WEBVTT 00:00:00.000 --> 00:00:02.445 align:middle line:90% [SQUEAKING] 00:00:02.445 --> 00:00:03.912 align:middle line:90% [RUSTLING] 00:00:03.912 --> 00:00:06.357 align:middle line:90% [CLICKING] 00:00:06.357 --> 00:00:12.175 align:middle line:90% 00:00:12.175 --> 00:00:13.300 align:middle line:90% ANKUR MOITRA: Hello, class. 00:00:13.300 --> 00:00:14.780 align:middle line:90% Welcome to 18.200. 00:00:14.780 --> 00:00:16.800 align:middle line:84% So if you're in the right place, this 00:00:16.800 --> 00:00:19.560 align:middle line:84% is Principles of Discrete Applied Math. 00:00:19.560 --> 00:00:21.340 align:middle line:84% I'll be one of your co-lecturers. 00:00:21.340 --> 00:00:24.540 align:middle line:84% I'm Ankur Moitra, and the other co-lecturer is Peter Shor So, 00:00:24.540 --> 00:00:28.440 align:middle line:84% Peter, do you want to stand up and take a bow? 00:00:28.440 --> 00:00:30.720 align:middle line:84% So the basic topics that we're going to be covering 00:00:30.720 --> 00:00:33.940 align:middle line:84% is we're going to cover topics like probability counting. 00:00:33.940 --> 00:00:36.040 align:middle line:84% This will be the beginning part of the class. 00:00:36.040 --> 00:00:38.760 align:middle line:84% Some of you may have seen these topics before. 00:00:38.760 --> 00:00:40.700 align:middle line:84% But really in the later parts of the class, 00:00:40.700 --> 00:00:42.860 align:middle line:84% we're going to cover some quite new things. 00:00:42.860 --> 00:00:44.240 align:middle line:84% We'll talk a lot about generating 00:00:44.240 --> 00:00:48.620 align:middle line:84% functions, number theory, cryptography, coding theory, 00:00:48.620 --> 00:00:52.180 align:middle line:84% things like noiseless coding, linear programming, 00:00:52.180 --> 00:00:54.240 align:middle line:90% and lots of other fun topics. 00:00:54.240 --> 00:00:56.103 align:middle line:90% I want to mention that-- 00:00:56.103 --> 00:00:57.520 align:middle line:84% I mean, the goal for this class is 00:00:57.520 --> 00:00:59.810 align:middle line:84% to try and teach you how to express proofs 00:00:59.810 --> 00:01:01.730 align:middle line:90% in an understandable way. 00:01:01.730 --> 00:01:03.610 align:middle line:84% This is one of these skills at MIT 00:01:03.610 --> 00:01:07.010 align:middle line:84% that you can pick up a little bit 00:01:07.010 --> 00:01:09.410 align:middle line:84% from various classes along the way, 00:01:09.410 --> 00:01:11.170 align:middle line:84% but sitting down and conscientiously 00:01:11.170 --> 00:01:15.090 align:middle line:84% thinking about what makes a proof easy to follow, 00:01:15.090 --> 00:01:18.355 align:middle line:84% how to make it more modular and understandable for the audience 00:01:18.355 --> 00:01:19.730 align:middle line:84% is something that's going to help 00:01:19.730 --> 00:01:22.930 align:middle line:84% you a lot with other parts of math, 00:01:22.930 --> 00:01:25.630 align:middle line:84% and also other parts of computer science in general. 00:01:25.630 --> 00:01:29.490 align:middle line:84% A lot of the things that go into good coding practices in terms 00:01:29.490 --> 00:01:31.410 align:middle line:84% of modularity are the same types of things 00:01:31.410 --> 00:01:35.250 align:middle line:84% we think about when we're writing good proofs. 00:01:35.250 --> 00:01:39.210 align:middle line:84% So that's enough boilerplate for me. 00:01:39.210 --> 00:01:42.370 align:middle line:84% I can only go so long without talking about math. 00:01:42.370 --> 00:01:44.490 align:middle line:84% So what we're going to be talking about today 00:01:44.490 --> 00:01:48.593 align:middle line:84% is we're going to talk about the pigeonhole principle. 00:01:48.593 --> 00:01:50.010 align:middle line:84% Actually, first let me hand it off 00:01:50.010 --> 00:01:52.135 align:middle line:84% to Susan, who's going to tell you a little bit more 00:01:52.135 --> 00:01:56.250 align:middle line:84% about the recitations, and then I'll take it away after that. 00:01:56.250 --> 00:01:58.100 align:middle line:84% SUSAN: Thank you very much, Ankur. 00:01:58.100 --> 00:02:01.820 align:middle line:84% So as Ankur noted, this is a CIM, a communication 00:02:01.820 --> 00:02:03.400 align:middle line:90% intensive class in the major. 00:02:03.400 --> 00:02:08.300 align:middle line:84% And the purpose of CIMs is to help you to communicate 00:02:08.300 --> 00:02:11.280 align:middle line:90% effectively as mathematicians. 00:02:11.280 --> 00:02:14.860 align:middle line:84% That's for CIMs in the math department. 00:02:14.860 --> 00:02:18.780 align:middle line:84% So communicating successfully in a new discipline 00:02:18.780 --> 00:02:21.800 align:middle line:84% requires expertise in several different domains. 00:02:21.800 --> 00:02:26.020 align:middle line:84% This is based on a model by Anne Beaufort. 00:02:26.020 --> 00:02:29.220 align:middle line:84% We will teach you in these different domains 00:02:29.220 --> 00:02:32.300 align:middle line:84% over the course of this semester in the recitations. 00:02:32.300 --> 00:02:35.300 align:middle line:84% So you need expertise, of course, with the content. 00:02:35.300 --> 00:02:37.580 align:middle line:84% And so the recitations will be designed 00:02:37.580 --> 00:02:39.860 align:middle line:84% to help you reinforce your understanding 00:02:39.860 --> 00:02:42.620 align:middle line:84% of the mathematics, in addition to 00:02:42.620 --> 00:02:44.540 align:middle line:90% other aspects of communication. 00:02:44.540 --> 00:02:47.200 align:middle line:84% To communicate successfully as a mathematician, 00:02:47.200 --> 00:02:51.640 align:middle line:84% you need to know the norms of mathematics, because they vary. 00:02:51.640 --> 00:02:54.820 align:middle line:84% There are differences between norms in CORE 6 00:02:54.820 --> 00:02:57.750 align:middle line:84% and norms in mathematics, and you'll become familiar 00:02:57.750 --> 00:02:58.650 align:middle line:90% with some of those. 00:02:58.650 --> 00:03:02.790 align:middle line:84% For example, what goes into the introduction of a paper? 00:03:02.790 --> 00:03:06.750 align:middle line:84% For the term paper, that will be basically a multi-page proof 00:03:06.750 --> 00:03:09.190 align:middle line:84% and we'll tell you the differences what 00:03:09.190 --> 00:03:11.430 align:middle line:90% goes in the introduction there. 00:03:11.430 --> 00:03:15.070 align:middle line:84% Also norms of wording, like how to use the word we. 00:03:15.070 --> 00:03:18.070 align:middle line:84% Mathematicians use we in different ways 00:03:18.070 --> 00:03:20.870 align:middle line:90% than some other disciplines. 00:03:20.870 --> 00:03:24.330 align:middle line:84% Also, to communicate successfully as a mathematician, 00:03:24.330 --> 00:03:28.830 align:middle line:84% you also need expertise with processes for generating 00:03:28.830 --> 00:03:30.210 align:middle line:90% effective communication. 00:03:30.210 --> 00:03:31.330 align:middle line:90% That includes drafting. 00:03:31.330 --> 00:03:34.990 align:middle line:84% We'll have multiple drafts built into the term paper, 00:03:34.990 --> 00:03:37.370 align:middle line:84% as well as things like how to use LaTeX. 00:03:37.370 --> 00:03:40.450 align:middle line:84% And you'll start learning that right off with the first p set. 00:03:40.450 --> 00:03:43.290 align:middle line:84% If you're unfamiliar with LaTeX, please reach out. 00:03:43.290 --> 00:03:46.310 align:middle line:90% We can help you in office hours. 00:03:46.310 --> 00:03:51.030 align:middle line:84% You also need expertise with rhetoric. 00:03:51.030 --> 00:03:54.040 align:middle line:90% And by this I mean. 00:03:54.040 --> 00:03:56.160 align:middle line:84% rhetorical strategies, strategies 00:03:56.160 --> 00:04:00.400 align:middle line:84% for accomplishing your goals as a communicator. 00:04:00.400 --> 00:04:04.200 align:middle line:84% For example, mathematicians, if they have a complicated proof, 00:04:04.200 --> 00:04:09.180 align:middle line:84% one way to communicate the logic clearly is to use modularity. 00:04:09.180 --> 00:04:12.560 align:middle line:84% Break the proof apart, pull out a lemma. 00:04:12.560 --> 00:04:15.800 align:middle line:84% Other strategies that you'll learn in recitation 00:04:15.800 --> 00:04:23.120 align:middle line:84% are how to structure the ordering of information 00:04:23.120 --> 00:04:26.680 align:middle line:84% within sentences, within sections, within the paper, 00:04:26.680 --> 00:04:29.400 align:middle line:84% to make the paper flow well, to reveal 00:04:29.400 --> 00:04:31.840 align:middle line:84% the flow of the underlying logic, 00:04:31.840 --> 00:04:34.660 align:middle line:84% so that readers can easily follow the logic, 00:04:34.660 --> 00:04:38.240 align:middle line:84% and also how to use what we call guiding text to guide readers 00:04:38.240 --> 00:04:40.360 align:middle line:90% through the logic. 00:04:40.360 --> 00:04:42.910 align:middle line:84% And we'll give other strategies in recitation also. 00:04:42.910 --> 00:04:46.000 align:middle line:90% 00:04:46.000 --> 00:04:47.000 align:middle line:90% ANKUR MOITRA: All right. 00:04:47.000 --> 00:04:48.240 align:middle line:90% So back to me. 00:04:48.240 --> 00:04:51.450 align:middle line:84% So I'm going to tell you about one of the first topics 00:04:51.450 --> 00:04:55.530 align:middle line:84% we're going to cover in class, which is a very 00:04:55.530 --> 00:04:58.050 align:middle line:90% simple but powerful idea. 00:04:58.050 --> 00:05:00.450 align:middle line:84% We're going to be discussing something called 00:05:00.450 --> 00:05:03.250 align:middle line:90% the pigeonhole principle. 