --> 00:10:27.810 align:middle line:90% So f of the set of 1, 2-- 00:10:27.810 --> 00:10:29.670 align:middle line:90% so 1 gets mapped to a. 00:10:29.670 --> 00:10:30.990 align:middle line:90% 2 gets mapped to a. 00:10:30.990 --> 00:10:34.470 align:middle line:90% So this is just the set 2. 00:10:34.470 --> 00:10:38.520 align:middle line:90% f of the set-- 00:10:38.520 --> 00:10:41.820 align:middle line:90% let's go with 1 and 3. 00:10:41.820 --> 00:10:43.980 align:middle line:90% 1 gets mapped to a. 00:10:43.980 --> 00:10:47.640 align:middle line:90% So this should be a. 00:10:47.640 --> 00:10:49.896 align:middle line:90% 1 gets mapped to a. 00:10:49.896 --> 00:10:52.515 align:middle line:84% And 3 gets mapped gets mapped to c. 00:10:52.515 --> 00:10:55.860 align:middle line:90% 00:10:55.860 --> 00:10:59.130 align:middle line:84% And so a couple of inverse images-- 00:10:59.130 --> 00:11:04.360 align:middle line:84% if I look at the inverse image of a, 00:11:04.360 --> 00:11:08.040 align:middle line:84% this is equal to the set of all guys that map to a. 00:11:08.040 --> 00:11:12.150 align:middle line:84% This is, well, 1 maps to a and 2 maps to a. 00:11:12.150 --> 00:11:14.100 align:middle line:90% So this is 1, 2. 00:11:14.100 --> 00:11:16.890 align:middle line:90% 00:11:16.890 --> 00:11:21.180 align:middle line:84% Now, if I look at the inverse image of a, 00:11:21.180 --> 00:11:27.420 align:middle line:84% c, d, what elements get mapped to a, c, and d? 00:11:27.420 --> 00:11:30.630 align:middle line:90% Well, 1 and 2 get mapped to a. 00:11:30.630 --> 00:11:32.170 align:middle line:90% 3 gets map to c. 00:11:32.170 --> 00:11:33.360 align:middle line:90% 4 gets mapped to d. 00:11:33.360 --> 00:11:36.390 align:middle line:84% So everything maps into a, C, and d. 00:11:36.390 --> 00:11:40.170 align:middle line:84% So this is just the original set or the set A-- 00:11:40.170 --> 00:11:42.780 align:middle line:90% 1, 2, 3, 4. 00:11:42.780 --> 00:11:49.630 align:middle line:90% 00:11:49.630 --> 00:11:55.570 align:middle line:84% All right, so there's more terminology. 00:11:55.570 --> 00:11:58.600 align:middle line:84% And this is what we will mean by when 00:11:58.600 --> 00:12:00.940 align:middle line:90% two sets can be paired off. 00:12:00.940 --> 00:12:02.590 align:middle line:90% Or this makes that more precise. 00:12:02.590 --> 00:12:07.024 align:middle line:90% 00:12:07.024 --> 00:12:19.160 align:middle line:84% Let f be a function from A to B. We say that f is injective, 00:12:19.160 --> 00:12:21.305 align:middle line:90% or I'll write one-to-one. 00:12:21.305 --> 00:12:27.712 align:middle line:84% It should be read one-to-one if f satisfies 00:12:27.712 --> 00:12:28.670 align:middle line:90% the following property. 00:12:28.670 --> 00:12:31.220 align:middle line:90% 00:12:31.220 --> 00:12:39.800 align:middle line:84% f of x1 equals f of x2 implies x1 equals x2. 00:12:39.800 --> 00:12:42.020 align:middle line:84% So injective, or one-to-one, means 00:12:42.020 --> 00:12:46.595 align:middle line:84% if I take two different inputs, I get two different outputs. 00:12:46.595 --> 00:12:49.560 align:middle line:90% 00:12:49.560 --> 00:12:51.220 align:middle line:84% That's essentially what this means. 00:12:51.220 --> 00:12:56.910 align:middle line:90% I mean, taking the equivalent-- 00:12:56.910 --> 00:13:04.930 align:middle line:84% so equivalently, from a logical standpoint, 00:13:04.930 --> 00:13:06.550 align:middle line:84% this statement implying this statement 00:13:06.550 --> 00:13:08.590 align:middle line:84% is equivalent to the negation of this statement 00:13:08.590 --> 00:13:11.060 align:middle line:84% implying the negation of this statement. 00:13:11.060 --> 00:13:17.620 align:middle line:84% So equivalently, this means x not equal to x2 implies f of x1 00:13:17.620 --> 00:13:20.440 align:middle line:90% not equal to f of x2. 00:13:20.440 --> 00:13:22.100 align:middle line:90% So maybe this is clear. 00:13:22.100 --> 00:13:25.150 align:middle line:84% If I were to define f as injective if it satisfies 00:13:25.150 --> 00:13:27.640 align:middle line:84% this property, maybe that would have 00:13:27.640 --> 00:13:32.293 align:middle line:84% been clearer that f takes two different elements to two 00:13:32.293 --> 00:13:33.085 align:middle line:90% different elements. 00:13:33.085 --> 00:13:36.790 align:middle line:90% 00:13:36.790 --> 00:13:40.540 align:middle line:84% But this condition here is typically easier to verify, 00:13:40.540 --> 00:13:42.775 align:middle line:84% or at least simpler to state and verify. 00:13:42.775 --> 00:13:45.390 align:middle line:90% 00:13:45.390 --> 00:14:04.870 align:middle line:84% So f is surjective or onto if the image of A 00:14:04.870 --> 00:14:11.450 align:middle line:84% is B. So let me write that statement out a little bit 00:14:11.450 --> 00:14:11.950 align:middle line:90% more. 00:14:11.950 --> 00:14:24.350 align:middle line:84% Everything in the set B gets mapped to by something from A. 00:14:24.350 --> 00:14:30.890 align:middle line:84% So equivalently, this says that for all y in B 00:14:30.890 --> 00:14:37.670 align:middle line:84% there exists an x in A so that f of x equals y. 00:14:37.670 --> 00:14:46.104 align:middle line:90% 00:14:46.104 --> 00:14:56.420 align:middle line:84% f is bijective if f is one-to-one and onto. 00:14:56.420 --> 00:15:01.870 align:middle line:84% So f is bijective if it's both injective and surjective. 00:15:01.870 --> 00:15:07.550 align:middle line:90% 00:15:07.550 --> 00:15:11.960 align:middle line:84% So for example, this map that we just drew over here 00:15:11.960 --> 00:15:16.710 align:middle line:90% that sends 1 to a, 2 to b-- 00:15:16.710 --> 00:15:20.340 align:middle line:84% I mean 2 to a, 3 to c, 4, to d, this 00:15:20.340 --> 00:15:22.620 align:middle line:84% is neither injective nor surjective. 00:15:22.620 --> 00:15:26.730 align:middle line:84% It's not injective because it takes two different elements 00:15:26.730 --> 00:15:29.280 align:middle line:90% to the image in B, namely a. 00:15:29.280 --> 00:15:31.180 align:middle line:90% So it takes 1 and 2 to a. 00:15:31.180 --> 00:15:33.780 align:middle line:84% It's not surjective because nothing gets 00:15:33.780 --> 00:15:38.235 align:middle line:90% mapped to the element b here. 00:15:38.235 --> 00:15:54.890 align:middle line:90% 00:15:54.890 --> 00:15:56.570 align:middle line:84% So we've seen that's not surjective. 00:15:56.570 --> 00:15:58.520 align:middle line:90% Of course, this map, if I take-- 00:15:58.520 --> 00:16:05.030 align:middle line:84% again, imagine this is 1, 2, 3 a, b, c, d. 00:16:05.030 --> 00:16:10.220 align:middle line:84% This map here, this function here that takes 1 to a, 2 00:16:10.220 --> 00:16:14.990 align:middle line:84% to b, 3, to c, this is actually injective but not surjective. 00:16:14.990 --> 00:16:24.820 align:middle line:90% 00:16:24.820 --> 00:16:28.770 align:middle line:84% And then of course, we could change this slightly. 00:16:28.770 --> 00:16:38.980 align:middle line:84% And 1 goes to a, 2 goes to b, and 3 goes to b. 00:16:38.980 --> 00:16:47.532 align:middle line:84% Then this is surjective but not injective. 00:16:47.532 --> 00:16:52.210 align:middle line:90% 00:16:52.210 --> 00:16:54.250 align:middle line:90% Now, the map that sends-- 00:16:54.250 --> 00:16:59.110 align:middle line:90% 00:16:59.110 --> 00:17:01.330 align:middle line:90% let's switch sides here-- 00:17:01.330 --> 00:17:06.730 align:middle line:90% a, b, and c, 1, 2, 3-- 00:17:06.730 --> 00:17:17.150 align:middle line:84% a to 1, b to 2, c to 3, this is a bijection, bijective. 00:17:17.150 --> 00:17:20.240 align:middle line:84% So if I say something's an injection or surjection 00:17:20.240 --> 00:17:23.510 align:middle line:84% or bijection, that just means it's a map that is injective, 00:17:23.510 --> 00:17:30.860 align:middle line:84% or, surjective or bijective respectively. 00:17:30.860 --> 00:17:54.320 align:middle line:90% 00:17:54.320 --> 00:18:04.620 align:middle line:90% And now, a definition-- 00:18:04.620 --> 00:18:05.870 align:middle line:90% so this is really-- 00:18:05.870 --> 00:18:07.370 align:middle line:84% there's not much to this definition, 00:18:07.370 --> 00:18:15.240 align:middle line:84% but just defining a couple of related functions 00:18:15.240 --> 00:18:18.490 align:middle line:84% that are related to a given function. 00:18:18.490 --> 00:18:27.540 align:middle line:84% If f goes from A to B, g goes from B to C, 00:18:27.540 --> 00:18:37.110 align:middle line:84% the composition g of f is the function which goes from A to C 00:18:37.110 --> 00:18:48.142 align:middle line:84% is defined by g of f of x equals g of f of x. 00:18:48.142 --> 00:18:52.214 align:middle line:90% 00:18:52.214 --> 00:18:57.130 align:middle line:90% And 2, if f is bijective-- 00:18:57.130 --> 00:19:00.250 align:middle line:90% 00:19:00.250 --> 00:19:02.770 align:middle line:84% so this is the composition of two functions. 00:19:02.770 --> 00:19:05.860 align:middle line:84% I didn't write the word composition, but g of f 00:19:05.860 --> 00:19:08.980 align:middle line:84% means the composition, or is referred to as a composition. 00:19:08.980 --> 00:19:15.850 align:middle line:84% If f is bijective, then we define the inverse function 00:19:15.850 --> 00:19:24.520 align:middle line:90% to f, B to A, by the following. 00:19:24.520 --> 00:19:27.740 align:middle line:90% 00:19:27.740 --> 00:19:35.990 align:middle line:84% If y is in B, then f inverse of y in A 00:19:35.990 --> 00:19:52.090 align:middle line:84% is the unique elements in A such that if I take this element, 00:19:52.090 --> 00:19:56.770 align:middle line:90% stick it into f, I get back y. 00:19:56.770 --> 00:20:00.610 align:middle line:84% So the inverse of a function only exists 00:20:00.610 --> 00:20:03.700 align:middle line:84% for bijective functions, or at least 00:20:03.700 --> 00:20:07.780 align:middle line:84% is only defined for bijective functions. 00:20:07.780 --> 00:20:08.660 align:middle line:90% Keep that in mind. 00:20:08.660 --> 00:20:13.690 align:middle line:84% Don't confuse that with the inverse image of sets. 00:20:13.690 --> 00:20:17.290 align:middle line:84% Although, those two notations look the same. 00:20:17.290 --> 00:20:22.245 align:middle line:84% You have an inverse f to the minus 1. 