00:05:03.250 --> 00:05:05.490 align:middle line:84% Now the pigeonhole principle, let 00:05:05.490 --> 00:05:08.290 align:middle line:84% me tell you the main takeaways before we get into it. 00:05:08.290 --> 00:05:09.870 align:middle line:90% It's a very simple idea. 00:05:09.870 --> 00:05:12.650 align:middle line:84% I'll tell you what exactly it is, and then we'll 00:05:12.650 --> 00:05:16.570 align:middle line:84% go through some neat applications of it. 00:05:16.570 --> 00:05:18.950 align:middle line:84% It turns out to be shockingly powerful, 00:05:18.950 --> 00:05:23.130 align:middle line:84% even though you'll think of it as a very obvious statement. 00:05:23.130 --> 00:05:26.710 align:middle line:84% I guess the other thing, which is important to keep in mind, 00:05:26.710 --> 00:05:29.330 align:middle line:90% it's also quite oddly named. 00:05:29.330 --> 00:05:31.090 align:middle line:84% I don't really know the etymology 00:05:31.090 --> 00:05:33.490 align:middle line:84% of why it's called the pigeonhole principle, 00:05:33.490 --> 00:05:35.830 align:middle line:90% but the basic idea-- 00:05:35.830 --> 00:05:37.630 align:middle line:84% let's explain it through an example, 00:05:37.630 --> 00:05:40.690 align:middle line:84% and then I'll tell you about where the pigeons come in. 00:05:40.690 --> 00:05:44.570 align:middle line:84% So here's a problem that many of us face every day. 00:05:44.570 --> 00:05:51.960 align:middle line:84% So imagine that we have five colors of socks in your drawer. 00:05:51.960 --> 00:05:56.540 align:middle line:90% 00:05:56.540 --> 00:05:58.660 align:middle line:84% And it's early in the morning, so you're 00:05:58.660 --> 00:06:02.420 align:middle line:84% trying to find a match that you can get dressed and go to work. 00:06:02.420 --> 00:06:05.580 align:middle line:84% And one of the basic problems you could ask 00:06:05.580 --> 00:06:10.140 align:middle line:84% is, how many socks do you have to pick out from your drawer 00:06:10.140 --> 00:06:14.620 align:middle line:84% to guarantee that even if you're very, very unlucky, no matter 00:06:14.620 --> 00:06:18.220 align:middle line:90% what, you must have a match? 00:06:18.220 --> 00:06:20.860 align:middle line:84% So this is really an example of the pigeonhole principle 00:06:20.860 --> 00:06:22.540 align:middle line:90% in action. 00:06:22.540 --> 00:06:24.700 align:middle line:84% We'll see why exactly it's the pigeonhole principle 00:06:24.700 --> 00:06:25.320 align:middle line:90% in a minute. 00:06:25.320 --> 00:06:26.920 align:middle line:90% But any ideas? 00:06:26.920 --> 00:06:28.660 align:middle line:84% So how many pairs, how many socks 00:06:28.660 --> 00:06:31.340 align:middle line:84% do I have to pull out so that I'm guaranteed a pair? 00:06:31.340 --> 00:06:32.460 align:middle line:90% Yeah? 00:06:32.460 --> 00:06:33.040 align:middle line:90% Six. 00:06:33.040 --> 00:06:34.180 align:middle line:90% Perfect. 00:06:34.180 --> 00:06:35.940 align:middle line:84% Because you could get very unlucky 00:06:35.940 --> 00:06:37.580 align:middle line:84% in the beginning and your first five 00:06:37.580 --> 00:06:39.860 align:middle line:84% could be each different colors, and then 00:06:39.860 --> 00:06:41.780 align:middle line:84% no matter what the sixth one is, it's 00:06:41.780 --> 00:06:45.340 align:middle line:84% going to match with one of the ones you've selected so far. 00:06:45.340 --> 00:06:50.350 align:middle line:84% This is really an example of a much more general principle. 00:06:50.350 --> 00:06:53.210 align:middle line:84% So let me explain it with pigeons, 00:06:53.210 --> 00:06:56.350 align:middle line:84% and then we'll talk about some more fun applications. 00:06:56.350 --> 00:07:02.830 align:middle line:84% So more generally, we can imagine 00:07:02.830 --> 00:07:04.420 align:middle line:90% that we've got n pigeons. 00:07:04.420 --> 00:07:07.110 align:middle line:90% 00:07:07.110 --> 00:07:10.950 align:middle line:90% I'll even draw them for you. 00:07:10.950 --> 00:07:16.330 align:middle line:84% And we're placing them in m pigeonholes. 00:07:16.330 --> 00:07:21.150 align:middle line:90% 00:07:21.150 --> 00:07:25.390 align:middle line:84% I don't know why they want to go in the pigeonholes, but they do. 00:07:25.390 --> 00:07:28.150 align:middle line:84% And let's imagine that n are a number of pigeons 00:07:28.150 --> 00:07:30.190 align:middle line:90% is strictly larger than m. 00:07:30.190 --> 00:07:41.350 align:middle line:84% Then what I claim is that there must be at least two pigeons 00:07:41.350 --> 00:07:44.510 align:middle line:90% in the same pigeonhole. 00:07:44.510 --> 00:07:46.140 align:middle line:90% That's the pigeonhole principle. 00:07:46.140 --> 00:07:49.040 align:middle line:90% 00:07:49.040 --> 00:07:52.380 align:middle line:84% Now we're going to talk a lot about how to write good proofs, 00:07:52.380 --> 00:07:54.520 align:middle line:84% and honestly, a lot of us professors 00:07:54.520 --> 00:07:57.600 align:middle line:84% spend a lot of time thinking about how to teach in class 00:07:57.600 --> 00:07:59.280 align:middle line:90% and make it engaging. 00:07:59.280 --> 00:08:02.333 align:middle line:84% I remember back when I used to teach 042, 00:08:02.333 --> 00:08:04.500 align:middle line:84% when I covered things like the pigeonhole principle, 00:08:04.500 --> 00:08:07.600 align:middle line:84% there was a bit of an arms race between faculty to come up with 00:08:07.600 --> 00:08:11.840 align:middle line:84% the best props for how to explain simple things, 00:08:11.840 --> 00:08:15.360 align:middle line:84% and I borrowed from one of my colleagues a whole box of fake 00:08:15.360 --> 00:08:19.520 align:middle line:84% dead pigeons, which was very strange because I was going down 00:08:19.520 --> 00:08:22.520 align:middle line:84% the elevators in the status center with my box of dead 00:08:22.520 --> 00:08:25.380 align:middle line:84% pigeons, and I got a lot of strange looks. 00:08:25.380 --> 00:08:27.745 align:middle line:84% People would ask me, what's going on? 00:08:27.745 --> 00:08:29.120 align:middle line:84% And then I had to explain to them 00:08:29.120 --> 00:08:30.780 align:middle line:84% that I'm teaching the pigeonhole principle. 00:08:30.780 --> 00:08:33.091 align:middle line:84% I don't know why it's called the pigeonhole principle. 00:08:33.091 --> 00:08:35.299 align:middle line:84% But then someone asked me when I was on the elevator, 00:08:35.299 --> 00:08:37.000 align:middle line:84% does that really help people understand 00:08:37.000 --> 00:08:38.600 align:middle line:90% the pigeonhole principle? 00:08:38.600 --> 00:08:40.539 align:middle line:84% And I got to say, I don't think it does. 00:08:40.539 --> 00:08:42.039 align:middle line:84% Once you get that question, you have 00:08:42.039 --> 00:08:46.450 align:middle line:84% to reevaluate your teaching preferences and priorities. 00:08:46.450 --> 00:08:48.590 align:middle line:84% But in any case, this is the pigeonhole principle. 00:08:48.590 --> 00:08:51.690 align:middle line:84% I don't know why the pigeons want to go into the pigeonholes, 00:08:51.690 --> 00:08:55.970 align:middle line:84% but you can see how the simple example of socks and finding 00:08:55.970 --> 00:08:59.270 align:middle line:84% a pair is an example of the pigeonhole principle. 00:08:59.270 --> 00:09:01.610 align:middle line:84% You just have to be careful about what are the pigeons 00:09:01.610 --> 00:09:02.710 align:middle line:90% and what are the holes. 00:09:02.710 --> 00:09:04.690 align:middle line:84% In this case, it's pretty obvious 00:09:04.690 --> 00:09:07.850 align:middle line:84% that the pigeons are individual socks 00:09:07.850 --> 00:09:11.098 align:middle line:84% and the holes are colors of different socks. 00:09:11.098 --> 00:09:12.890 align:middle line:84% But there are other applications that we're 00:09:12.890 --> 00:09:15.570 align:middle line:84% going to see where it's a little bit more subtle how exactly 00:09:15.570 --> 00:09:17.410 align:middle line:90% to get this principle to work. 00:09:17.410 --> 00:09:19.650 align:middle line:84% Let me tell you one generalization, which is also 00:09:19.650 --> 00:09:23.930 align:middle line:84% powerful, which is, in general, imagine I had way more pigeons 00:09:23.930 --> 00:09:25.450 align:middle line:90% than pigeonholes. 00:09:25.450 --> 00:09:27.570 align:middle line:84% It turns out you can guarantee that there's 00:09:27.570 --> 00:09:33.490 align:middle line:84% at least the ceiling of n over m pigeons in the same pigeonhole. 00:09:33.490 --> 00:09:35.748 align:middle line:90% So this is a very simple idea. 00:09:35.748 --> 00:09:37.290 align:middle line:84% Let's actually try and prove it, even 00:09:37.290 --> 00:09:39.690 align:middle line:90% though it might seem obvious. 00:09:39.690 --> 00:09:43.810 align:middle line:84% So it turns out there's a very simple way to prove it. 00:09:43.810 --> 00:09:45.890 align:middle line:84% This is one example of a proof technique 00:09:45.890 --> 00:09:48.930 align:middle line:84% we're going to see all throughout class, which 00:09:48.930 --> 00:09:51.390 align:middle line:90% is a proof by contradiction. 00:09:51.390 --> 00:09:54.530 align:middle line:90% 00:09:54.530 --> 00:09:56.210 align:middle line:84% And I'm going to try and practice 00:09:56.210 --> 00:09:58.450 align:middle line:84% what I preach in today's lecture and show you 00:09:58.450 --> 00:10:01.610 align:middle line:84% what are the right ways to write proofs. 00:10:01.610 --> 00:10:04.330 align:middle line:84% This won't be quite as formal as you do for your writing 00:10:04.330 --> 00:10:06.490 align:middle line:90% exercise, but pretty close. 