00:20:22.245 --> 00:20:26.530 align:middle line:84% f to the minus 1, if it's a function, 00:20:26.530 --> 00:20:28.460 align:middle line:90% that is the inverse function. 00:20:28.460 --> 00:20:31.900 align:middle line:84% However, if I'm taking f to the minus 1 of a set, 00:20:31.900 --> 00:20:35.770 align:middle line:84% that means the inverse image of that set as defined over there. 00:20:35.770 --> 00:20:39.890 align:middle line:90% 00:20:39.890 --> 00:20:45.650 align:middle line:84% So bijections, meaning bijective functions, 00:20:45.650 --> 00:20:51.380 align:middle line:84% will be what we mean when we say two sets can be paired off. 00:20:51.380 --> 00:20:54.050 align:middle line:90% It's what we mean when we-- 00:20:54.050 --> 00:20:59.960 align:middle line:84% or at least what Cantor's answer to that original question 00:20:59.960 --> 00:21:04.480 align:middle line:84% was, when did two sets have the same size. 00:21:04.480 --> 00:21:14.150 align:middle line:84% So this is the notion of cardinality that I alluded to. 00:21:14.150 --> 00:21:30.740 align:middle line:84% So we say two sets, A and B, have 00:21:30.740 --> 00:21:44.010 align:middle line:84% the same cardinality if there exists 00:21:44.010 --> 00:22:00.880 align:middle line:84% a bijection or a bijective function f from A to B. 00:22:00.880 --> 00:22:14.500 align:middle line:84% And so let me just make some notation here. so this is not 00:22:14.500 --> 00:22:19.420 align:middle line:84% really new objects I'm defining or operations. 00:22:19.420 --> 00:22:20.900 align:middle line:90% This is just some notation. 00:22:20.900 --> 00:22:24.850 align:middle line:84% So when two sets have the same cardinality, we write-- 00:22:24.850 --> 00:22:42.030 align:middle line:90% 00:22:42.030 --> 00:22:45.267 align:middle line:84% so this is just a shorthand way of writing that two functions 00:22:45.267 --> 00:22:46.350 align:middle line:90% have the same cardinality. 00:22:46.350 --> 00:22:49.770 align:middle line:84% You should not necessarily think this means taking 00:22:49.770 --> 00:22:51.630 align:middle line:90% the absolute value of a set. 00:22:51.630 --> 00:22:54.060 align:middle line:84% That doesn't mean anything, all right? 00:22:54.060 --> 00:22:56.280 align:middle line:84% This is just shorthand notation for saying two 00:22:56.280 --> 00:22:57.840 align:middle line:90% sets have the same cardinality. 00:22:57.840 --> 00:23:02.180 align:middle line:90% 00:23:02.180 --> 00:23:15.600 align:middle line:84% If A has the same cardinality as the set 1, 2, 3 up to n, 00:23:15.600 --> 00:23:20.060 align:middle line:90% we write A equals n. 00:23:20.060 --> 00:23:22.670 align:middle line:84% That's just shorthand for saying that a set has 00:23:22.670 --> 00:23:27.860 align:middle line:84% the same cardinality as the natural numbers up to n. 00:23:27.860 --> 00:23:38.380 align:middle line:90% 00:23:38.380 --> 00:23:50.230 align:middle line:84% So if there exists an injective function-- or I 00:23:50.230 --> 00:23:53.530 align:middle line:84% will often say either function or map. 00:23:53.530 --> 00:23:56.620 align:middle line:84% You should take those as synonymous. 00:23:56.620 --> 00:24:00.340 align:middle line:84% There exists an injective function f-- 00:24:00.340 --> 00:24:06.800 align:middle line:84% if there exists an injective function f from A to B, 00:24:06.800 --> 00:24:12.540 align:middle line:90% we write this thing. 00:24:12.540 --> 00:24:15.390 align:middle line:84% Again, do not read this as taking the absolute value 00:24:15.390 --> 00:24:17.940 align:middle line:84% of some set and that absolute value 00:24:17.940 --> 00:24:20.580 align:middle line:84% is less than or equal to the absolute value of the other set 00:24:20.580 --> 00:24:21.870 align:middle line:90% because that's meaningless. 00:24:21.870 --> 00:24:24.780 align:middle line:84% We haven't said what that means even. 00:24:24.780 --> 00:24:29.590 align:middle line:90% 00:24:29.590 --> 00:24:31.000 align:middle line:90% This is just shorthand notation. 00:24:31.000 --> 00:24:40.230 align:middle line:90% 00:24:40.230 --> 00:24:52.440 align:middle line:84% And if there exists an injection from A to B 00:24:52.440 --> 00:25:04.385 align:middle line:84% but they don't have the same cardinality, we write this. 00:25:04.385 --> 00:25:08.040 align:middle line:90% 00:25:08.040 --> 00:25:12.210 align:middle line:84% So even though this notation of these absolute value 00:25:12.210 --> 00:25:14.850 align:middle line:84% things being on the outside of the set 00:25:14.850 --> 00:25:19.770 align:middle line:84% makes you think absolute value or should be interpreted 00:25:19.770 --> 00:25:22.080 align:middle line:84% as absolute value, it's OK to think 00:25:22.080 --> 00:25:27.130 align:middle line:84% as a certain ordering being there, 00:25:27.130 --> 00:25:31.930 align:middle line:90% as A having smaller size than B. 00:25:31.930 --> 00:25:35.530 align:middle line:84% We write this because there being an injection from one 00:25:35.530 --> 00:25:41.080 align:middle line:84% set to the other means I can pair off elements of A 00:25:41.080 --> 00:25:44.470 align:middle line:84% with some elements of B. Maybe I don't 00:25:44.470 --> 00:25:47.170 align:middle line:84% get all of the elements of B. But I can pair off 00:25:47.170 --> 00:25:50.500 align:middle line:84% some of the elements of A with some of the elements of B. 00:25:50.500 --> 00:25:53.860 align:middle line:84% For example, that first map we wrote up there, 00:25:53.860 --> 00:25:59.360 align:middle line:84% that says that the set 1, 2, 3 in size 00:25:59.360 --> 00:26:02.060 align:middle line:84% is less than or equal to the size of a, b, c, 00:26:02.060 --> 00:26:04.820 align:middle line:84% d because we found an injection, an injective map 00:26:04.820 --> 00:26:07.625 align:middle line:84% from the first set to the second set. 00:26:07.625 --> 00:26:10.350 align:middle line:90% 00:26:10.350 --> 00:26:16.560 align:middle line:84% This third map would say that the size of the set a, b,c is 00:26:16.560 --> 00:26:21.210 align:middle line:84% equal to 3, written here, or shorthand written here. 00:26:21.210 --> 00:26:24.080 align:middle line:90% 00:26:24.080 --> 00:26:32.650 align:middle line:84% And in fact, that first map that we wrote up there, again, is-- 00:26:32.650 --> 00:26:37.720 align:middle line:84% so from first map, this says that the set of 1, 2, 3-- 00:26:37.720 --> 00:26:41.170 align:middle line:84% in our shorthand notation, absolute value 00:26:41.170 --> 00:26:45.370 align:middle line:84% is less than the size of a, b, c, d. 00:26:45.370 --> 00:26:49.150 align:middle line:84% So don't think of these as saying the absolute value. 00:26:49.150 --> 00:26:52.135 align:middle line:84% It's best to maybe think of this as saying the size of. 00:26:52.135 --> 00:26:55.070 align:middle line:90% 00:26:55.070 --> 00:27:01.040 align:middle line:84% All right, so if there exists an injective map or function 00:27:01.040 --> 00:27:03.920 align:middle line:84% from one set to another, then the size of A 00:27:03.920 --> 00:27:07.587 align:middle line:84% is less than or equal to the size of B. If the size of A 00:27:07.587 --> 00:27:09.170 align:middle line:84% is less than or equal to the size of B 00:27:09.170 --> 00:27:11.390 align:middle line:84% but the size of A and B are not the same, 00:27:11.390 --> 00:27:15.830 align:middle line:84% we write the size of A is less than the size of B. 00:27:15.830 --> 00:27:17.525 align:middle line:90% So best to think of those-- 00:27:17.525 --> 00:27:20.120 align:middle line:90% 00:27:20.120 --> 00:27:21.770 align:middle line:84% this absolute value looking thing 00:27:21.770 --> 00:27:25.700 align:middle line:84% as being shorthand for saying the words size of. 00:27:25.700 --> 00:27:32.330 align:middle line:90% 00:27:32.330 --> 00:27:34.040 align:middle line:84% Now, I'm not going to prove this. 00:27:34.040 --> 00:27:39.820 align:middle line:84% It goes a little bit beyond the scope of this class. 00:27:39.820 --> 00:27:45.550 align:middle line:84% But let me just say that this ordering, this inequality, 00:27:45.550 --> 00:27:50.010 align:middle line:84% these symbols that we're writing does 00:27:50.010 --> 00:27:53.800 align:middle line:84% bear some sort of semblance to the ordering of real numbers 00:27:53.800 --> 00:27:57.060 align:middle line:84% in that if I have two real numbers, one is less than 00:27:57.060 --> 00:27:59.380 align:middle line:84% or equal to the other and vice versa. 00:27:59.380 --> 00:28:02.220 align:middle line:84% So A is less than or equal to B and B is less than 00:28:02.220 --> 00:28:04.680 align:middle line:84% or equal to A, then A must equal B. 00:28:04.680 --> 00:28:08.460 align:middle line:84% That is, in fact, true also for this elementary notion 00:28:08.460 --> 00:28:10.200 align:middle line:90% of size of sets. 00:28:10.200 --> 00:28:13.990 align:middle line:90% And this is-- 00:28:13.990 --> 00:28:21.070 align:middle line:84% I mean, it shouldn't surprise you for finite sets 00:28:21.070 --> 00:28:21.820 align:middle line:90% necessarily. 00:28:21.820 --> 00:28:32.750 align:middle line:84% If I have a pairing of A, if A is no bigger than n and n is 00:28:32.750 --> 00:28:37.100 align:middle line:84% no bigger than the size of A, then a should have n elements. 00:28:37.100 --> 00:28:39.590 align:middle line:84% But it takes a little bit more to prove 00:28:39.590 --> 00:28:42.650 align:middle line:90% for sets which are not finite. 00:28:42.650 --> 00:28:45.450 align:middle line:90% So I forgot to write this down. 00:28:45.450 --> 00:28:49.970 align:middle line:84% We also say that if the size of A 00:28:49.970 --> 00:28:53.480 align:middle line:84% is equal to the size of this finite set 1 through n, 00:28:53.480 --> 00:28:58.235 align:middle line:90% we say A is finite. 