00:10:06.490 --> 00:10:08.170 align:middle line:84% You can see that the first thing I'm 00:10:08.170 --> 00:10:10.130 align:middle line:84% doing when I write a proof by contradiction 00:10:10.130 --> 00:10:12.810 align:middle line:84% is I'm putting very clearly the fact 00:10:12.810 --> 00:10:14.850 align:middle line:84% that it's a proof by contradiction. 00:10:14.850 --> 00:10:16.250 align:middle line:84% You don't want your readers to be 00:10:16.250 --> 00:10:19.610 align:middle line:84% confused until the very end where the pieces come together. 00:10:19.610 --> 00:10:21.370 align:middle line:84% You want as much as possible that they 00:10:21.370 --> 00:10:23.210 align:middle line:84% see the blueprint of how the proof is 00:10:23.210 --> 00:10:27.010 align:middle line:84% supposed to go before they get through all of it, right? 00:10:27.010 --> 00:10:29.510 align:middle line:84% And so what we're going to do is we're going to suppose not. 00:10:29.510 --> 00:10:33.596 align:middle line:84% We'll suppose that the pigeonhole principle is false. 00:10:33.596 --> 00:10:37.810 align:middle line:84% Then what I can do is I can really just count 00:10:37.810 --> 00:10:40.020 align:middle line:84% the number of pigeons and the number of holes 00:10:40.020 --> 00:10:41.240 align:middle line:90% in two different ways. 00:10:41.240 --> 00:10:45.780 align:middle line:84% So what I can do is n is my number of pigeons. 00:10:45.780 --> 00:10:48.480 align:middle line:84% Now each pigeon has to be assigned to some hole. 00:10:48.480 --> 00:10:52.220 align:middle line:84% So I can sum over all of the different holes h 00:10:52.220 --> 00:10:53.940 align:middle line:90% that I'm interested in. 00:10:53.940 --> 00:10:57.940 align:middle line:84% And I can look at Nh3, where Nh3 here is just 00:10:57.940 --> 00:11:02.260 align:middle line:84% defined to be the number of pigeons that 00:11:02.260 --> 00:11:05.700 align:middle line:90% are in some particular hole h. 00:11:05.700 --> 00:11:08.020 align:middle line:84% So it's just the count of how many pigeons ended up 00:11:08.020 --> 00:11:09.620 align:middle line:90% in that hole. 00:11:09.620 --> 00:11:12.460 align:middle line:84% And all I'm doing is I'm just counting the different pigeons 00:11:12.460 --> 00:11:13.280 align:middle line:90% in a different way. 00:11:13.280 --> 00:11:15.500 align:middle line:90% So that's definitely equal to n. 00:11:15.500 --> 00:11:18.580 align:middle line:84% But now, by assumption, since the pigeonhole principle 00:11:18.580 --> 00:11:24.660 align:middle line:84% is false, I'm assuming that each one of the holes 00:11:24.660 --> 00:11:28.460 align:middle line:84% has at most one pigeon in it, right? 00:11:28.460 --> 00:11:32.260 align:middle line:84% And then the total number of holes I get is m. 00:11:32.260 --> 00:11:34.920 align:middle line:84% So now what I claim is I have a contradiction. 00:11:34.920 --> 00:11:35.660 align:middle line:90% Why? 00:11:35.660 --> 00:11:38.550 align:middle line:84% Because I've just shown for you that if the pigeonhole principle 00:11:38.550 --> 00:11:42.470 align:middle line:84% is false, then n is less than or equal to m. 00:11:42.470 --> 00:11:44.430 align:middle line:84% But what was my starting assumption 00:11:44.430 --> 00:11:48.230 align:middle line:84% was just that n is strictly larger than m. 00:11:48.230 --> 00:11:49.750 align:middle line:90% So this is my proof. 00:11:49.750 --> 00:11:53.450 align:middle line:90% But m is larger than n. 00:11:53.450 --> 00:11:55.110 align:middle line:90% That's one of the assumptions. 00:11:55.110 --> 00:11:56.770 align:middle line:90% And so we get a contradiction. 00:11:56.770 --> 00:12:02.550 align:middle line:90% 00:12:02.550 --> 00:12:03.330 align:middle line:90% That's it. 00:12:03.330 --> 00:12:05.477 align:middle line:84% That's our first proof in this class. 00:12:05.477 --> 00:12:07.310 align:middle line:84% This is an example of how you might write up 00:12:07.310 --> 00:12:08.570 align:middle line:90% a proof by contradiction. 00:12:08.570 --> 00:12:11.030 align:middle line:84% A lot of times in class they'll be a lot more complicated 00:12:11.030 --> 00:12:13.690 align:middle line:84% than this, but they all follow this basic blueprint. 00:12:13.690 --> 00:12:14.900 align:middle line:90% Yes? 00:12:14.900 --> 00:12:16.150 align:middle line:90% AUDIENCE: Is n greater than m? 00:12:16.150 --> 00:12:17.510 align:middle line:90% ANKUR MOITRA: Ah, sorry. 00:12:17.510 --> 00:12:18.610 align:middle line:90% n is greater than m. 00:12:18.610 --> 00:12:22.830 align:middle line:90% Thank you n is greater than m. 00:12:22.830 --> 00:12:24.610 align:middle line:84% Any other questions, or that was it? 00:12:24.610 --> 00:12:25.530 align:middle line:90% OK, good. 00:12:25.530 --> 00:12:28.470 align:middle line:84% Feel free to interject any time with questions. 00:12:28.470 --> 00:12:28.970 align:middle line:90% All right. 00:12:28.970 --> 00:12:31.370 align:middle line:84% Let me show you one other principle in action. 00:12:31.370 --> 00:12:33.390 align:middle line:84% So let's talk about some cooler applications 00:12:33.390 --> 00:12:35.480 align:middle line:84% of the pigeonhole principle, now that we 00:12:35.480 --> 00:12:37.860 align:middle line:84% know what it is and we know why it's true, 00:12:37.860 --> 00:12:42.000 align:middle line:84% and we've seen some simple illustrations of it. 00:12:42.000 --> 00:12:46.200 align:middle line:84% What I claim is that a lot of times, writing a good proof 00:12:46.200 --> 00:12:49.360 align:middle line:84% is also about finding the right abstraction for how 00:12:49.360 --> 00:12:52.240 align:middle line:84% to think about something, how to model it. 00:12:52.240 --> 00:12:56.880 align:middle line:84% So let me give you a simple lemma that we're going to prove. 00:12:56.880 --> 00:12:59.160 align:middle line:84% So first, this lemma isn't going to sound 00:12:59.160 --> 00:13:01.360 align:middle line:84% like the pigeonhole principle, but we'll turn it 00:13:01.360 --> 00:13:03.520 align:middle line:90% into the pigeonhole principle. 00:13:03.520 --> 00:13:06.100 align:middle line:84% Let's imagine you're going to a party. 00:13:06.100 --> 00:13:09.600 align:middle line:84% Let's say you're going to the Super Bowl party this weekend. 00:13:09.600 --> 00:13:13.520 align:middle line:90% And you've got n people. 00:13:13.520 --> 00:13:15.640 align:middle line:84% There are n total people, including yourself, 00:13:15.640 --> 00:13:16.880 align:middle line:90% at the party. 00:13:16.880 --> 00:13:21.620 align:middle line:84% What I claim is that no matter who's there, 00:13:21.620 --> 00:13:24.840 align:middle line:84% there are always two people, at least 00:13:24.840 --> 00:13:36.600 align:middle line:84% two people who know the same number of other people. 00:13:36.600 --> 00:13:39.610 align:middle line:90% 00:13:39.610 --> 00:13:43.410 align:middle line:84% This is a mathematical fact about parties. 00:13:43.410 --> 00:13:46.130 align:middle line:84% This doesn't quite look like the pigeonhole principle, 00:13:46.130 --> 00:13:47.770 align:middle line:90% but I claim it is. 00:13:47.770 --> 00:13:49.730 align:middle line:84% And moreover, what we really need 00:13:49.730 --> 00:13:52.700 align:middle line:84% is we need the right way to think about this. 00:13:52.700 --> 00:13:57.330 align:middle line:84% And we'll practice this type of skill a lot in recitations. 00:13:57.330 --> 00:13:59.210 align:middle line:84% This is an example of something which 00:13:59.210 --> 00:14:01.490 align:middle line:84% is a lot easier to prove when you have the right way 00:14:01.490 --> 00:14:03.250 align:middle line:90% to visualize it. 00:14:03.250 --> 00:14:07.450 align:middle line:84% So it turns out that the right way to visualize it 00:14:07.450 --> 00:14:11.250 align:middle line:90% is to draw a graph. 00:14:11.250 --> 00:14:13.690 align:middle line:84% Namely, something with nodes that 00:14:13.690 --> 00:14:15.430 align:middle line:90% represent people at the party. 00:14:15.430 --> 00:14:18.130 align:middle line:84% So there are n different nodes in my graph. 00:14:18.130 --> 00:14:20.130 align:middle line:84% And then there are edges that represent 00:14:20.130 --> 00:14:23.090 align:middle line:84% different relationships, whether the two people know each other 00:14:23.090 --> 00:14:24.370 align:middle line:90% or not. 00:14:24.370 --> 00:14:28.170 align:middle line:84% So for example, I could have that the people in my party 00:14:28.170 --> 00:14:30.550 align:middle line:84% look like they define this graph. 00:14:30.550 --> 00:14:34.340 align:middle line:84% I've got four people in my party n equals 4. 00:14:34.340 --> 00:14:39.140 align:middle line:84% Again, as I mentioned, each of these nodes are people. 00:14:39.140 --> 00:14:40.980 align:middle line:90% They're individual people. 00:14:40.980 --> 00:14:46.640 align:middle line:84% And these edges represent people who know each other. 00:14:46.640 --> 00:14:50.300 align:middle line:90% 00:14:50.300 --> 00:14:53.192 align:middle line:84% And then when I draw in all my edges, 00:14:53.192 --> 00:14:56.380 align:middle line:84% maybe I'll end up with a graph like this. 00:14:56.380 --> 00:14:58.620 align:middle line:84% And you can see that there are two people who 00:14:58.620 --> 00:15:02.100 align:middle line:84% know the same number of people at the party, like this node 00:15:02.100 --> 00:15:04.140 align:middle line:84% right here, and this node right here. 00:15:04.140 --> 00:15:06.260 align:middle line:84% The number of edges that are incident 00:15:06.260 --> 00:15:09.040 align:middle line:84% to that node, which represents how many other people they know 00:15:09.040 --> 00:15:10.580 align:middle line:90% is both two. 00:15:10.580 --> 00:15:13.580 align:middle line:84% So my lemma is true in this simple example. 