00:28:58.235 --> 00:29:02.450 align:middle line:90% 00:29:02.450 --> 00:29:04.040 align:middle line:84% So this theorem that I'm stating here 00:29:04.040 --> 00:29:18.860 align:middle line:84% is the Cantor Schroeder Bernstein theorem, 00:29:18.860 --> 00:29:24.410 align:middle line:84% which states that if the size of A 00:29:24.410 --> 00:29:30.200 align:middle line:84% is less than or equal to the size of B and the size of B 00:29:30.200 --> 00:29:37.885 align:middle line:84% is less than or equal to the size of A, then the size of A 00:29:37.885 --> 00:29:43.060 align:middle line:84% equals the size of B. So again, if you're thinking of these 00:29:43.060 --> 00:29:46.960 align:middle line:84% in the context of real numbers, one being less than 00:29:46.960 --> 00:29:49.330 align:middle line:84% or equal to the other and vice versa, 00:29:49.330 --> 00:29:51.820 align:middle line:84% of course, that implies that those two real numbers are 00:29:51.820 --> 00:29:52.960 align:middle line:90% equal to each other. 00:29:52.960 --> 00:29:55.180 align:middle line:84% But we're not talking about real numbers. 00:29:55.180 --> 00:29:58.150 align:middle line:84% Again, this is just shorthand notation 00:29:58.150 --> 00:30:03.130 align:middle line:84% for saying there exists a bijective map from A to B. This 00:30:03.130 --> 00:30:06.800 align:middle line:84% means there exists a bijective map from A to the set. 00:30:06.800 --> 00:30:11.200 align:middle line:84% This means there exists an injective map from A into this, 00:30:11.200 --> 00:30:14.380 align:middle line:90% from a to this set. 00:30:14.380 --> 00:30:17.080 align:middle line:84% So this is not a statement about real numbers. 00:30:17.080 --> 00:30:20.620 align:middle line:84% This is a statement about cardinality, all right? 00:30:20.620 --> 00:30:26.275 align:middle line:90% 00:30:26.275 --> 00:30:26.775 align:middle line:90% OK. 00:30:26.775 --> 00:30:29.900 align:middle line:90% 00:30:29.900 --> 00:30:40.930 align:middle line:84% So finite sets are sets that you can count if you had n fingers. 00:30:40.930 --> 00:30:50.150 align:middle line:84% Now, we would like to be able to define what it means 00:30:50.150 --> 00:30:52.370 align:middle line:90% to be able to count a set. 00:30:52.370 --> 00:30:54.530 align:middle line:84% So what do I mean by count a set? 00:30:54.530 --> 00:30:57.890 align:middle line:84% Meaning, if I had infinite time, I 00:30:57.890 --> 00:31:01.130 align:middle line:84% could go through the set counting them-- 00:31:01.130 --> 00:31:08.180 align:middle line:84% 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and so on. 00:31:08.180 --> 00:31:09.770 align:middle line:90% I don't need to go anymore. 00:31:09.770 --> 00:31:12.560 align:middle line:90% 00:31:12.560 --> 00:31:16.320 align:middle line:84% But what does that process of counting mean? 00:31:16.320 --> 00:31:18.890 align:middle line:84% That means, for each element of the set 00:31:18.890 --> 00:31:21.800 align:middle line:84% that I'm trying to count, I can pair it off 00:31:21.800 --> 00:31:24.080 align:middle line:90% with a natural number-- 00:31:24.080 --> 00:31:26.100 align:middle line:90% 1, 2, 3, and so on. 00:31:26.100 --> 00:31:31.740 align:middle line:84% So this is how we define countable sets. 00:31:31.740 --> 00:31:37.650 align:middle line:84% So if A has the same size as the natural numbers, 00:31:37.650 --> 00:31:44.940 align:middle line:84% meaning there exists a bijection from A to the natural numbers, 00:31:44.940 --> 00:31:55.380 align:middle line:84% then we say A is countably infinite. 00:31:55.380 --> 00:32:02.070 align:middle line:90% 00:32:02.070 --> 00:32:10.870 align:middle line:90% If A is finite, or countably-- 00:32:10.870 --> 00:32:14.890 align:middle line:84% I also use a lot of shorthand, but it's not clever shorthand. 00:32:14.890 --> 00:32:17.890 align:middle line:84% So you should just be able to sound what-- 00:32:17.890 --> 00:32:20.350 align:middle line:84% if you just sound that out, you'll get what word I mean, 00:32:20.350 --> 00:32:24.480 align:middle line:90% countably, countably infinite. 00:32:24.480 --> 00:32:30.450 align:middle line:90% 00:32:30.450 --> 00:32:38.080 align:middle line:90% We say A is countable. 00:32:38.080 --> 00:32:41.500 align:middle line:84% So countably infinite means I need all the natural numbers 00:32:41.500 --> 00:32:45.640 align:middle line:84% to be able to count off the elements of the set A. 00:32:45.640 --> 00:32:48.850 align:middle line:84% Countable means maybe I stop after some point 00:32:48.850 --> 00:32:50.890 align:middle line:90% and I've counted all them. 00:32:50.890 --> 00:32:55.030 align:middle line:84% So A is finite or it's countably infinite. 00:32:55.030 --> 00:32:57.520 align:middle line:84% Otherwise, if a set is not countable, 00:32:57.520 --> 00:32:58.840 align:middle line:90% we say it's uncountable. 00:32:58.840 --> 00:33:18.220 align:middle line:90% 00:33:18.220 --> 00:33:27.850 align:middle line:84% OK, so let's take a look at a couple of countable sets 00:33:27.850 --> 00:33:32.440 align:middle line:84% that maybe don't come off as being countable. 00:33:32.440 --> 00:33:38.032 align:middle line:90% 00:33:38.032 --> 00:33:44.880 align:middle line:84% The set of-- actually, that's a homework problem. 00:33:44.880 --> 00:33:52.200 align:middle line:84% The set of even integers is countable. 00:33:52.200 --> 00:33:54.840 align:middle line:90% 00:33:54.840 --> 00:34:00.060 align:middle line:84% The set of odd-- so I said integers a minute ago. 00:34:00.060 --> 00:34:02.280 align:middle line:84% Maybe I should just say natural numbers. 00:34:02.280 --> 00:34:11.500 align:middle line:84% The set of odd natural numbers is countable. 00:34:11.500 --> 00:34:16.630 align:middle line:84% So in fact-- so this is just an aside. 00:34:16.630 --> 00:34:24.090 align:middle line:90% 00:34:24.090 --> 00:34:27.440 align:middle line:84% So these are two disjoint subsets 00:34:27.440 --> 00:34:30.290 align:middle line:84% that make up the natural numbers, even 00:34:30.290 --> 00:34:31.940 align:middle line:90% and odd natural numbers. 00:34:31.940 --> 00:34:36.739 align:middle line:84% And they both have the same size as the set that they make up. 00:34:36.739 --> 00:34:42.610 align:middle line:84% So it's almost like saying the cardinality 00:34:42.610 --> 00:34:44.949 align:middle line:84% of the natural numbers is twice the cardinality 00:34:44.949 --> 00:34:47.679 align:middle line:84% of the natural numbers, since you would think 00:34:47.679 --> 00:34:51.130 align:middle line:84% that the size of N should be the size of this set 00:34:51.130 --> 00:34:53.597 align:middle line:84% plus the size of this set, since they are disjoint 00:34:53.597 --> 00:34:54.639 align:middle line:90% and they make up the set. 00:34:54.639 --> 00:34:57.640 align:middle line:90% 00:34:57.640 --> 00:35:02.760 align:middle line:84% But that's not how cardinality works. 00:35:02.760 --> 00:35:07.560 align:middle line:84% You don't just add cardinalities to get the cardinalities. 00:35:07.560 --> 00:35:17.230 align:middle line:84% So this is a subtle, interesting thing about cardinalities 00:35:17.230 --> 00:35:19.600 align:middle line:84% So Richard Feynman, who won the Nobel Prize 00:35:19.600 --> 00:35:26.620 align:middle line:84% in the '60s for his work on QED, described this 00:35:26.620 --> 00:35:44.310 align:middle line:84% as saying there are twice as many numbers as numbers. 00:35:44.310 --> 00:35:48.990 align:middle line:90% 00:35:48.990 --> 00:35:57.610 align:middle line:90% OK, so let's prove this. 00:35:57.610 --> 00:35:59.650 align:middle line:90% So what does this mean? 00:35:59.650 --> 00:36:03.750 align:middle line:84% This means we have to be able to find a bijective function 00:36:03.750 --> 00:36:10.680 align:middle line:84% from this set to this set or from this set to this set, one 00:36:10.680 --> 00:36:11.310 align:middle line:90% or the other. 00:36:11.310 --> 00:36:14.240 align:middle line:90% 00:36:14.240 --> 00:36:22.660 align:middle line:90% And so in fact, maybe I should-- 00:36:22.660 --> 00:36:25.360 align:middle line:84% well, I'll say something about that in a minute. 00:36:25.360 --> 00:36:28.620 align:middle line:90% 00:36:28.620 --> 00:36:30.300 align:middle line:90% Well, let me pause. 00:36:30.300 --> 00:36:34.680 align:middle line:84% Let me pause this just for a minute. 00:36:34.680 --> 00:36:38.520 align:middle line:84% And let me make a few comments about cardinality, 00:36:38.520 --> 00:36:40.350 align:middle line:90% which maybe I should have. 00:36:40.350 --> 00:36:43.540 align:middle line:90% 00:36:43.540 --> 00:36:53.070 align:middle line:84% So this you can think of as a little theorem. 00:36:53.070 --> 00:37:00.460 align:middle line:84% If I have two sets, A has the same size as B, 00:37:00.460 --> 00:37:04.570 align:middle line:84% then B has the same size as A. So remember, I mean, 00:37:04.570 --> 00:37:08.470 align:middle line:84% what I just said in English is not exactly 00:37:08.470 --> 00:37:09.790 align:middle line:90% what those symbols mean. 00:37:09.790 --> 00:37:14.290 align:middle line:84% Remember, this means there exists a bijection from A to B 00:37:14.290 --> 00:37:17.650 align:middle line:84% This means there exists a bijection from B to A. 00:37:17.650 --> 00:37:23.210 align:middle line:84% So what is the proof of this statement? 00:37:23.210 --> 00:37:24.950 align:middle line:84% So let's start off with the hypothesis. 00:37:24.950 --> 00:37:27.470 align:middle line:90% 00:37:27.470 --> 00:37:42.130 align:middle line:84% Suppose then there exists a bijective map, 00:37:42.130 --> 00:37:57.210 align:middle line:84% bijective function f from A to B. 00:37:57.210 --> 00:38:00.840 align:middle line:84% Now, if I have a bijection from A to B, then 00:38:00.840 --> 00:38:03.118 align:middle line:84% what would be a bijection going from B to A? 00:38:03.118 --> 00:38:04.410 align:middle line:90% That would just be the inverse. 00:38:04.410 --> 00:38:09.340 align:middle line:90% 00:38:09.340 --> 00:38:11.970 align:middle line:84% So this is not the inverse image of sets, 00:38:11.970 --> 00:38:13.870 align:middle line:84% like we described, but the actual inverse, 00:38:13.870 --> 00:38:15.525 align:middle line:90% which we defined over there. 00:38:15.525 --> 00:38:22.980 align:middle line:90% 00:38:22.980 --> 00:38:24.330 align:middle line:90% Is a bijection. 