00:15:13.580 --> 00:15:16.140 align:middle line:84% And now I can understand, through this visual, 00:15:16.140 --> 00:15:19.780 align:middle line:84% how exactly a graph relates to this lemma 00:15:19.780 --> 00:15:23.300 align:middle line:90% that I asserted to begin with. 00:15:23.300 --> 00:15:25.620 align:middle line:84% So now we can actually prove it once we 00:15:25.620 --> 00:15:28.380 align:middle line:90% have this abstraction in hand. 00:15:28.380 --> 00:15:31.710 align:middle line:84% So let's talk about how that works. 00:15:31.710 --> 00:15:35.110 align:middle line:84% Let me define one more quantity which will be of interest for us 00:15:35.110 --> 00:15:37.750 align:middle line:90% today just called the degree. 00:15:37.750 --> 00:15:41.370 align:middle line:84% That's just the number of edges that are incident to a node. 00:15:41.370 --> 00:15:46.950 align:middle line:90% 00:15:46.950 --> 00:16:00.010 align:middle line:84% So the degree of a node is the number of edges incident to it. 00:16:00.010 --> 00:16:04.390 align:middle line:90% 00:16:04.390 --> 00:16:08.350 align:middle line:84% That share an end point, that have that node as an point. 00:16:08.350 --> 00:16:13.190 align:middle line:84% So now let's try and prove this using the pigeonhole principle. 00:16:13.190 --> 00:16:14.967 align:middle line:84% In fact, the crucial thing in applying 00:16:14.967 --> 00:16:17.550 align:middle line:84% the pigeonhole principle is to figure out what are the pigeons 00:16:17.550 --> 00:16:19.870 align:middle line:90% and what are the holes. 00:16:19.870 --> 00:16:22.810 align:middle line:84% Does anyone have any idea what we should make the pigeons be, 00:16:22.810 --> 00:16:24.613 align:middle line:84% and what we should make the holes be 00:16:24.613 --> 00:16:25.780 align:middle line:90% to try and prove this lemma? 00:16:25.780 --> 00:16:31.520 align:middle line:90% 00:16:31.520 --> 00:16:32.240 align:middle line:90% Yeah? 00:16:32.240 --> 00:16:38.080 align:middle line:84% AUDIENCE: Maybe nodes are the holes and the connections are-- 00:16:38.080 --> 00:16:39.340 align:middle line:90% ANKUR MOITRA: OK. 00:16:39.340 --> 00:16:41.960 align:middle line:90% We could try that way. 00:16:41.960 --> 00:16:44.260 align:middle line:84% I might have a little bit of difficulty that way. 00:16:44.260 --> 00:16:47.040 align:middle line:90% Yeah? 00:16:47.040 --> 00:16:48.540 align:middle line:90% Yeah, exactly. 00:16:48.540 --> 00:16:49.280 align:middle line:90% OK. 00:16:49.280 --> 00:16:51.820 align:middle line:84% Because where do I want the collision? 00:16:51.820 --> 00:16:54.980 align:middle line:84% So the collision happens between two pigeons 00:16:54.980 --> 00:16:56.640 align:middle line:90% who land in the same hole. 00:16:56.640 --> 00:17:00.360 align:middle line:84% So what I want is I want two people to collide. 00:17:00.360 --> 00:17:02.020 align:middle line:84% And what does it mean to collide? 00:17:02.020 --> 00:17:04.640 align:middle line:84% I want them to have the same degree. 00:17:04.640 --> 00:17:06.520 align:middle line:84% So that tells me what should be the pigeons 00:17:06.520 --> 00:17:08.480 align:middle line:90% and what should be the holes. 00:17:08.480 --> 00:17:14.369 align:middle line:84% So what I claim is that the degree of each node. 00:17:14.369 --> 00:17:17.560 align:middle line:90% 00:17:17.560 --> 00:17:20.640 align:middle line:84% Well, first of all, it's an integer 00:17:20.640 --> 00:17:29.090 align:middle line:84% that's in this set from 0, 1, all the way up to n minus 1. 00:17:29.090 --> 00:17:37.556 align:middle line:84% And each node is going to be this pigeon. 00:17:37.556 --> 00:17:40.570 align:middle line:84% And the way that I'm going to choose which particular 00:17:40.570 --> 00:17:45.610 align:middle line:84% hole it gets mapped to is what the associated degree is. 00:17:45.610 --> 00:17:46.930 align:middle line:90% So I have a hole. 00:17:46.930 --> 00:17:51.570 align:middle line:84% Each name is an integer from up to n minus 1. 00:17:51.570 --> 00:17:53.290 align:middle line:84% And then each particular node that 00:17:53.290 --> 00:17:56.490 align:middle line:84% represents a person is going to get mapped to whatever 00:17:56.490 --> 00:17:59.050 align:middle line:90% their degree is. 00:17:59.050 --> 00:18:02.650 align:middle line:90% So sounds like I'm done. 00:18:02.650 --> 00:18:03.710 align:middle line:90% Am I done here? 00:18:03.710 --> 00:18:05.830 align:middle line:84% Can I just appeal to the pigeonhole principle 00:18:05.830 --> 00:18:07.640 align:middle line:84% and say that two nodes must collide? 00:18:07.640 --> 00:18:12.410 align:middle line:90% 00:18:12.410 --> 00:18:13.290 align:middle line:90% Yeah? 00:18:13.290 --> 00:18:18.345 align:middle line:84% AUDIENCE: [INAUDIBLE] the nodes and the set of possible degrees. 00:18:18.345 --> 00:18:19.470 align:middle line:90% ANKUR MOITRA: That's right. 00:18:19.470 --> 00:18:22.410 align:middle line:84% So I'm definitely not done right now, 00:18:22.410 --> 00:18:24.260 align:middle line:84% because the pigeonhole principle only 00:18:24.260 --> 00:18:27.220 align:middle line:84% works when n is strictly larger than m. 00:18:27.220 --> 00:18:29.500 align:middle line:84% That's when I picked out six socks from my drawer 00:18:29.500 --> 00:18:31.180 align:middle line:90% and I had five colors. 00:18:31.180 --> 00:18:34.420 align:middle line:84% I really needed one more pigeon than the number of holes. 00:18:34.420 --> 00:18:35.720 align:middle line:90% And the trouble is here. 00:18:35.720 --> 00:18:37.200 align:middle line:90% How many pigeons do I have? 00:18:37.200 --> 00:18:39.060 align:middle line:84% Well, it's each node in the graph. 00:18:39.060 --> 00:18:41.860 align:middle line:84% That's the number of the people at the party that's n. 00:18:41.860 --> 00:18:43.200 align:middle line:90% How many holes do I have? 00:18:43.200 --> 00:18:45.740 align:middle line:84% Well, unfortunately, that's also n 00:18:45.740 --> 00:18:49.980 align:middle line:84% because it's the integers from 0 to n minus 1. 00:18:49.980 --> 00:18:52.540 align:middle line:90% But now we need a clever idea. 00:18:52.540 --> 00:18:57.540 align:middle line:84% So I wonder, is it possible to have a node of degree 0 00:18:57.540 --> 00:19:00.670 align:middle line:84% and a node of degree n minus 1 at the same time? 00:19:00.670 --> 00:19:04.340 align:middle line:90% 00:19:04.340 --> 00:19:05.420 align:middle line:90% So who can help me out? 00:19:05.420 --> 00:19:05.920 align:middle line:90% Yeah? 00:19:05.920 --> 00:19:10.197 align:middle line:84% AUDIENCE: If you know n people, that means everybody. 00:19:10.197 --> 00:19:11.780 align:middle line:84% Then there's a contradiction because-- 00:19:11.780 --> 00:19:12.988 align:middle line:90% ANKUR MOITRA: That's perfect. 00:19:12.988 --> 00:19:13.800 align:middle line:90% Exactly right. 00:19:13.800 --> 00:19:16.020 align:middle line:84% Let me say it again because I have the mic. 00:19:16.020 --> 00:19:17.820 align:middle line:84% So if there's someone at the party who 00:19:17.820 --> 00:19:21.450 align:middle line:84% knows n minus 1 people, well, there's n nodes, 00:19:21.450 --> 00:19:24.350 align:middle line:84% and that means he's connected to every other node. 00:19:24.350 --> 00:19:26.230 align:middle line:84% And that means there isn't anyone 00:19:26.230 --> 00:19:28.430 align:middle line:90% at the party who knows no one. 00:19:28.430 --> 00:19:33.910 align:middle line:84% So actually, these two endpoints are mutually exclusive. 00:19:33.910 --> 00:19:37.270 align:middle line:90% They can't both happen. 00:19:37.270 --> 00:19:40.090 align:middle line:84% So if I wanted to write this out formally as a proof, 00:19:40.090 --> 00:19:43.230 align:middle line:84% I would actually have to do a case analysis here. 00:19:43.230 --> 00:19:46.590 align:middle line:84% In order to create my contradiction, I would say, 00:19:46.590 --> 00:19:50.710 align:middle line:84% case number one, there is a node of degree n minus 1, 00:19:50.710 --> 00:19:54.310 align:middle line:84% in which case that means there is no node of degree 0 00:19:54.310 --> 00:19:58.248 align:middle line:84% and I'm back down to having n minus 1 holes and n pigeons, 00:19:58.248 --> 00:20:00.290 align:middle line:84% and then I can appeal to the pigeonhole principle 00:20:00.290 --> 00:20:01.670 align:middle line:90% and call it a day. 00:20:01.670 --> 00:20:05.710 align:middle line:84% And in case two where there's no node of degree n minus 1, 00:20:05.710 --> 00:20:10.310 align:middle line:84% again, I only have n minus 1 possibilities for the holes. 00:20:10.310 --> 00:20:13.430 align:middle line:84% So I'm not going to be able to write up the full blown proofs 00:20:13.430 --> 00:20:15.230 align:middle line:90% when we do them in lecture. 00:20:15.230 --> 00:20:16.990 align:middle line:84% But you can see that to really make 00:20:16.990 --> 00:20:19.880 align:middle line:84% this a full bulletproof proof, there 00:20:19.880 --> 00:20:22.880 align:middle line:84% are a lot of steps that go into how exactly you can lay out 00:20:22.880 --> 00:20:25.000 align:middle line:90% the proof. 00:20:25.000 --> 00:20:29.160 align:middle line:84% So I'll just stop here and say that can't 00:20:29.160 --> 00:20:37.080 align:middle line:90% be both degree n minus 1 and 0. 00:20:37.080 --> 00:20:40.880 align:middle line:84% And that lets us appeal to the pigeonhole principle, 00:20:40.880 --> 00:20:41.470 align:middle line:90% so we're good. 00:20:41.470 --> 00:20:43.720 align:middle line:84% So this is another example of the pigeonhole principle 00:20:43.720 --> 00:20:46.498 align:middle line:84% in action, even though it doesn't look like it at first 00:20:46.498 --> 00:20:48.040 align:middle line:84% because you have to figure out what's 00:20:48.040 --> 00:20:50.880 align:middle line:84% the right abstraction-- namely a graph for how 00:20:50.880 --> 00:20:54.120 align:middle line:84% to think about and visualize what's actually happening 00:20:54.120 --> 00:20:56.400 align:middle line:90% underneath this lemma. 00:20:56.400 --> 00:20:57.940 align:middle line:90% Let's do another example. 00:20:57.940 --> 00:21:01.880 align:middle line:84% We're going to get harder and harder, unfortunately. 00:21:01.880 --> 00:21:04.320 align:middle line:84% So let me give you another of my favorite examples 00:21:04.320 --> 00:21:08.280 align:middle line:84% of the pigeonhole principle in action. 00:21:08.280 --> 00:21:10.660 align:middle line:84% Which is a very easy to state problem, 00:21:10.660 --> 00:21:14.520 align:middle line:84% but turns out to be very difficult from a complexity 00:21:14.520 --> 00:21:15.500 align:middle line:90% standpoint. 00:21:15.500 --> 00:21:19.250 align:middle line:90% 00:21:19.250 --> 00:21:23.330 align:middle line:84% So let's imagine, let me tell you the statement of this lemma 00:21:23.330 --> 00:21:26.130 align:middle line:84% and then we'll get into how exactly to prove it. 00:21:26.130 --> 00:21:34.810 align:middle line:84% Suppose I give you 20 four digit numbers. 00:21:34.810 --> 00:21:35.760 align:middle line:90% Positive numbers. 00:21:35.760 --> 00:21:41.810 align:middle line:90% 00:21:41.810 --> 00:21:47.130 align:middle line:84% Let's just call them x1 up to x20. 00:21:47.130 --> 00:21:56.750 align:middle line:84% Then what I claim is that there are two disjoint sets of them. 00:21:56.750 --> 00:21:59.850 align:middle line:90% 00:21:59.850 --> 00:22:06.510 align:middle line:84% And I'll say specifically nonempty sets with the same sum. 00:22:06.510 --> 00:22:10.010 align:middle line:90% 00:22:10.010 --> 00:22:12.050 align:middle line:90% So what am I asking here? 00:22:12.050 --> 00:22:14.910 align:middle line:84% So I give you any 20 four digit numbers. 00:22:14.910 --> 00:22:17.380 align:middle line:84% They have to be strictly positive. 00:22:17.380 --> 00:22:20.100 align:middle line:84% And what I'm going to do is I'm going 00:22:20.100 --> 00:22:25.780 align:middle line:84% to take some subset of them, maybe x1 plus x3 plus x5, 00:22:25.780 --> 00:22:27.900 align:middle line:84% and then I'll take some other subset of them 00:22:27.900 --> 00:22:31.740 align:middle line:84% that doesn't overlap in any of those integers, x2 00:22:31.740 --> 00:22:34.020 align:middle line:90% and x4 and x6. 00:22:34.020 --> 00:22:36.520 align:middle line:84% And what I claim is that there's a way of doing this, 00:22:36.520 --> 00:22:39.460 align:middle line:84% so that the sum of the integers in one set 00:22:39.460 --> 00:22:43.740 align:middle line:84% is equal to the sum of the integers in the other set. 00:22:43.740 --> 00:22:46.740 align:middle line:84% Now, moreover, this would be a trivial lemma 00:22:46.740 --> 00:22:49.100 align:middle line:84% if I let these sets be empty, because I could just 00:22:49.100 --> 00:22:52.380 align:middle line:84% take the empty set for one and the empty set for the other. 00:22:52.380 --> 00:22:53.560 align:middle line:90% They're definitely disjoint. 00:22:53.560 --> 00:22:55.380 align:middle line:84% They don't share anything in common. 00:22:55.380 --> 00:22:59.100 align:middle line:84% And their sum is zero, so this would be a really boring way 00:22:59.100 --> 00:23:00.740 align:middle line:90% for this lemma to be true. 00:23:00.740 --> 00:23:02.580 align:middle line:84% What I'm saying by nonempty is I'm 00:23:02.580 --> 00:23:04.280 align:middle line:90% ruling out that trivial case. 00:23:04.280 --> 00:23:05.780 align:middle line:84% There's always an interesting way 00:23:05.780 --> 00:23:08.140 align:middle line:84% to do this where you've really created 00:23:08.140 --> 00:23:13.110 align:middle line:84% a positive sum for these integers two different ways. 00:23:13.110 --> 00:23:13.880 align:middle line:90% All right. 00:23:13.880 --> 00:23:15.630 align:middle line:84% So let's try and prove this, and then I'll 00:23:15.630 --> 00:23:18.990 align:middle line:84% tell you some interesting facts about this very simple 00:23:18.990 --> 00:23:20.710 align:middle line:90% statement. 00:23:20.710 --> 00:23:23.310 align:middle line:84% And again, what we have to do to write up 00:23:23.310 --> 00:23:25.990 align:middle line:84% a proof here is we have to do the usual things 00:23:25.990 --> 00:23:28.730 align:middle line:84% where we have to figure out what the pigeons are, 00:23:28.730 --> 00:23:30.430 align:middle line:90% what the holes are. 00:23:30.430 --> 00:23:32.190 align:middle line:84% But moreover, we have to figure out 00:23:32.190 --> 00:23:35.750 align:middle line:84% the right mathematical notation for expressing what's 00:23:35.750 --> 00:23:39.065 align:middle line:90% going on in this problem. 00:23:39.065 --> 00:23:43.030 align:middle line:84% And so, in fact, a lot of times the first step 00:23:43.030 --> 00:23:46.030 align:middle line:84% is really to figure out some mathematical way 00:23:46.030 --> 00:23:48.550 align:middle line:84% to express the statement right here 00:23:48.550 --> 00:23:50.830 align:middle line:84% that I've written out in English. 00:23:50.830 --> 00:23:53.910 align:middle line:84% So another way to think about this 00:23:53.910 --> 00:23:58.510 align:middle line:84% is that I claim there are sets I and J, 00:23:58.510 --> 00:24:03.270 align:middle line:84% which are subsets of the integers from 1 up to 20. 00:24:03.270 --> 00:24:05.610 align:middle line:84% What are these subsets supposed to represent? 00:24:05.610 --> 00:24:09.270 align:middle line:84% Just which of the numbers I'm taking in my sum. 00:24:09.270 --> 00:24:14.340 align:middle line:84% I want that these sets are both nonempty. 00:24:14.340 --> 00:24:19.160 align:middle line:84% So I and J had better not be equal to the empty set. 00:24:19.160 --> 00:24:21.320 align:middle line:84% I need that they're disjoint, which 00:24:21.320 --> 00:24:25.400 align:middle line:84% just says that the intersection between I and J 00:24:25.400 --> 00:24:27.960 align:middle line:90% is equal to the empty set. 00:24:27.960 --> 00:24:29.640 align:middle line:84% So this is just the mathematical way 00:24:29.640 --> 00:24:32.680 align:middle line:84% of expressing all of the things I have here. 00:24:32.680 --> 00:24:35.640 align:middle line:84% And now for the most important piece of the puzzle, 00:24:35.640 --> 00:24:41.480 align:middle line:84% I want that the sum over all of the x sub I's in this first set 00:24:41.480 --> 00:24:48.560 align:middle line:84% is equal to the sum of all of the x sub J's in the second set. 00:24:48.560 --> 00:24:51.720 align:middle line:84% So a lot of times it's hard to write a mathematical proof 00:24:51.720 --> 00:24:54.280 align:middle line:84% unless you have the right mathematical notation 00:24:54.280 --> 00:24:55.640 align:middle line:90% to begin with. 00:24:55.640 --> 00:24:58.560 align:middle line:84% Because even though I could have expressed this in English 00:24:58.560 --> 00:25:00.780 align:middle line:84% now to actually prove this statement, 00:25:00.780 --> 00:25:04.080 align:middle line:84% I'm going to have to do a bunch of manipulations on these sets. 00:25:04.080 --> 00:25:07.880 align:middle line:84% And this will be a lot easier with this notation in hand. 00:25:07.880 --> 00:25:09.330 align:middle line:84% So another thing that we're going 00:25:09.330 --> 00:25:11.730 align:middle line:84% to teach you a lot in recitation is how exactly 00:25:11.730 --> 00:25:15.730 align:middle line:84% to set up convenient notation so that your proofs will be easier 00:25:15.730 --> 00:25:16.405 align:middle line:90% to write. 00:25:16.405 --> 00:25:20.210 align:middle line:90% 00:25:20.210 --> 00:25:22.360 align:middle line:84% So now how exactly are we going to prove this? 00:25:22.360 --> 00:25:28.570 align:middle line:90% 00:25:28.570 --> 00:25:30.278 align:middle line:84% Let's see if you guys have any intuition. 00:25:30.278 --> 00:25:32.820 align:middle line:84% So obviously we're going to be using the pigeonhole principle 00:25:32.820 --> 00:25:35.490 align:middle line:84% because I've only taught you one thing so far, so we better 00:25:35.490 --> 00:25:36.970 align:middle line:90% use it. 00:25:36.970 --> 00:25:42.370 align:middle line:84% But what should we use as the pigeons? 00:25:42.370 --> 00:25:44.090 align:middle line:90% Yeah? 00:25:44.090 --> 00:25:51.010 align:middle line:84% AUDIENCE: The sum is divided by 200,000. 00:25:51.010 --> 00:25:51.970 align:middle line:90% ANKUR MOITRA: OK. 00:25:51.970 --> 00:25:52.710 align:middle line:90% You're right. 00:25:52.710 --> 00:25:54.490 align:middle line:90% But for the holes, I think. 00:25:54.490 --> 00:25:55.350 align:middle line:90% So let's be careful. 00:25:55.350 --> 00:25:55.490 align:middle line:90% Yeah. 00:25:55.490 --> 00:25:57.070 align:middle line:84% So I'll take your answer for the holes, 00:25:57.070 --> 00:25:58.445 align:middle line:84% because the way to think about it 00:25:58.445 --> 00:26:01.250 align:middle line:84% is that we want a collision among the pigeons. 00:26:01.250 --> 00:26:04.177 align:middle line:84% So we really want the collision among what objects? 00:26:04.177 --> 00:26:05.510 align:middle line:90% Was there an answer in the back? 00:26:05.510 --> 00:26:07.980 align:middle line:90% Yeah? 00:26:07.980 --> 00:26:09.380 align:middle line:90% Yeah. 00:26:09.380 --> 00:26:10.180 align:middle line:90% I like that. 00:26:10.180 --> 00:26:12.580 align:middle line:84% I'm going to strike number of and let's 00:26:12.580 --> 00:26:14.740 align:middle line:84% just say subsets, because really, 00:26:14.740 --> 00:26:17.040 align:middle line:84% it's the subsets that I care about. 