00:38:24.330 --> 00:38:37.440 align:middle line:90% 00:38:37.440 --> 00:39:01.440 align:middle line:90% So B has the same size as A. 00:39:01.440 --> 00:39:11.640 align:middle line:84% One other statement-- so if A and B have the same size and B 00:39:11.640 --> 00:39:15.070 align:middle line:90% and C have the same size-- 00:39:15.070 --> 00:39:16.500 align:middle line:90% so again, A, B, and C are sets. 00:39:16.500 --> 00:39:19.050 align:middle line:84% I should have written that at the beginning. 00:39:19.050 --> 00:39:21.015 align:middle line:84% But from this context, you should understand 00:39:21.015 --> 00:39:23.040 align:middle line:90% that A, B, and C are sets-- 00:39:23.040 --> 00:39:27.885 align:middle line:90% then A and C have the same size. 00:39:27.885 --> 00:39:38.450 align:middle line:90% 00:39:38.450 --> 00:39:40.200 align:middle line:90% So let's do a proof of that. 00:39:40.200 --> 00:39:43.950 align:middle line:84% So let's start with the hypothesis, 00:39:43.950 --> 00:39:57.770 align:middle line:84% meaning A has the same size as B and B has the same size as C. 00:39:57.770 --> 00:39:59.900 align:middle line:84% So what did these two statements mean 00:39:59.900 --> 00:40:01.400 align:middle line:90% in terms of the definition? 00:40:01.400 --> 00:40:04.460 align:middle line:84% That means there exists a bijection from A to B 00:40:04.460 --> 00:40:18.310 align:middle line:84% and a bijection from B to C. And this statement-- so 00:40:18.310 --> 00:40:19.220 align:middle line:90% let me finish this. 00:40:19.220 --> 00:40:20.970 align:middle line:84% And then I'll say what I was going to say. 00:40:20.970 --> 00:40:33.340 align:middle line:84% Then there exists bijections f from A to B and g from B to C. 00:40:33.340 --> 00:40:39.070 align:middle line:84% So perhaps I should have said a few more words 00:40:39.070 --> 00:40:41.860 align:middle line:90% about why this is true. 00:40:41.860 --> 00:40:45.910 align:middle line:84% But I'm going to leave this as an exercise 00:40:45.910 --> 00:40:49.150 align:middle line:84% just to pause the lecture and do it yourself. 00:40:49.150 --> 00:40:54.190 align:middle line:90% And what I'll write shortly-- 00:40:54.190 --> 00:40:55.900 align:middle line:84% for this case, I'll actually prove 00:40:55.900 --> 00:40:58.960 align:middle line:84% that the thing I'm going to define is bijection. 00:40:58.960 --> 00:41:00.200 align:middle line:90% That should help you. 00:41:00.200 --> 00:41:06.070 align:middle line:84% So let me draw over here off to the side. 00:41:06.070 --> 00:41:12.917 align:middle line:84% We have, again, 1, 2, 3, a, b, c. 00:41:12.917 --> 00:41:16.740 align:middle line:90% 00:41:16.740 --> 00:41:18.630 align:middle line:90% So think of this as f. 00:41:18.630 --> 00:41:22.080 align:middle line:84% This is my set A. This is my set B. 00:41:22.080 --> 00:41:26.850 align:middle line:84% And then I have alpha, beta, gamma. 00:41:26.850 --> 00:41:30.810 align:middle line:84% And a gets mapped to alpha, b, gets mapped to beta, 00:41:30.810 --> 00:41:33.160 align:middle line:90% c gets mapped to gamma. 00:41:33.160 --> 00:41:39.060 align:middle line:84% So what would be the map going from A down to the set C? 00:41:39.060 --> 00:41:42.060 align:middle line:90% Well, perhaps the composition. 00:41:42.060 --> 00:41:45.210 align:middle line:84% 1 gets mapped to a, which gets mapped to alpha. 00:41:45.210 --> 00:41:46.890 align:middle line:90% So 1 gets mapped to alpha. 00:41:46.890 --> 00:41:48.450 align:middle line:90% 2 gets mapped to beta. 00:41:48.450 --> 00:41:50.310 align:middle line:90% 3 gets mapped to gamma. 00:41:50.310 --> 00:41:53.130 align:middle line:84% How do I build this function out of the things that I know, 00:41:53.130 --> 00:41:56.010 align:middle line:90% namely this f and g? 00:41:56.010 --> 00:41:58.590 align:middle line:84% Well, this function going from A to C 00:41:58.590 --> 00:42:02.105 align:middle line:84% is just the composition of these two functions. 00:42:02.105 --> 00:42:03.230 align:middle line:90% So this is off to the side. 00:42:03.230 --> 00:42:04.563 align:middle line:90% This is not a part of the proof. 00:42:04.563 --> 00:42:08.170 align:middle line:90% 00:42:08.170 --> 00:42:27.060 align:middle line:84% Let h go from A to C be the function g of f of x. 00:42:27.060 --> 00:42:30.311 align:middle line:90% 00:42:30.311 --> 00:42:35.650 align:middle line:84% So I claim that this function is a bijection. 00:42:35.650 --> 00:42:51.960 align:middle line:84% So we want to prove that it's one-to-one and onto, all right? 00:42:51.960 --> 00:42:55.095 align:middle line:90% So let's do one-to-one first. 00:42:55.095 --> 00:43:01.810 align:middle line:90% 00:43:01.810 --> 00:43:03.910 align:middle line:84% So we this is the part where I'm going 00:43:03.910 --> 00:43:06.650 align:middle line:84% to put this in parentheses, meaning what we're doing now. 00:43:06.650 --> 00:43:11.260 align:middle line:84% So we're going to prove that h is one-to-one. 00:43:11.260 --> 00:43:16.240 align:middle line:84% So that means we have to verify the definition that I've 00:43:16.240 --> 00:43:17.650 align:middle line:90% erased. 00:43:17.650 --> 00:43:19.610 align:middle line:90% So remember the definition. 00:43:19.610 --> 00:43:25.290 align:middle line:90% So let's write it all out. 00:43:25.290 --> 00:43:29.860 align:middle line:90% 00:43:29.860 --> 00:43:36.940 align:middle line:90% We first show h is one-to-one. 00:43:36.940 --> 00:43:41.080 align:middle line:84% And what this means, according to the definition that I'm now 00:43:41.080 --> 00:43:57.190 align:middle line:84% writing over, if h of x1 equals h of x2, then x1 equals x2. 00:43:57.190 --> 00:43:58.570 align:middle line:84% So this is what we want to prove. 00:43:58.570 --> 00:44:02.230 align:middle line:84% This is the definition of h being one-to-one. 00:44:02.230 --> 00:44:06.770 align:middle line:90% 00:44:06.770 --> 00:44:07.685 align:middle line:90% So let's start. 00:44:07.685 --> 00:44:10.530 align:middle line:90% 00:44:10.530 --> 00:44:16.320 align:middle line:84% If h of x1 equals h of x2, then, in terms 00:44:16.320 --> 00:44:19.500 align:middle line:84% of how we've defined h, which is g 00:44:19.500 --> 00:44:28.400 align:middle line:84% of f of x, so the composition, then this means g of f of x1 00:44:28.400 --> 00:44:30.065 align:middle line:90% equals g of f of x2. 00:44:30.065 --> 00:44:32.860 align:middle line:90% 00:44:32.860 --> 00:44:35.400 align:middle line:90% That's just what h is. 00:44:35.400 --> 00:44:40.780 align:middle line:90% Now, this statement here-- 00:44:40.780 --> 00:44:44.280 align:middle line:90% now g we know is a bijection. 00:44:44.280 --> 00:44:46.840 align:middle line:90% We know g is one-to-one. 00:44:46.840 --> 00:44:52.180 align:middle line:84% And since g of something equals g of something else and g 00:44:52.180 --> 00:45:00.320 align:middle line:84% is one-to-one, this implies that f of x1 equals f of x2. 00:45:00.320 --> 00:45:04.740 align:middle line:90% This is since g is one-to-one. 00:45:04.740 --> 00:45:08.800 align:middle line:90% 00:45:08.800 --> 00:45:11.440 align:middle line:84% So starting with this and the definition of h 00:45:11.440 --> 00:45:15.810 align:middle line:84% being the composition, we conclude that f of x1 00:45:15.810 --> 00:45:20.070 align:middle line:84% has to equal f of x2 because g is one-to-one. 00:45:20.070 --> 00:45:23.280 align:middle line:84% Now, since f is also one-to-one-- 00:45:23.280 --> 00:45:26.220 align:middle line:84% we are assuming that f and g are bijections. 00:45:26.220 --> 00:45:33.240 align:middle line:84% Since f is one-to-one, this implies x1 equals x2, 00:45:33.240 --> 00:45:36.420 align:middle line:90% since f is one-to-one. 00:45:36.420 --> 00:45:41.880 align:middle line:90% 00:45:41.880 --> 00:45:44.400 align:middle line:84% And this is what we wanted to prove. 00:45:44.400 --> 00:45:48.330 align:middle line:84% We wanted to start with assuming h of x1 equals h of x2 00:45:48.330 --> 00:45:51.300 align:middle line:84% and prove x1 equals x2, which is what we've done. 00:45:51.300 --> 00:45:57.900 align:middle line:84% So thus, we've proven that h is one-to-one. 00:45:57.900 --> 00:46:00.885 align:middle line:84% Now, we have to show that it's surjective, that it's onto. 00:46:00.885 --> 00:46:12.580 align:middle line:90% 00:46:12.580 --> 00:46:18.160 align:middle line:84% h of A equals C. And again, I'll write out what this means. 00:46:18.160 --> 00:46:21.800 align:middle line:90% This means for all y in C-- 00:46:21.800 --> 00:46:24.850 align:middle line:84% all right, let me call it [? z-- ?] 00:46:24.850 --> 00:46:32.680 align:middle line:84% there exists an x in A such that h of x equals z. 00:46:32.680 --> 00:46:37.850 align:middle line:90% 00:46:37.850 --> 00:46:40.610 align:middle line:84% Now, we used the fact that f and g 00:46:40.610 --> 00:46:46.230 align:middle line:84% were injective to conclude that the composition is injective. 00:46:46.230 --> 00:46:47.887 align:middle line:84% So it stands to reason that we're 00:46:47.887 --> 00:46:49.970 align:middle line:84% going to use the fact that they're both surjective 00:46:49.970 --> 00:46:51.980 align:middle line:90% to prove that h is surjective. 00:46:51.980 --> 00:46:55.020 align:middle line:90% 00:46:55.020 --> 00:46:57.870 align:middle line:84% So we need to prove that for all z in C 00:46:57.870 --> 00:47:00.540 align:middle line:84% there exists an x in A as such that h of x equals z. 00:47:00.540 --> 00:47:05.310 align:middle line:84% So let z be in C. Now, we need to find some 00:47:05.310 --> 00:47:07.650 align:middle line:90% x in A that gets mapped to z. 00:47:07.650 --> 00:47:12.060 align:middle line:84% We'll use that g and f are both surjective. 00:47:12.060 --> 00:47:16.530 align:middle line:90% 00:47:16.530 --> 00:47:19.245 align:middle line:84% And what's the picture that goes along with this? 00:47:19.245 --> 00:47:26.100 align:middle line:90% 00:47:26.100 --> 00:47:33.110 align:middle line:84% Here's the sets C, B, and A. We have some element of z 00:47:33.110 --> 00:47:39.330 align:middle line:84% And we know that since g is surjective, 00:47:39.330 --> 00:47:47.100 align:middle line:84% there's some element in B that gets mapped to z by G. 00:47:47.100 --> 00:47:50.940 align:middle line:84% But now, since f going from A to B is surjective, 00:47:50.940 --> 00:47:54.468 align:middle line:84% there exists some x which maps to y. 00:47:54.468 --> 00:47:57.112 align:middle line:90% 00:47:57.112 --> 00:47:58.570 align:middle line:84% And then that's the whole argument. 00:47:58.570 --> 00:48:00.880 align:middle line:90% That's it. 00:48:00.880 --> 00:48:01.990 align:middle line:90% I drew this picture. 00:48:01.990 --> 00:48:05.170 align:middle line:84% But now, I just need to turn it into English 00:48:05.170 --> 00:48:08.710 align:middle line:84% using the properties and assumptions that I have. 00:48:08.710 --> 00:48:13.980 align:middle line:90% 00:48:13.980 --> 00:48:25.380 align:middle line:84% Since g is surjective, there exist a y in B such that g of y 00:48:25.380 --> 00:48:26.220 align:middle line:90% equals z. 00:48:26.220 --> 00:48:37.660 align:middle line:90% 00:48:37.660 --> 00:48:50.900 align:middle line:84% Since f is surjective, there exists an x in A so that f of x 00:48:50.900 --> 00:48:51.680 align:middle line:90% equals y. 00:48:51.680 --> 00:48:54.780 align:middle line:90% 00:48:54.780 --> 00:48:59.100 align:middle line:84% But then if I look at h of x, this 00:48:59.100 --> 00:49:04.960 align:middle line:84% is equal to g of f of x, which is equal to g of y, 00:49:04.960 --> 00:49:09.540 align:middle line:84% which is equal to z by how we've defined y. 00:49:09.540 --> 00:49:13.300 align:middle line:84% Remember, g of y equals z, how we found y and how we found x. 00:49:13.300 --> 00:49:16.380 align:middle line:90% Remember, f of x equals this y. 00:49:16.380 --> 00:49:23.070 align:middle line:84% And therefore, this map h is onto. 00:49:23.070 --> 00:49:29.470 align:middle line:84% And therefore, h is a bijection, proving this theorem. 00:49:29.470 --> 00:49:32.438 align:middle line:90% So that should help you. 00:49:32.438 --> 00:49:39.150 align:middle line:84% I'll give you many exercises to prove 00:49:39.150 --> 00:49:42.630 align:middle line:84% that the inverse of a bijection is a bijection. 00:49:42.630 --> 00:49:49.910 align:middle line:90% 00:49:49.910 --> 00:49:57.600 align:middle line:84% So back to what we were doing to begin with. 00:49:57.600 --> 00:50:01.910 align:middle line:84% We want to show the set of even natural numbers has 00:50:01.910 --> 00:50:04.830 align:middle line:84% the same size as the natural numbers, 00:50:04.830 --> 00:50:07.830 align:middle line:84% has the same cardinality, as the natural numbers and the same 00:50:07.830 --> 00:50:08.550 align:middle line:90% for the odd ones. 00:50:08.550 --> 00:50:10.590 align:middle line:84% So I'm just going to do the odd ones. 00:50:10.590 --> 00:50:13.530 align:middle line:84% Again, this will be a small exercise for you 00:50:13.530 --> 00:50:19.640 align:middle line:84% to do to prove number 2, the statement for the odd ones. 00:50:19.640 --> 00:50:21.380 align:middle line:84% So I'm going to do the even ones. 00:50:21.380 --> 00:50:24.470 align:middle line:90% And you can do the odd ones. 00:50:24.470 --> 00:50:28.720 align:middle line:84% You should, if you plan on studying more math, 00:50:28.720 --> 00:50:35.050 align:middle line:84% get used to the instructor, professor, research paper 00:50:35.050 --> 00:50:40.580 align:middle line:84% writer, textbook author giving out little exercises 00:50:40.580 --> 00:50:42.290 align:middle line:84% to make sure that you're following along 00:50:42.290 --> 00:50:47.980 align:middle line:84% and that you can do some minor task at some point 00:50:47.980 --> 00:50:49.870 align:middle line:90% during the discussion. 00:50:49.870 --> 00:50:52.790 align:middle line:90% 00:50:52.790 --> 00:50:56.840 align:middle line:90% So we're going to find-- 00:50:56.840 --> 00:51:14.260 align:middle line:84% so we want to show that the natural numbers has 00:51:14.260 --> 00:51:16.990 align:middle line:84% the same size as the even natural numbers, which 00:51:16.990 --> 00:51:19.600 align:middle line:84% is the same as this statement by that first theorem I wrote up 00:51:19.600 --> 00:51:20.100 align:middle line:90% there. 00:51:20.100 --> 00:51:24.510 align:middle line:90% 00:51:24.510 --> 00:51:27.360 align:middle line:84% Now, that means we need to find a bijection 00:51:27.360 --> 00:51:35.140 align:middle line:84% from the natural numbers to the set of even natural numbers. 00:51:35.140 --> 00:51:38.820 align:middle line:90% This should be not too bad. 00:51:38.820 --> 00:51:43.315 align:middle line:84% I mean, so again, this is off to the side. 00:51:43.315 --> 00:51:44.565 align:middle line:90% This is not part of the proof. 00:51:44.565 --> 00:52:02.570 align:middle line:90% 00:52:02.570 --> 00:52:07.610 align:middle line:84% What would be the map going from these guys to these guys? 00:52:07.610 --> 00:52:10.790 align:middle line:84% Well, I mean, there's several you could choose. 00:52:10.790 --> 00:52:13.560 align:middle line:84% But maybe the simplest is 1 gets mapped to 2, 00:52:13.560 --> 00:52:16.670 align:middle line:84% 2 gets mapped to 4, 3 gets mapped to 6, 00:52:16.670 --> 00:52:22.850 align:middle line:84% 4 gets mapped to 8, 5 to 10, 6 to 12, and so on, and so on. 00:52:22.850 --> 00:52:25.730 align:middle line:90% Now, what is that map? 00:52:25.730 --> 00:52:29.570 align:middle line:84% And now, I'll continue the proof. 00:52:29.570 --> 00:52:35.120 align:middle line:90% 00:52:35.120 --> 00:52:48.840 align:middle line:84% Let f be the function into the even natural numbers defined 00:52:48.840 --> 00:52:54.870 align:middle line:90% by f of n equals 2 times n. 00:52:54.870 --> 00:52:59.110 align:middle line:90% And so n is a natural number. 00:52:59.110 --> 00:53:03.150 align:middle line:84% So this is just formally writing the function that takes 1 to 2, 00:53:03.150 --> 00:53:08.800 align:middle line:84% 2 to 4, 3 to 6, 4, to 8, and so on, and so on. 00:53:08.800 --> 00:53:12.760 align:middle line:90% I claim f is a bijection. 00:53:12.760 --> 00:53:16.345 align:middle line:84% So I have to show that f is one-to-one and onto. 00:53:16.345 --> 00:53:23.250 align:middle line:90% 00:53:23.250 --> 00:53:28.980 align:middle line:84% So we'll first show f is one-to-one. 00:53:28.980 --> 00:53:31.930 align:middle line:90% 00:53:31.930 --> 00:53:36.480 align:middle line:84% So that means that I have to assume f of n1 equals f of n2 00:53:36.480 --> 00:53:39.510 align:middle line:90% and conclude that n1 equals n2. 00:53:39.510 --> 00:53:42.390 align:middle line:84% So let me actually write out what this means again for you 00:53:42.390 --> 00:53:44.460 align:middle line:90% to say that f is one-to-one. 00:53:44.460 --> 00:53:46.140 align:middle line:90% Is one to one-- i.e. 00:53:46.140 --> 00:53:55.860 align:middle line:84% if f of n1 equals f of n2, then n1 equals n2. 00:53:55.860 --> 00:53:58.560 align:middle line:84% But this is easy to verify for this function 00:53:58.560 --> 00:54:07.430 align:middle line:84% that we've written down because if f of n1 equals f of n2-- so 00:54:07.430 --> 00:54:10.430 align:middle line:84% remember, this is what we want to show, all right? 00:54:10.430 --> 00:54:15.410 align:middle line:84% So to show it, I start with my assumption. 00:54:15.410 --> 00:54:18.710 align:middle line:84% And I need to conclude n1 equals n2. 00:54:18.710 --> 00:54:24.560 align:middle line:84% So suppose f of n1 equals f of n2, 00:54:24.560 --> 00:54:26.600 align:middle line:90% my assumption, my hypothesis. 00:54:26.600 --> 00:54:28.220 align:middle line:84% And I need to conclude n1 equals n2. 00:54:28.220 --> 00:54:30.880 align:middle line:90% 00:54:30.880 --> 00:54:33.805 align:middle line:84% Then this implies, by the definition of f, 00:54:33.805 --> 00:54:37.420 align:middle line:84% 2 times n2 equals 2 times n2, which, 00:54:37.420 --> 00:54:41.500 align:middle line:84% by algebra of just eliminating the 2's, n1 equals n2, which 00:54:41.500 --> 00:54:44.650 align:middle line:90% is the conclusion that I want. 00:54:44.650 --> 00:54:49.060 align:middle line:84% So I've proven the statement that if f of n1 equals f of n2, 00:54:49.060 --> 00:54:50.470 align:middle line:90% then n1 equals n2. 00:54:50.470 --> 00:54:53.510 align:middle line:90% Therefore, f is one-to-one. 00:54:53.510 --> 00:54:59.320 align:middle line:90% So thus, f is one-to-one. 00:54:59.320 --> 00:55:11.620 align:middle line:84% So now, we want to show f is onto, surjective. 00:55:11.620 --> 00:55:14.500 align:middle line:90% I'll write this out again. i.e. 00:55:14.500 --> 00:55:28.190 align:middle line:84% For all elements that n are an even number, 00:55:28.190 --> 00:55:33.800 align:middle line:84% there exists an n such that f of n equals m. 00:55:33.800 --> 00:55:40.710 align:middle line:90% 00:55:40.710 --> 00:55:47.900 align:middle line:90% Now, let m be an even integer. 00:55:47.900 --> 00:55:52.500 align:middle line:84% And so not to confuse us, let me write 2 times k. 00:55:52.500 --> 00:55:54.230 align:middle line:90% This is the same set. 00:55:54.230 --> 00:55:57.290 align:middle line:84% I'm just using a different dummy variable instead of n 00:55:57.290 --> 00:55:59.705 align:middle line:84% in my description of the even natural numbers. 00:55:59.705 --> 00:56:05.790 align:middle line:90% 00:56:05.790 --> 00:56:07.410 align:middle line:90% And let's do that here as well. 00:56:07.410 --> 00:56:15.470 align:middle line:90% 00:56:15.470 --> 00:56:17.030 align:middle line:84% Again, this is not changing anything. 00:56:17.030 --> 00:56:18.530 align:middle line:84% This is just changing the letter I'm 00:56:18.530 --> 00:56:24.820 align:middle line:84% using to describe the set, which is inconsequential. 00:56:24.820 --> 00:56:29.530 align:middle line:84% But I don't want you to get the false impression that somehow 00:56:29.530 --> 00:56:30.955 align:middle line:90% I'm not doing anything at all. 00:56:30.955 --> 00:56:34.250 align:middle line:90% 00:56:34.250 --> 00:56:39.770 align:middle line:84% OK, so suppose I have an even integer, then there exists-- 00:56:39.770 --> 00:56:42.290 align:middle line:84% simply by the definition of this set, 00:56:42.290 --> 00:56:50.880 align:middle line:84% there exists an n natural number so that m equals 2 times n. 00:56:50.880 --> 00:56:53.710 align:middle line:90% 00:56:53.710 --> 00:56:59.380 align:middle line:84% Then f of this natural number, which is 2 times n, equals m. 00:56:59.380 --> 00:57:02.875 align:middle line:84% And therefore, I get something that maps to m. 00:57:02.875 --> 00:57:12.490 align:middle line:90% 00:57:12.490 --> 00:57:14.020 align:middle line:90% Therefore, f is onto. 00:57:14.020 --> 00:57:17.860 align:middle line:84% Therefore, f is a bijection and the two sets 00:57:17.860 --> 00:57:19.840 align:middle line:90% have the same cardinality. 