00:26:17.040 --> 00:26:18.420 align:middle line:90% Each one of them is a pigeon. 00:26:18.420 --> 00:26:19.540 align:middle line:90% But perfect. 00:26:19.540 --> 00:26:20.220 align:middle line:90% All right. 00:26:20.220 --> 00:26:22.900 align:middle line:84% So that's exactly what the pigeons will be. 00:26:22.900 --> 00:26:26.580 align:middle line:84% They'll just be the appropriate subsets. 00:26:26.580 --> 00:26:31.220 align:middle line:84% So in particular, I want them to be nonempty subsets. 00:26:31.220 --> 00:26:33.460 align:middle line:84% That's definitely one of the constraints 00:26:33.460 --> 00:26:37.340 align:middle line:84% I have in this constraint on I and J here. 00:26:37.340 --> 00:26:40.280 align:middle line:84% I'll have to worry about this disjointness constraint later, 00:26:40.280 --> 00:26:42.220 align:middle line:90% and we'll get to that. 00:26:42.220 --> 00:26:44.680 align:middle line:84% And then exactly as you guys already mentioned, 00:26:44.680 --> 00:26:47.500 align:middle line:84% the holes are going to be the possible sums. 00:26:47.500 --> 00:26:50.220 align:middle line:90% That's exactly right. 00:26:50.220 --> 00:26:53.540 align:middle line:84% So what can we say about the possible sums? 00:26:53.540 --> 00:26:57.500 align:middle line:84% So let's say that the number of possible sums-- 00:26:57.500 --> 00:26:59.300 align:middle line:90% let's just call that m. 00:26:59.300 --> 00:27:02.140 align:middle line:84% I can just come up with an upper bound, because I just 00:27:02.140 --> 00:27:03.780 align:middle line:84% need to show that the number of holes 00:27:03.780 --> 00:27:06.580 align:middle line:84% is smaller than the number of pigeons. 00:27:06.580 --> 00:27:09.540 align:middle line:84% And definitely an upper bound on this 00:27:09.540 --> 00:27:13.820 align:middle line:84% is the largest that the sum can be, which is the sum from I 00:27:13.820 --> 00:27:17.220 align:middle line:90% equals 1 to 20 of x sub i. 00:27:17.220 --> 00:27:24.580 align:middle line:84% And this is comfortably less than 20 times 00:27:24.580 --> 00:27:29.220 align:middle line:84% 10 to the 4, which is strictly less than 10 to the 6. 00:27:29.220 --> 00:27:30.400 align:middle line:90% Was there a question here? 00:27:30.400 --> 00:27:30.900 align:middle line:90% No? 00:27:30.900 --> 00:27:31.820 align:middle line:90% OK. 00:27:31.820 --> 00:27:32.660 align:middle line:90% All right. 00:27:32.660 --> 00:27:34.980 align:middle line:84% So now we're in business because we can try and appeal 00:27:34.980 --> 00:27:37.230 align:middle line:84% to the pigeonhole principle again. 00:27:37.230 --> 00:27:40.073 align:middle line:84% But we have to be a little bit careful here. 00:27:40.073 --> 00:27:46.060 align:middle line:84% So what this is going to do is that the number of nonempty 00:27:46.060 --> 00:27:53.580 align:middle line:84% subsets, well, this is equal to 2 to the 20 minus 1. 00:27:53.580 --> 00:27:56.380 align:middle line:84% So the way to think about this-- and we'll talk more about this 00:27:56.380 --> 00:27:58.060 align:middle line:90% when we get into counting-- 00:27:58.060 --> 00:28:00.080 align:middle line:84% is really through what's called a bijection. 00:28:00.080 --> 00:28:02.540 align:middle line:84% We'll do lots of examples of bijections. 00:28:02.540 --> 00:28:04.310 align:middle line:84% But the way to think about that is 00:28:04.310 --> 00:28:08.870 align:middle line:84% that each subset of the integers from 1 to 20 00:28:08.870 --> 00:28:11.390 align:middle line:84% is really associated with a bit string. 00:28:11.390 --> 00:28:13.650 align:middle line:84% The first coordinate, which is 1 or 0, 00:28:13.650 --> 00:28:16.870 align:middle line:84% tells you whether the first element is in the set or not. 00:28:16.870 --> 00:28:19.070 align:middle line:84% The second entry in that string tells you 00:28:19.070 --> 00:28:22.150 align:middle line:84% whether the second element is in it, and so on. 00:28:22.150 --> 00:28:25.210 align:middle line:84% So a priori, there are 2 to the 20 possibilities, 00:28:25.210 --> 00:28:27.150 align:middle line:84% because for each coordinate I have 00:28:27.150 --> 00:28:29.670 align:middle line:84% the choice of whether I include it or don't include it 00:28:29.670 --> 00:28:31.110 align:middle line:90% in my subset. 00:28:31.110 --> 00:28:35.390 align:middle line:84% This minus 1 is because I want the subset to be nonempty. 00:28:35.390 --> 00:28:39.630 align:middle line:84% So I have to rule out the subset that takes nothing. 00:28:39.630 --> 00:28:44.870 align:middle line:84% So this is still comfortably much larger than 10 to the 6. 00:28:44.870 --> 00:28:48.630 align:middle line:84% So now I'm in good shape because n is strictly larger than m, 00:28:48.630 --> 00:28:50.510 align:middle line:90% but there's a catch. 00:28:50.510 --> 00:28:53.350 align:middle line:90% So what exactly does this imply? 00:28:53.350 --> 00:28:56.390 align:middle line:84% It implies when I use the pigeonhole principle 00:28:56.390 --> 00:29:05.200 align:middle line:84% that there exists nonempty A which is not equal to B, 00:29:05.200 --> 00:29:07.240 align:middle line:84% because they are different pigeons that 00:29:07.240 --> 00:29:09.320 align:middle line:90% end up in the same pigeonhole. 00:29:09.320 --> 00:29:11.600 align:middle line:90% And yet, the sum is equal. 00:29:11.600 --> 00:29:16.160 align:middle line:84% The sum of all of the xi's in the set A 00:29:16.160 --> 00:29:23.520 align:middle line:84% is equal to the sum of all of the xj's in the set B. 00:29:23.520 --> 00:29:25.600 align:middle line:90% So I'm not done yet. 00:29:25.600 --> 00:29:27.840 align:middle line:90% Why am I not done? 00:29:27.840 --> 00:29:29.493 align:middle line:90% Yes? 00:29:29.493 --> 00:29:31.160 align:middle line:84% AUDIENCE: Because it says might not be-- 00:29:31.160 --> 00:29:33.035 align:middle line:84% ANKUR MOITRA: The sets might not be disjoint. 00:29:33.035 --> 00:29:36.160 align:middle line:84% So I haven't proven what I promised you because I 00:29:36.160 --> 00:29:38.000 align:middle line:90% didn't meet this condition. 00:29:38.000 --> 00:29:41.720 align:middle line:84% Also notice something about how I wrote this proof. 00:29:41.720 --> 00:29:46.412 align:middle line:90% So I did not use I and J here. 00:29:46.412 --> 00:29:48.640 align:middle line:84% Because at the end of the day, what I want to show 00:29:48.640 --> 00:29:51.480 align:middle line:84% is that there exists this I and J, 00:29:51.480 --> 00:29:53.920 align:middle line:84% and I already know that I'm not done. 00:29:53.920 --> 00:29:57.080 align:middle line:84% So I left myself some slack in the notation 00:29:57.080 --> 00:30:00.690 align:middle line:84% for me to later choose what I and J should be. 00:30:00.690 --> 00:30:02.830 align:middle line:84% If I had made these things be I and J, 00:30:02.830 --> 00:30:06.725 align:middle line:84% I would have really confused my audience. 00:30:06.725 --> 00:30:09.990 align:middle line:84% So how exactly can we now make these things disjoint? 00:30:09.990 --> 00:30:12.090 align:middle line:90% I claim we're almost home. 00:30:12.090 --> 00:30:15.010 align:middle line:84% We just have to fix what A and B are. 00:30:15.010 --> 00:30:17.050 align:middle line:84% So anyone who hasn't answered yet 00:30:17.050 --> 00:30:21.410 align:middle line:84% have a fix for how I can fix A and B? 00:30:21.410 --> 00:30:24.810 align:middle line:84% So I've gotten most of it that are nonempty. 00:30:24.810 --> 00:30:25.750 align:middle line:90% Have you answered yet? 00:30:25.750 --> 00:30:26.250 align:middle line:90% No? 00:30:26.250 --> 00:30:26.750 align:middle line:90% OK. 00:30:26.750 --> 00:30:27.890 align:middle line:90% All right. 00:30:27.890 --> 00:30:29.850 align:middle line:84% AUDIENCE: Take the intersection of A and B 00:30:29.850 --> 00:30:32.685 align:middle line:84% and you remove that common section from both sets. 00:30:32.685 --> 00:30:33.810 align:middle line:90% ANKUR MOITRA: That's right. 00:30:33.810 --> 00:30:34.750 align:middle line:90% That's right. 00:30:34.750 --> 00:30:37.130 align:middle line:84% So here, let's think about a baby version of this first 00:30:37.130 --> 00:30:40.690 align:middle line:84% before we explain it with all of the full blown mathematical 00:30:40.690 --> 00:30:41.930 align:middle line:90% notation. 00:30:41.930 --> 00:30:44.850 align:middle line:84% A and B might not be disjoint because maybe they 00:30:44.850 --> 00:30:46.930 align:middle line:90% share one element in common. 00:30:46.930 --> 00:30:49.850 align:middle line:90% Maybe they both contain x1. 00:30:49.850 --> 00:30:52.770 align:middle line:84% But then the point is that these two sums are equal 00:30:52.770 --> 00:30:55.050 align:middle line:84% and they both have an x1 in them. 00:30:55.050 --> 00:30:57.340 align:middle line:84% So I can just remove that common element, 00:30:57.340 --> 00:31:00.140 align:middle line:84% and then I'm removing that x1 from the sum. 00:31:00.140 --> 00:31:02.060 align:middle line:84% When I take it out, I've made the sets 00:31:02.060 --> 00:31:04.060 align:middle line:84% be disjoint because they no longer share 00:31:04.060 --> 00:31:05.700 align:middle line:90% that element in common. 00:31:05.700 --> 00:31:07.300 align:middle line:84% And yet, I haven't changed the fact 00:31:07.300 --> 00:31:09.340 align:middle line:84% that the sums are equal, even though I've 00:31:09.340 --> 00:31:12.500 align:middle line:90% changed the sums themselves. 00:31:12.500 --> 00:31:14.240 align:middle line:90% So that's the rest of the proof. 00:31:14.240 --> 00:31:15.440 align:middle line:90% You're exactly right. 