00:57:19.840 --> 00:57:29.260 align:middle line:90% 00:57:29.260 --> 00:57:42.170 align:middle line:90% All right, so let's-- 00:57:42.170 --> 00:57:42.890 align:middle line:90% where are we at? 00:57:42.890 --> 00:57:44.450 align:middle line:90% Where should I write? 00:57:44.450 --> 00:57:45.350 align:middle line:90% Let's write here. 00:57:45.350 --> 00:57:50.907 align:middle line:90% 00:57:50.907 --> 00:57:52.990 align:middle line:84% Maybe I'll leave that up because I don't want to-- 00:57:52.990 --> 00:58:11.170 align:middle line:90% 00:58:11.170 --> 00:58:13.990 align:middle line:84% now, using this-- and I'll probably put this 00:58:13.990 --> 00:58:15.790 align:middle line:90% in the homework-- 00:58:15.790 --> 00:58:21.370 align:middle line:90% I mean, one can also show-- 00:58:21.370 --> 00:58:24.040 align:middle line:90% 00:58:24.040 --> 00:58:27.940 align:middle line:84% I should say using this, but one can also 00:58:27.940 --> 00:58:35.410 align:middle line:84% show that the integers have the same size 00:58:35.410 --> 00:58:40.290 align:middle line:84% as the natural numbers, which, again, is a little bit 00:58:40.290 --> 00:58:42.990 align:middle line:84% surprising since the natural numbers are 00:58:42.990 --> 00:58:48.540 align:middle line:90% a strict subset of Z. 00:58:48.540 --> 00:58:50.680 align:middle line:90% So what's the proof? 00:58:50.680 --> 00:58:53.820 align:middle line:84% So I'm going to draw a picture, then 00:58:53.820 --> 00:58:55.620 align:middle line:84% I'm going to write down the function, 00:58:55.620 --> 00:58:59.192 align:middle line:84% and then I'm going to leave it as-- 00:58:59.192 --> 00:59:01.400 align:middle line:84% actually, I'm going to put it in the homework for you 00:59:01.400 --> 00:59:04.400 align:middle line:84% to verify that this function is one-to-one and onto. 00:59:04.400 --> 00:59:23.110 align:middle line:84% So let's say there's as many natural numbers 00:59:23.110 --> 00:59:40.270 align:middle line:84% as I want to write and as many integers as I want to write. 00:59:40.270 --> 00:59:46.510 align:middle line:84% OK, so what would be a way of mapping 00:59:46.510 --> 00:59:51.340 align:middle line:84% the integers in a one-to-one and onto fashion 00:59:51.340 --> 00:59:52.435 align:middle line:90% onto the natural numbers? 00:59:52.435 --> 00:59:54.990 align:middle line:90% 00:59:54.990 --> 00:59:58.480 align:middle line:90% Well, what we could do-- 00:59:58.480 --> 01:00:05.330 align:middle line:84% first off, let's send 0 to 1 just to get 0 out of the way. 01:00:05.330 --> 01:00:08.390 align:middle line:84% And from then on, now we just need 01:00:08.390 --> 01:00:10.760 align:middle line:84% to find a way to map the positive integers 01:00:10.760 --> 01:00:14.570 align:middle line:84% and the negative integers onto the natural numbers bigger 01:00:14.570 --> 01:00:17.020 align:middle line:90% than 2. 01:00:17.020 --> 01:00:22.870 align:middle line:84% And in some way, mentally, we should 01:00:22.870 --> 01:00:27.910 align:middle line:84% feel like we can do this because we kind of did it over there, 01:00:27.910 --> 01:00:29.770 align:middle line:90% but not explicitly. 01:00:29.770 --> 01:00:41.290 align:middle line:84% So how about we take 1 to 2, 2 to 4, 3 to 6? 01:00:41.290 --> 01:00:44.860 align:middle line:90% 4 would then go to 8. 01:00:44.860 --> 01:00:53.995 align:middle line:84% And we'll take 1 to 3, minus 2 to 5. 01:00:53.995 --> 01:00:56.620 align:middle line:90% I'm getting crossed. 01:00:56.620 --> 01:01:01.850 align:middle line:84% And then minus 3 would get sent to 7. 01:01:01.850 --> 01:01:04.940 align:middle line:84% So you see that the positive integers 01:01:04.940 --> 01:01:07.700 align:middle line:84% get mapped to the even natural numbers 01:01:07.700 --> 01:01:15.480 align:middle line:84% and the negative integers get mapped to the odd natural 01:01:15.480 --> 01:01:16.770 align:middle line:90% numbers bigger than 1. 01:01:16.770 --> 01:01:20.160 align:middle line:90% 01:01:20.160 --> 01:01:21.910 align:middle line:84% So I'm not even going to write the proof. 01:01:21.910 --> 01:01:28.400 align:middle line:84% This will be part of the homework. 01:01:28.400 --> 01:01:33.970 align:middle line:84% Now, it is a bit surprising that there are twice 01:01:33.970 --> 01:01:36.910 align:middle line:84% as many numbers as numbers, not too 01:01:36.910 --> 01:01:42.070 align:middle line:84% surprising since these subsets, if you picture them 01:01:42.070 --> 01:01:45.250 align:middle line:84% as I've been doing as subsets of the real line, 01:01:45.250 --> 01:01:46.690 align:middle line:84% you know they're kind of discrete. 01:01:46.690 --> 01:01:50.050 align:middle line:84% So you should be able to count them. 01:01:50.050 --> 01:01:55.450 align:middle line:84% What is not-- what is more surprising are-- 01:01:55.450 --> 01:01:59.270 align:middle line:84% is whether or not can count subsets that, in some sense, 01:01:59.270 --> 01:02:00.340 align:middle line:90% are not discrete. 01:02:00.340 --> 01:02:04.250 align:middle line:84% For example, what about the rational numbers? 01:02:04.250 --> 01:02:10.990 align:middle line:90% So this is a theorem that-- 01:02:10.990 --> 01:02:14.110 align:middle line:84% and let me just look at those positive rational numbers. 01:02:14.110 --> 01:02:16.130 align:middle line:84% In fact, I could take all rational numbers. 01:02:16.130 --> 01:02:21.600 align:middle line:84% But for the statement of this theorem, 01:02:21.600 --> 01:02:31.700 align:middle line:84% if I look at those rational numbers which are positive, 01:02:31.700 --> 01:02:36.790 align:middle line:84% then this has the same size as the natural numbers. 01:02:36.790 --> 01:02:42.195 align:middle line:84% You can count the positive rational numbers, which is just 01:02:42.195 --> 01:02:48.310 align:middle line:84% a bit crazy to me because here, at least for let's say 01:02:48.310 --> 01:02:51.580 align:middle line:84% the integers, once I'm at an integer, 01:02:51.580 --> 01:02:55.805 align:middle line:84% I can move to a next biggest one and count that one in some way, 01:02:55.805 --> 01:02:56.305 align:middle line:90% right? 01:02:56.305 --> 01:02:59.810 align:middle line:90% 01:02:59.810 --> 01:03:06.068 align:middle line:84% And so that makes it believable that I can count the integers. 01:03:06.068 --> 01:03:08.360 align:middle line:84% The integers have the same size as the natural numbers, 01:03:08.360 --> 01:03:10.050 align:middle line:84% even though that, at first glance, 01:03:10.050 --> 01:03:13.490 align:middle line:84% it looks like there's twice as many. 01:03:13.490 --> 01:03:17.780 align:middle line:84% But for rational numbers, between any two 01:03:17.780 --> 01:03:19.850 align:middle line:84% rational numbers, there's another rational number 01:03:19.850 --> 01:03:20.990 align:middle line:90% in between them. 01:03:20.990 --> 01:03:26.840 align:middle line:84% You just take the average of those two rational numbers. 01:03:26.840 --> 01:03:30.290 align:middle line:84% So now, this idea of being at a rational number 01:03:30.290 --> 01:03:34.620 align:middle line:84% and then moving to the next big one, you can't do that now. 01:03:34.620 --> 01:03:38.960 align:middle line:84% So now, it's a little bit up in the air at least whether or not 01:03:38.960 --> 01:03:41.930 align:middle line:90% can count the rational numbers. 01:03:41.930 --> 01:03:44.757 align:middle line:84% And what I'm saying is that indeed you can. 01:03:44.757 --> 01:03:46.340 align:middle line:84% And this will be part of the homework. 01:03:46.340 --> 01:03:49.410 align:middle line:90% 01:03:49.410 --> 01:03:57.500 align:middle line:84% I will at least give you an idea of how the proof will 01:03:57.500 --> 01:03:59.720 align:middle line:84% go, what we will actually be able to write down, 01:03:59.720 --> 01:04:03.800 align:middle line:90% a map based on a simple fact. 01:04:03.800 --> 01:04:12.530 align:middle line:84% So let me not write down proof, but let me write down a remark. 01:04:12.530 --> 01:04:15.790 align:middle line:84% So we'll actually be able to write down a map based 01:04:15.790 --> 01:04:19.000 align:middle line:84% on one simple fact, which is this fundamental theorem 01:04:19.000 --> 01:04:22.750 align:middle line:90% of arithmetic, which says that-- 01:04:22.750 --> 01:04:24.265 align:middle line:90% so this is just discussion now. 01:04:24.265 --> 01:04:27.015 align:middle line:90% 01:04:27.015 --> 01:04:29.670 align:middle line:84% More stuff will be written in the homework about this. 01:04:29.670 --> 01:04:31.650 align:middle line:90% So just try to follow along. 01:04:31.650 --> 01:04:33.330 align:middle line:84% So the fundamental theorem of arithmetic 01:04:33.330 --> 01:04:35.880 align:middle line:84% says that if you have a positive natural number, 01:04:35.880 --> 01:04:40.410 align:middle line:84% you can write it in a unique way as a product of prime numbers. 01:04:40.410 --> 01:04:45.071 align:middle line:84% Now, for rational numbers, using that, 01:04:45.071 --> 01:04:48.920 align:middle line:84% that means you can write every rational number uniquely 01:04:48.920 --> 01:04:53.170 align:middle line:84% as a product of prime numbers divided 01:04:53.170 --> 01:04:55.690 align:middle line:84% by another product of prime numbers 01:04:55.690 --> 01:04:58.840 align:middle line:90% where no two prime numbers-- 01:04:58.840 --> 01:05:01.570 align:middle line:84% where the prime numbers up top and the prime numbers 01:05:01.570 --> 01:05:04.040 align:middle line:84% at the bottom, none are in common, 01:05:04.040 --> 01:05:07.580 align:middle line:84% meaning you've simplified as much as possible. 01:05:07.580 --> 01:05:13.095 align:middle line:90% So instead of 15 over 3-- 01:05:13.095 --> 01:05:14.790 align:middle line:90% no, that's not good. 01:05:14.790 --> 01:05:16.650 align:middle line:90% Let's say 15 over 30. 01:05:16.650 --> 01:05:19.900 align:middle line:90% You have 1/2. 01:05:19.900 --> 01:05:23.650 align:middle line:90% So the map that I would take-- 01:05:23.650 --> 01:05:40.020 align:middle line:84% so what I'm saying here is that every rational number 01:05:40.020 --> 01:05:46.480 align:middle line:90% can be written as some product. 01:05:46.480 --> 01:05:50.290 align:middle line:84% So let me not use that notation, but some product p 01:05:50.290 --> 01:06:07.330 align:middle line:84% r1 pN rN q1 s1 qn sM where p1, p2, and pN, these are all 01:06:07.330 --> 01:06:07.830 align:middle line:90% primes. 01:06:07.830 --> 01:06:10.650 align:middle line:90% q1 up to qN, they're all primes. 01:06:10.650 --> 01:06:15.220 align:middle line:84% r1, rN, these are all exponents, positive natural numbers. 01:06:15.220 --> 01:06:21.120 align:middle line:84% So rj, sk, these are natural numbers. 01:06:21.120 --> 01:06:23.880 align:middle line:90% p1 pN q1 to qM are primes. 01:06:23.880 --> 01:06:35.780 align:middle line:84% And for all jk, qj does not equal pk. 01:06:35.780 --> 01:06:39.440 align:middle line:84% So there's no prime that appears both up here and down here. 01:06:39.440 --> 01:06:42.050 align:middle line:84% We've already simplified that away. 01:06:42.050 --> 01:06:56.400 align:middle line:90% 01:06:56.400 --> 01:07:03.272 align:middle line:84% So just so you don't think I'm fooling you, 01:07:03.272 --> 01:07:04.980 align:middle line:84% 9/2, which is a positive rational number, 01:07:04.980 --> 01:07:07.040 align:middle line:90% this is 3 squared over 2, yeah? 01:07:07.040 --> 01:07:13.070 align:middle line:90% 01:07:13.070 --> 01:07:14.280 align:middle line:90% I'm not going to do any more. 01:07:14.280 --> 01:07:14.780 align:middle line:90% That's it. 01:07:14.780 --> 01:07:20.420 align:middle line:90% 01:07:20.420 --> 01:07:25.930 align:middle line:84% So the map that we will take from this rational number 01:07:25.930 --> 01:07:30.790 align:middle line:84% to a natural number will be this gets 01:07:30.790 --> 01:07:44.360 align:middle line:84% mapped to the natural number p1 to r1, pN to rN times now q1 01:07:44.360 --> 01:07:53.780 align:middle line:90% to 2s 1 minus 1 qM 2Sn minus 1. 01:07:53.780 --> 01:07:57.290 align:middle line:84% So basically, I map it to the integer that 01:07:57.290 --> 01:08:02.690 align:middle line:84% has this expansion in terms of prime numbers, where now 01:08:02.690 --> 01:08:04.610 align:middle line:84% the exponents of these prime numbers 01:08:04.610 --> 01:08:07.850 align:middle line:84% is even depending on these exponents on top 01:08:07.850 --> 01:08:11.780 align:middle line:84% and the exponents of this one are odd, 01:08:11.780 --> 01:08:15.290 align:middle line:84% depending on this exponent on top. 01:08:15.290 --> 01:08:24.080 align:middle line:84% So for example, 3 squared over 2, this would get mapped to 3 01:08:24.080 --> 01:08:26.780 align:middle line:90% to the 4 times 2. 01:08:26.780 --> 01:08:36.859 align:middle line:90% 01:08:36.859 --> 01:08:38.899 align:middle line:84% So what we'll do in the homework is show 01:08:38.899 --> 01:08:41.390 align:middle line:84% that this map is, in fact, a bijection. 01:08:41.390 --> 01:08:58.279 align:middle line:90% 01:08:58.279 --> 01:09:02.720 align:middle line:84% So those two theorems you will prove in the homework. 01:09:02.720 --> 01:09:05.330 align:middle line:90% They won't be too bad. 01:09:05.330 --> 01:09:07.970 align:middle line:90% I will leave enough hints. 01:09:07.970 --> 01:09:11.255 align:middle line:90% So we've dealt with z. 01:09:11.255 --> 01:09:13.550 align:middle line:90% We've dealt with q, essentially. 01:09:13.550 --> 01:09:26.260 align:middle line:90% 01:09:26.260 --> 01:09:31.520 align:middle line:84% I mean, so let me actually write down-- 01:09:31.520 --> 01:09:35.290 align:middle line:84% this is really a corollary of these two theorems 01:09:35.290 --> 01:09:38.700 align:middle line:84% here, which I haven't proved, but you 01:09:38.700 --> 01:09:40.200 align:middle line:90% will prove in the homework. 01:09:40.200 --> 01:09:48.670 align:middle line:84% So this says that the rationals are countable, 01:09:48.670 --> 01:09:52.330 align:middle line:84% all the rationales, not just the positive ones. 01:09:52.330 --> 01:09:56.270 align:middle line:90% And what's the proof? 01:09:56.270 --> 01:10:02.330 align:middle line:90% So we know that-- 01:10:02.330 --> 01:10:08.970 align:middle line:90% 01:10:08.970 --> 01:10:13.930 align:middle line:84% so I'm going to give you a sketch. 01:10:13.930 --> 01:10:16.123 align:middle line:84% Should I sketch it or should I write it all out? 01:10:16.123 --> 01:10:17.665 align:middle line:84% I'm running a little bit out of time. 01:10:17.665 --> 01:10:24.170 align:middle line:90% 01:10:24.170 --> 01:10:28.730 align:middle line:84% So maybe I will just tell you why this is true. 01:10:28.730 --> 01:10:30.830 align:middle line:84% I mean, all the details are essentially here. 01:10:30.830 --> 01:10:33.163 align:middle line:84% I'm just not going to write it as carefully as I've been 01:10:33.163 --> 01:10:35.010 align:middle line:90% writing down the proofs before. 01:10:35.010 --> 01:10:40.430 align:middle line:84% So let me write this as the proof sketch. 01:10:40.430 --> 01:10:43.410 align:middle line:90% 01:10:43.410 --> 01:10:46.900 align:middle line:90% Of course, you can write these-- 01:10:46.900 --> 01:10:49.570 align:middle line:84% you can actually use the definitions and write this out. 01:10:49.570 --> 01:10:51.800 align:middle line:90% But this is the essential idea. 01:10:51.800 --> 01:11:05.800 align:middle line:84% So we have that the size of rational numbers-- 01:11:05.800 --> 01:11:25.890 align:middle line:84% so let me-- this has the same size as the rational numbers 01:11:25.890 --> 01:11:27.290 align:middle line:90% which are negative. 01:11:27.290 --> 01:11:36.940 align:middle line:90% 01:11:36.940 --> 01:11:41.320 align:middle line:90% And how do we establish this? 01:11:41.320 --> 01:11:45.070 align:middle line:84% Or instead of using the same letter, 01:11:45.070 --> 01:11:55.120 align:middle line:84% let's say r, since f of Q equals minus Q 01:11:55.120 --> 01:12:06.810 align:middle line:84% is a bijection from the first set to the second set. 01:12:06.810 --> 01:12:24.829 align:middle line:90% 01:12:24.829 --> 01:12:27.440 align:middle line:84% If I just take a rational number, positive 1, 01:12:27.440 --> 01:12:31.160 align:middle line:84% take its minus, then I get an element of the second set. 01:12:31.160 --> 01:12:33.260 align:middle line:90% And this is a bijection. 01:12:33.260 --> 01:12:36.100 align:middle line:90% 01:12:36.100 --> 01:12:49.450 align:middle line:84% OK, so thus, since this has the same size 01:12:49.450 --> 01:12:55.380 align:middle line:84% as the natural numbers, and by that theorem 01:12:55.380 --> 01:12:57.560 align:middle line:84% we proved over there, the size of this set 01:12:57.560 --> 01:12:59.100 align:middle line:84% is the same as the natural numbers-- 01:12:59.100 --> 01:13:01.897 align:middle line:90% so it is countable-- 01:13:01.897 --> 01:13:09.580 align:middle line:90% 01:13:09.580 --> 01:13:20.885 align:middle line:84% then there exists bijections f going from Q-- 01:13:20.885 --> 01:13:22.745 align:middle line:84% so the positive rational numbers-- 01:13:22.745 --> 01:13:27.340 align:middle line:90% 01:13:27.340 --> 01:13:48.210 align:middle line:84% to the natural numbers and g going from the negative ones 01:13:48.210 --> 01:13:51.630 align:middle line:90% to the natural numbers. 01:13:51.630 --> 01:13:53.805 align:middle line:90% So the picture here is-- 01:13:53.805 --> 01:13:59.010 align:middle line:90% 01:13:59.010 --> 01:14:02.910 align:middle line:84% well, let's not draw the picture yet. 01:14:02.910 --> 01:14:05.190 align:middle line:84% OK, so I have these two bijections 01:14:05.190 --> 01:14:08.340 align:middle line:84% from the positive rationales to the integers-- 01:14:08.340 --> 01:14:12.090 align:middle line:84% this one from the negative rationales 01:14:12.090 --> 01:14:14.880 align:middle line:90% to the natural numbers. 01:14:14.880 --> 01:14:17.960 align:middle line:84% How do I get a map that goes from all of q 01:14:17.960 --> 01:14:19.570 align:middle line:90% now to the natural numbers? 01:14:19.570 --> 01:14:23.050 align:middle line:84% Well, let's go in between and go to the integers. 01:14:23.050 --> 01:14:33.550 align:middle line:84% Then I define a function h, which goes now from all of Q 01:14:33.550 --> 01:14:48.970 align:middle line:84% to the integers by h of x equals 0 if x equals 0; 01:14:48.970 --> 01:14:56.990 align:middle line:84% equals f of x if x is positive; and negative g 01:14:56.990 --> 01:15:00.520 align:middle line:90% of x if x is negative. 01:15:00.520 --> 01:15:06.320 align:middle line:90% 01:15:06.320 --> 01:15:11.010 align:middle line:90% And h is a bijection. 01:15:11.010 --> 01:15:15.330 align:middle line:84% So everything up to this point has 01:15:15.330 --> 01:15:18.520 align:middle line:84% been completely fine with the exception 01:15:18.520 --> 01:15:21.850 align:middle line:84% of me verifying that this map is a bijection 01:15:21.850 --> 01:15:23.540 align:middle line:90% and this map is a bijection. 01:15:23.540 --> 01:15:24.940 align:middle line:84% So that's the only parts that I'm 01:15:24.940 --> 01:15:27.550 align:middle line:90% leaving out for you to verify. 01:15:27.550 --> 01:15:29.800 align:middle line:90% Then h is a bijection. 01:15:29.800 --> 01:15:38.410 align:middle line:84% So Q has the same cardinality as integers, which we've shown 01:15:38.410 --> 01:15:41.635 align:middle line:84% has the same cardinality as the natural numbers. 01:15:41.635 --> 01:15:44.560 align:middle line:90% 01:15:44.560 --> 01:15:48.610 align:middle line:84% And therefore, the rationals have the same cardinality 01:15:48.610 --> 01:15:49.760 align:middle line:90% as the natural numbers. 01:15:49.760 --> 01:15:51.490 align:middle line:90% So they are countably infinite. 