00:31:15.440 --> 00:31:17.470 align:middle line:84% So let's finish up with that proof. 00:31:17.470 --> 00:31:22.980 align:middle line:90% 00:31:22.980 --> 00:31:36.700 align:middle line:84% So the problem is A and B need not be disjoint. 00:31:36.700 --> 00:31:38.860 align:middle line:84% Now I'm deliberately trying to explain 00:31:38.860 --> 00:31:41.660 align:middle line:84% a bit of how I think about the pedagogy of explaining 00:31:41.660 --> 00:31:44.500 align:middle line:84% these proofs, not just the proofs themselves. 00:31:44.500 --> 00:31:49.220 align:middle line:84% See, one of the things I could have done for proving this is I 00:31:49.220 --> 00:31:55.310 align:middle line:84% could have just told you what I want I and J to be. 00:31:55.310 --> 00:31:56.910 align:middle line:84% So in fact, what I'm going to do is 00:31:56.910 --> 00:32:01.910 align:middle line:84% I'm going to define I is equal to A minus B 00:32:01.910 --> 00:32:04.990 align:middle line:90% and J is equal to B minus A. 00:32:04.990 --> 00:32:08.410 align:middle line:84% So what this notation means is I take my set A 00:32:08.410 --> 00:32:13.270 align:middle line:84% and I remove every element that shows up in B from that set. 00:32:13.270 --> 00:32:16.550 align:middle line:84% And whatever I have left over with is I. Now, notice, 00:32:16.550 --> 00:32:20.310 align:middle line:84% B does not have to be a subset of A for this to make sense. 00:32:20.310 --> 00:32:23.610 align:middle line:84% If B contains an element that isn't an A, 00:32:23.610 --> 00:32:25.090 align:middle line:90% I just ignore that element. 00:32:25.090 --> 00:32:26.170 align:middle line:90% It wasn't there before. 00:32:26.170 --> 00:32:28.110 align:middle line:90% And I keep it out of my set. 00:32:28.110 --> 00:32:32.870 align:middle line:84% But if B and A both contain the element, then I remove it. 00:32:32.870 --> 00:32:35.510 align:middle line:84% So you can think about this pictorially. 00:32:35.510 --> 00:32:40.030 align:middle line:84% Maybe these two sets A and B, they look like this. 00:32:40.030 --> 00:32:42.810 align:middle line:84% And so what is my set I going to look like? 00:32:42.810 --> 00:32:46.190 align:middle line:84% It's going to be exactly this shaded region right here. 00:32:46.190 --> 00:32:49.310 align:middle line:84% Because I'm taking out everything that belongs to B 00:32:49.310 --> 00:32:52.390 align:middle line:84% and removing it from my set so that I create two disjoint 00:32:52.390 --> 00:32:53.360 align:middle line:90% things. 00:32:53.360 --> 00:32:58.880 align:middle line:84% So this right here would be I and this right here would be J. 00:32:58.880 --> 00:33:02.240 align:middle line:84% So let's think a little bit about pedagogy again. 00:33:02.240 --> 00:33:06.160 align:middle line:84% You see how this visual helps me understand what's going on 00:33:06.160 --> 00:33:08.200 align:middle line:90% mathematically in the notation. 00:33:08.200 --> 00:33:11.560 align:middle line:84% If I just defined for you what A minus B is, 00:33:11.560 --> 00:33:14.180 align:middle line:84% that's a perfectly valid definition. 00:33:14.180 --> 00:33:16.280 align:middle line:84% But a lot of times these visuals can help people 00:33:16.280 --> 00:33:18.223 align:middle line:90% understand them more easily. 00:33:18.223 --> 00:33:19.640 align:middle line:84% And if you notice, the other thing 00:33:19.640 --> 00:33:22.120 align:middle line:84% I did in terms of writing up this proof, 00:33:22.120 --> 00:33:25.480 align:middle line:84% I could have actually skipped this entire discussion 00:33:25.480 --> 00:33:28.360 align:middle line:84% about what happens when I choose A and B. I could 00:33:28.360 --> 00:33:31.200 align:middle line:90% have jumped first to I and J. 00:33:31.200 --> 00:33:34.020 align:middle line:84% I could have said A and B come from this pigeonhole principle, 00:33:34.020 --> 00:33:38.293 align:middle line:84% and then I could have defined I as A minus B and J as B minus A. 00:33:38.293 --> 00:33:39.960 align:middle line:84% But then you would have missed the point 00:33:39.960 --> 00:33:42.240 align:middle line:90% about why I needed to do that. 00:33:42.240 --> 00:33:45.360 align:middle line:84% So in fact, when you're writing proofs, not everything you write 00:33:45.360 --> 00:33:48.280 align:middle line:84% is mathematically necessary, but a lot of times 00:33:48.280 --> 00:33:51.290 align:middle line:84% it's pedagogically very useful because you 00:33:51.290 --> 00:33:55.530 align:middle line:84% can walk the reader through not just the formal verification 00:33:55.530 --> 00:33:58.890 align:middle line:84% that the statement is true, but how exactly you arrived 00:33:58.890 --> 00:34:03.110 align:middle line:84% at the proof yourself and the missteps you made along the way 00:34:03.110 --> 00:34:05.930 align:middle line:84% but that led you to the right answer. 00:34:05.930 --> 00:34:10.969 align:middle line:84% All right, so finishing this up, the rest of it is just notation. 00:34:10.969 --> 00:34:13.030 align:middle line:84% So when we look at this expression, 00:34:13.030 --> 00:34:17.449 align:middle line:84% the sum over all xi in the set A, 00:34:17.449 --> 00:34:20.250 align:middle line:84% we can break this up into two pieces. 00:34:20.250 --> 00:34:26.489 align:middle line:84% It's the sum for all I that are in both A and B of xi. 00:34:26.489 --> 00:34:31.250 align:middle line:84% And it's the sum of all xi for I that 00:34:31.250 --> 00:34:35.330 align:middle line:84% belongs to my set A. I. Capital I, which 00:34:35.330 --> 00:34:37.130 align:middle line:90% is the things that belong here. 00:34:37.130 --> 00:34:39.670 align:middle line:84% So I'm just taking the sum over this entire set A, 00:34:39.670 --> 00:34:41.370 align:middle line:84% and I'm breaking it up into this piece 00:34:41.370 --> 00:34:44.770 align:middle line:84% right here and this piece right here. 00:34:44.770 --> 00:34:48.889 align:middle line:84% But now, by assumption, I know, using the pigeonhole principle, 00:34:48.889 --> 00:34:55.340 align:middle line:84% that this is the same thing as the sum from all I and B of xi, 00:34:55.340 --> 00:34:57.640 align:middle line:84% and I can break this sum up the same way. 00:34:57.640 --> 00:35:02.940 align:middle line:84% It's the sum of all I in the intersection of xi, 00:35:02.940 --> 00:35:09.260 align:middle line:84% plus the sum of all of these things in my set capital J. 00:35:09.260 --> 00:35:12.180 align:middle line:84% So I'm just breaking up the sum to different ways. 00:35:12.180 --> 00:35:14.020 align:middle line:84% And now when I take this equation 00:35:14.020 --> 00:35:15.740 align:middle line:84% that the whole thing is equal, I can just 00:35:15.740 --> 00:35:19.420 align:middle line:84% cross off these two pieces because they're the same thing. 00:35:19.420 --> 00:35:21.300 align:middle line:90% And that gives me what I want. 00:35:21.300 --> 00:35:23.700 align:middle line:84% These sets I and J are definitely 00:35:23.700 --> 00:35:26.060 align:middle line:84% disjoint because I've made them be disjoint, 00:35:26.060 --> 00:35:28.300 align:middle line:90% and they have the same sum. 00:35:28.300 --> 00:35:30.460 align:middle line:84% You should double check that you never end up 00:35:30.460 --> 00:35:33.220 align:middle line:84% with these things being nonempty. 00:35:33.220 --> 00:35:36.860 align:middle line:84% That's a technicality you have to check, but it is true. 00:35:36.860 --> 00:35:39.660 align:middle line:84% And it follows from the assumptions. 00:35:39.660 --> 00:35:42.800 align:middle line:84% But that's the end of our next proof. 00:35:42.800 --> 00:35:43.700 align:middle line:90% All right? 00:35:43.700 --> 00:35:45.620 align:middle line:90% So are there any questions? 00:35:45.620 --> 00:35:47.470 align:middle line:90% Makes sense? 00:35:47.470 --> 00:35:47.970 align:middle line:90% Yeah? 00:35:47.970 --> 00:35:51.380 align:middle line:90% 00:35:51.380 --> 00:35:52.770 align:middle line:90% AUDIENCE: The minimum number-- 00:35:52.770 --> 00:35:54.593 align:middle line:84% ANKUR MOITRA: It's about the minimum. 00:35:54.593 --> 00:35:56.510 align:middle line:84% But since you asked, let me tell you something 00:35:56.510 --> 00:35:57.690 align:middle line:90% interesting about this. 00:35:57.690 --> 00:35:59.970 align:middle line:84% This lemma is very easy to state. 00:35:59.970 --> 00:36:01.110 align:middle line:90% It's not so bad to prove. 00:36:01.110 --> 00:36:02.830 align:middle line:90% We proved it pretty quickly. 00:36:02.830 --> 00:36:05.430 align:middle line:84% What's kind of shocking is I can write down 00:36:05.430 --> 00:36:09.350 align:middle line:84% 20 four digit numbers for you, and you 00:36:09.350 --> 00:36:12.070 align:middle line:84% would have no way-- you could prove to yourself using 00:36:12.070 --> 00:36:15.630 align:middle line:84% this proof right here that there are two subsets whose sum is 00:36:15.630 --> 00:36:19.790 align:middle line:84% the same, but it would be very hard to actually find them. 00:36:19.790 --> 00:36:22.390 align:middle line:84% So you can do this, for example, with populations 00:36:22.390 --> 00:36:25.230 align:middle line:84% of different countries, we know that there 00:36:25.230 --> 00:36:28.190 align:middle line:84% are subsets of countries that have literally 00:36:28.190 --> 00:36:30.110 align:middle line:90% the same number of people. 00:36:30.110 --> 00:36:33.630 align:middle line:84% But it would be exhaustive to try and search 00:36:33.630 --> 00:36:37.590 align:middle line:84% all possible subsets to figure out which pair collide. 00:36:37.590 --> 00:36:41.270 align:middle line:84% And we don't actually know any faster algorithms. 