01:15:51.490 --> 01:16:01.220 align:middle line:90% 01:16:01.220 --> 01:16:06.725 align:middle line:90% So a natural question is-- 01:16:06.725 --> 01:16:09.973 align:middle line:90% 01:16:09.973 --> 01:16:12.390 align:middle line:84% I mean, is there anything bigger than the natural numbers? 01:16:12.390 --> 01:16:14.280 align:middle line:84% Because everything I've written down, 01:16:14.280 --> 01:16:16.245 align:middle line:84% this has the size of the natural numbers. 01:16:16.245 --> 01:16:19.510 align:middle line:90% 01:16:19.510 --> 01:16:22.930 align:middle line:84% We haven't really defined the real numbers well enough yet 01:16:22.930 --> 01:16:32.830 align:middle line:84% to make any sort of claim like that about the real numbers. 01:16:32.830 --> 01:16:40.790 align:middle line:84% And now, is there just any set that 01:16:40.790 --> 01:16:43.440 align:middle line:84% is bigger in size than the natural numbers? 01:16:43.440 --> 01:16:47.620 align:middle line:90% The answer to that was unknown. 01:16:47.620 --> 01:16:53.850 align:middle line:84% And it's pretty strikingly yes, in fact. 01:16:53.850 --> 01:16:58.200 align:middle line:90% 01:16:58.200 --> 01:17:04.640 align:middle line:90% So let me phrase that question. 01:17:04.640 --> 01:17:13.760 align:middle line:90% 01:17:13.760 --> 01:17:26.050 align:middle line:84% Does there exist a set A such that A has strictly 01:17:26.050 --> 01:17:28.750 align:middle line:84% bigger cardinality than the natural numbers, so A 01:17:28.750 --> 01:17:29.680 align:middle line:90% is uncountable? 01:17:29.680 --> 01:17:34.410 align:middle line:90% 01:17:34.410 --> 01:17:38.730 align:middle line:84% So to answer this question astoundingly yes, let me first 01:17:38.730 --> 01:17:40.995 align:middle line:90% define for a general object. 01:17:40.995 --> 01:17:45.710 align:middle line:90% 01:17:45.710 --> 01:17:54.365 align:middle line:84% If A is a set, we define script p of A. 01:17:54.365 --> 01:18:08.480 align:middle line:84% This is the power set of A. This is a set that consists 01:18:08.480 --> 01:18:15.210 align:middle line:84% of all subsets of A. So this is a set of all sets B such that B 01:18:15.210 --> 01:18:31.860 align:middle line:84% is a subset of A. So for example, if A is the empty set, 01:18:31.860 --> 01:18:32.685 align:middle line:90% the power set-- 01:18:32.685 --> 01:18:35.690 align:middle line:90% 01:18:35.690 --> 01:18:37.340 align:middle line:90% so A is empty. 01:18:37.340 --> 01:18:40.860 align:middle line:84% What are the subsets of the empty set? 01:18:40.860 --> 01:18:46.030 align:middle line:84% Well, there's only one subset of the empty set, the empty set. 01:18:46.030 --> 01:18:48.160 align:middle line:84% And even though the empty set has no members, 01:18:48.160 --> 01:18:49.585 align:middle line:90% its power set has one member. 01:18:49.585 --> 01:18:53.730 align:middle line:90% 01:18:53.730 --> 01:18:56.730 align:middle line:84% If A equals 1, then the power set 01:18:56.730 --> 01:19:01.390 align:middle line:84% of A, the set of all subsets, consists of the empty set 01:19:01.390 --> 01:19:03.910 align:middle line:90% and the set itself, 1. 01:19:03.910 --> 01:19:06.940 align:middle line:90% 01:19:06.940 --> 01:19:10.180 align:middle line:90% And let's do one more. 01:19:10.180 --> 01:19:17.860 align:middle line:84% If A is the set 1, 2, then the power set of A, 01:19:17.860 --> 01:19:22.860 align:middle line:84% this is the set which consists of the empty set, 01:19:22.860 --> 01:19:26.830 align:middle line:84% because this is a subset of this set, 01:19:26.830 --> 01:19:30.840 align:middle line:84% the set consisting of 1, because this is a subset of this set, 01:19:30.840 --> 01:19:37.170 align:middle line:90% or 2, 1, 2. 01:19:37.170 --> 01:19:38.910 align:middle line:90% Now, notice something. 01:19:38.910 --> 01:19:41.730 align:middle line:90% This set had cardinality. 01:19:41.730 --> 01:19:44.640 align:middle line:84% Strictly speaking, I should have defined that 01:19:44.640 --> 01:19:47.580 align:middle line:90% as having cardinality 0. 01:19:47.580 --> 01:19:50.940 align:middle line:84% Yet, its power set has cardinality 1. 01:19:50.940 --> 01:19:55.560 align:middle line:84% A has size 1 because it's in one-to-one correspondence 01:19:55.560 --> 01:19:57.450 align:middle line:90% with just 1. 01:19:57.450 --> 01:20:00.870 align:middle line:84% And its power set has two elements. 01:20:00.870 --> 01:20:04.310 align:middle line:90% So it has cardinality 2. 01:20:04.310 --> 01:20:05.310 align:middle line:90% I could count them off-- 01:20:05.310 --> 01:20:07.290 align:middle line:90% 1, 2. 01:20:07.290 --> 01:20:10.170 align:middle line:90% A has size 2, has two elements. 01:20:10.170 --> 01:20:12.490 align:middle line:84% And the power set of A has size 4. 01:20:12.490 --> 01:20:16.590 align:middle line:84% It has four different elements-- the empty set, 1, 2, 1, 2. 01:20:16.590 --> 01:20:26.020 align:middle line:84% So what one can prove in general and which will appear 01:20:26.020 --> 01:20:33.130 align:middle line:84% on the homework is if A has size n-- 01:20:33.130 --> 01:20:37.330 align:middle line:84% so it's a finite set of size n, meaning it's in one-to-one 01:20:37.330 --> 01:20:40.660 align:middle line:84% correspondence with the numbers 1 up to n-- 01:20:40.660 --> 01:20:48.010 align:middle line:84% then its power set is also finite 01:20:48.010 --> 01:20:51.850 align:middle line:90% and has cardinality 2 to the n. 01:20:51.850 --> 01:20:56.250 align:middle line:84% And you can prove by induction, if you like, that 2 to the n 01:20:56.250 --> 01:20:57.540 align:middle line:90% is always bigger than n. 01:20:57.540 --> 01:21:00.240 align:middle line:90% 01:21:00.240 --> 01:21:07.340 align:middle line:84% And so a theorem I'll prove next time, which 01:21:07.340 --> 01:21:14.260 align:middle line:84% will finish our discussion of sets and cardinality, 01:21:14.260 --> 01:21:16.840 align:middle line:84% and then we'll move on to the real numbers, 01:21:16.840 --> 01:21:25.150 align:middle line:84% is this theorem due to Cantor, which 01:21:25.150 --> 01:21:30.800 align:middle line:84% says that not just for finite sets 01:21:30.800 --> 01:21:33.350 align:middle line:84% do we have the power set being, in some sense, strictly 01:21:33.350 --> 01:21:36.680 align:middle line:84% bigger than the original set, but for any set. 01:21:36.680 --> 01:21:44.970 align:middle line:84% So this theorem due to Cantor is if A is a set, 01:21:44.970 --> 01:21:50.130 align:middle line:84% then the cardinality of A is strictly smaller 01:21:50.130 --> 01:21:54.350 align:middle line:84% than the cardinality of its power set. 01:21:54.350 --> 01:21:58.910 align:middle line:84% So this definitely answers our question. 01:21:58.910 --> 01:22:01.400 align:middle line:84% Does there exist a set with cardinality bigger 01:22:01.400 --> 01:22:02.940 align:middle line:90% than the natural numbers? 01:22:02.940 --> 01:22:04.565 align:middle line:84% So let me write this as a remark. 01:22:04.565 --> 01:22:08.550 align:middle line:90% 01:22:08.550 --> 01:22:13.740 align:middle line:84% In fact, maybe I'll state this as another theorem, which 01:22:13.740 --> 01:22:16.650 align:middle line:84% just follows immediately from this getting 01:22:16.650 --> 01:22:21.870 align:middle line:84% used over and over again, is that the cardinality 01:22:21.870 --> 01:22:23.550 align:middle line:84% of the natural numbers, this is less 01:22:23.550 --> 01:22:27.210 align:middle line:84% than the cardinality of the power 01:22:27.210 --> 01:22:30.450 align:middle line:84% set of the natural numbers, which is less 01:22:30.450 --> 01:22:33.780 align:middle line:84% than the cardinality of the power 01:22:33.780 --> 01:22:37.140 align:middle line:84% set of the power set of the natural numbers, right? 01:22:37.140 --> 01:22:39.810 align:middle line:84% I can now take this as a set and take its power set. 01:22:39.810 --> 01:22:43.150 align:middle line:84% And I get a new set with bigger size, 01:22:43.150 --> 01:22:48.120 align:middle line:84% which is less than the power set of the power set where-- 01:22:48.120 --> 01:22:53.100 align:middle line:84% how many do we have now-- of the power set of natural numbers, 01:22:53.100 --> 01:22:54.030 align:middle line:90% and so on. 01:22:54.030 --> 01:23:08.570 align:middle line:90% 01:23:08.570 --> 01:23:25.540 align:middle line:84% And formally, this means there's an infinity of infinitudes. 01:23:25.540 --> 01:23:30.490 align:middle line:84% There's an infinite number of infinite sizes, 01:23:30.490 --> 01:23:32.260 align:middle line:90% one getting strictly-- 01:23:32.260 --> 01:23:34.675 align:middle line:84% in some sense, strictly bigger than the previous size. 01:23:34.675 --> 01:23:37.470 align:middle line:90% 01:23:37.470 --> 01:23:43.540 align:middle line:90% And maybe you're wondering-- 01:23:43.540 --> 01:23:45.120 align:middle line:84% there's one more question that's kind 01:23:45.120 --> 01:23:54.090 align:middle line:84% of sandwiched in between here, pun intended, is-- 01:23:54.090 --> 01:23:56.820 align:middle line:90% let's look at this first guy. 01:23:56.820 --> 01:24:06.020 align:middle line:90% 01:24:06.020 --> 01:24:19.930 align:middle line:84% Does there exist a set A such that it has size bigger 01:24:19.930 --> 01:24:23.470 align:middle line:84% than natural numbers-- so it's uncountable-- but has 01:24:23.470 --> 01:24:25.150 align:middle line:84% size strictly smaller than the power 01:24:25.150 --> 01:24:27.700 align:middle line:90% set of the natural numbers. 01:24:27.700 --> 01:24:39.426 align:middle line:84% And this question is called the continuum hypothesis, 01:24:39.426 --> 01:24:41.720 align:middle line:84% hypothesis because it is independent from one 01:24:41.720 --> 01:24:48.440 align:middle line:84% of the standard axiomatic treatments of set theory. 01:24:48.440 --> 01:24:51.080 align:middle line:84% But we will not touch this question. 01:24:51.080 --> 01:24:54.195 align:middle line:84% This is beyond the scope of this class, 01:24:54.195 --> 01:24:55.820 align:middle line:84% but it's an interesting question that's 01:24:55.820 --> 01:24:58.990 align:middle line:84% out there that people just don't know. 01:24:58.990 --> 01:25:07.000 align:middle line:90%