00:36:41.270 --> 00:36:43.350 align:middle line:84% It's related to some things in cryptography. 00:36:43.350 --> 00:36:43.850 align:middle line:90% All right. 00:36:43.850 --> 00:36:45.800 align:middle line:90% Any other questions? 00:36:45.800 --> 00:36:47.640 align:middle line:90% Makes sense so far? 00:36:47.640 --> 00:36:50.360 align:middle line:90% All right. 00:36:50.360 --> 00:36:50.980 align:middle line:90% All right. 00:36:50.980 --> 00:36:54.680 align:middle line:84% So we got to do our hardest result today. 00:36:54.680 --> 00:36:59.440 align:middle line:84% And then I'll do a fun end for one of my favorite applications 00:36:59.440 --> 00:37:01.720 align:middle line:90% of the pigeonhole principle. 00:37:01.720 --> 00:37:04.320 align:middle line:90% So let's do our hardest theorem. 00:37:04.320 --> 00:37:06.880 align:middle line:84% And I'm going to tell you it's the hardest result 00:37:06.880 --> 00:37:09.520 align:middle line:84% because I'm going to call it a theorem. 00:37:09.520 --> 00:37:11.200 align:middle line:84% And we'll be talking about extensions 00:37:11.200 --> 00:37:13.380 align:middle line:90% of this on your first p sets. 00:37:13.380 --> 00:37:16.480 align:middle line:84% You'll get more practice with this proof. 00:37:16.480 --> 00:37:18.480 align:middle line:84% But let me tell you about a beautiful result 00:37:18.480 --> 00:37:20.200 align:middle line:90% in combinatorics. 00:37:20.200 --> 00:37:21.280 align:middle line:90% All right. 00:37:21.280 --> 00:37:26.663 align:middle line:84% What I claim is that in any permutation-- 00:37:26.663 --> 00:37:28.580 align:middle line:84% I'll tell you what I mean by that in a second. 00:37:28.580 --> 00:37:31.680 align:middle line:84% We'll be talking much more about permutations later. 00:37:31.680 --> 00:37:35.440 align:middle line:84% Any permutation of the integers 1, 00:37:35.440 --> 00:37:41.360 align:middle line:84% 2, 3, all the way up to something n times m plus 1. 00:37:41.360 --> 00:37:58.730 align:middle line:84% What I claim is that there is an increasing subsequence of length 00:37:58.730 --> 00:38:13.250 align:middle line:84% m plus 1, or a decreasing subsequence of length n plus 1. 00:38:13.250 --> 00:38:13.850 align:middle line:90% OK? 00:38:13.850 --> 00:38:15.810 align:middle line:84% So I owe you a bunch of things to explain 00:38:15.810 --> 00:38:17.230 align:middle line:90% what this theorem means. 00:38:17.230 --> 00:38:17.970 align:middle line:90% Yes? 00:38:17.970 --> 00:38:20.050 align:middle line:84% AUDIENCE: Is subsequence [INAUDIBLE]? 00:38:20.050 --> 00:38:22.230 align:middle line:84% ANKUR MOITRA: I'll get to that, but no, it's not. 00:38:22.230 --> 00:38:23.990 align:middle line:84% So that's one of the things I owe you. 00:38:23.990 --> 00:38:26.250 align:middle line:84% But first, let's start with the simpler thing. 00:38:26.250 --> 00:38:27.770 align:middle line:90% So I owe you this. 00:38:27.770 --> 00:38:30.130 align:middle line:90% What do I mean by a permutation? 00:38:30.130 --> 00:38:32.750 align:middle line:84% So what I mean is you can take these integers 1, 00:38:32.750 --> 00:38:36.930 align:middle line:84% 2, 3 all the way up to nm plus 1. 00:38:36.930 --> 00:38:39.930 align:middle line:84% n and m are just parameters in this theorem statement. 00:38:39.930 --> 00:38:42.920 align:middle line:84% And instead of considering them all in increasing order. 00:38:42.920 --> 00:38:45.740 align:middle line:84% I can put all of the numbers in any order I want. 00:38:45.740 --> 00:38:48.600 align:middle line:84% So basically, I take this deck of cards and I shuffle it, 00:38:48.600 --> 00:38:51.260 align:middle line:84% and I consider any possible ordering, that's 00:38:51.260 --> 00:38:52.600 align:middle line:90% what I mean by permutation. 00:38:52.600 --> 00:38:54.100 align:middle line:84% We'll talk more about them formally 00:38:54.100 --> 00:38:56.060 align:middle line:90% later when we get to counting. 00:38:56.060 --> 00:38:59.460 align:middle line:84% But I take any ordering for these integers. 00:38:59.460 --> 00:39:03.700 align:middle line:84% And what I claim is that there's some of subsequence where 00:39:03.700 --> 00:39:08.740 align:middle line:84% the numbers are going up by one at least one each time, 00:39:08.740 --> 00:39:10.060 align:middle line:90% and that's long. 00:39:10.060 --> 00:39:13.860 align:middle line:84% Or if not, there is some long subsequence 00:39:13.860 --> 00:39:17.900 align:middle line:84% where the numbers are going down at least by one each time. 00:39:17.900 --> 00:39:19.860 align:middle line:84% So let's do an example, because I should really 00:39:19.860 --> 00:39:23.500 align:middle line:84% be more precise by what I mean by a subsequence. 00:39:23.500 --> 00:39:27.940 align:middle line:84% So let's take, for example, m equals n equals 3. 00:39:27.940 --> 00:39:30.300 align:middle line:84% So then I'm really just asking for a permutation 00:39:30.300 --> 00:39:32.480 align:middle line:90% of the integers from 1 up to 10. 00:39:32.480 --> 00:39:35.140 align:middle line:84% So some shuffling of that deck of cards. 00:39:35.140 --> 00:39:36.700 align:middle line:90% Let's take an example. 00:39:36.700 --> 00:39:49.230 align:middle line:84% We could take 3, 2, 1, 6, 5, 4, 9, 8, 7, and then 10. 00:39:49.230 --> 00:39:54.030 align:middle line:84% So now an example of a long increasing sequence 00:39:54.030 --> 00:39:55.630 align:middle line:90% is this one right here. 00:39:55.630 --> 00:40:02.710 align:middle line:84% I take 3, then I take 6, then I take 9, and then I take 10. 00:40:02.710 --> 00:40:05.330 align:middle line:84% So this sequence does not have to be contiguous. 00:40:05.330 --> 00:40:07.230 align:middle line:84% These positions I'm interested in 00:40:07.230 --> 00:40:09.470 align:middle line:84% don't have to be right next to each other. 00:40:09.470 --> 00:40:11.390 align:middle line:84% But what I want is that when I consider 00:40:11.390 --> 00:40:14.310 align:middle line:84% the sequence of integers just in those positions where I've 00:40:14.310 --> 00:40:17.310 align:middle line:84% put the arrows, that the corresponding sequence 00:40:17.310 --> 00:40:21.310 align:middle line:84% I get is strictly increasing as I go up the arrows. 00:40:21.310 --> 00:40:23.290 align:middle line:84% So that's an increasing subsequence. 00:40:23.290 --> 00:40:27.150 align:middle line:84% I can also think about having a decreasing one like 3, 2, 1. 00:40:27.150 --> 00:40:29.510 align:middle line:84% And that's the longest increasing subsequence 00:40:29.510 --> 00:40:31.150 align:middle line:90% I can get here. 00:40:31.150 --> 00:40:33.430 align:middle line:84% So you can see that this example illustrates, 00:40:33.430 --> 00:40:36.190 align:middle line:84% A, it helps me understand what exactly 00:40:36.190 --> 00:40:39.800 align:middle line:84% these abstract definitions mean, like a permutation 00:40:39.800 --> 00:40:41.960 align:middle line:90% and an increasing subsequence. 00:40:41.960 --> 00:40:44.680 align:middle line:84% It also tells me something even more powerful, 00:40:44.680 --> 00:40:47.680 align:middle line:84% which is that this theorem really is tight. 00:40:47.680 --> 00:40:49.920 align:middle line:84% Because in the case where n equals m 00:40:49.920 --> 00:40:53.440 align:middle line:84% equals 3, what I'm promising you is 00:40:53.440 --> 00:40:57.460 align:middle line:84% that there is a length for increasing subsequence, 00:40:57.460 --> 00:41:00.840 align:middle line:84% or there's a length for decreasing subsequence. 00:41:00.840 --> 00:41:04.960 align:middle line:84% And even though I found one that has length 4 and is increasing, 00:41:04.960 --> 00:41:07.000 align:middle line:84% what I claim in this simple example 00:41:07.000 --> 00:41:10.000 align:middle line:84% is that I can't create anything longer, 00:41:10.000 --> 00:41:13.360 align:middle line:84% and I can't even create a decreasing subsequence that's 00:41:13.360 --> 00:41:18.420 align:middle line:84% longer than 3, because if I did 3, 2, 1, then I'm in trouble. 00:41:18.420 --> 00:41:20.100 align:middle line:90% Everything after it is larger. 00:41:20.100 --> 00:41:22.620 align:middle line:84% If I do 6, 5, 4 again, I'm in trouble. 00:41:22.620 --> 00:41:24.840 align:middle line:90% Everything after it is larger. 00:41:24.840 --> 00:41:25.340 align:middle line:90% All right. 00:41:25.340 --> 00:41:27.880 align:middle line:90% Any questions? 00:41:27.880 --> 00:41:29.440 align:middle line:84% So I claim that we can prove this 00:41:29.440 --> 00:41:31.160 align:middle line:90% using the pigeonhole principle. 00:41:31.160 --> 00:41:34.440 align:middle line:84% But we're going to have the same difficulties we usually have, 00:41:34.440 --> 00:41:37.410 align:middle line:84% which is where are where the pigeons and where are the holes. 00:41:37.410 --> 00:41:39.810 align:middle line:84% In fact, it'll be even more abstract 00:41:39.810 --> 00:41:43.490 align:middle line:84% because if you try and guess things like the sequences are 00:41:43.490 --> 00:41:46.135 align:middle line:84% the pigeons or the positions are the pigeons, 00:41:46.135 --> 00:41:47.510 align:middle line:84% you're going to have a hard time. 00:41:47.510 --> 00:41:50.010 align:middle line:84% That's not how this proof is going to go. 00:41:50.010 --> 00:41:54.410 align:middle line:84% Instead, we're going to need a very delicate definition 00:41:54.410 --> 00:41:59.010 align:middle line:84% to tell us where the pigeons are and how this relates 00:41:59.010 --> 00:42:00.430 align:middle line:90% to the pigeonhole principle. 00:42:00.430 --> 00:42:04.770 align:middle line:90% 00:42:04.770 --> 00:42:09.370 align:middle line:84% And the idea, basically, is that I can't-- 00:42:09.370 --> 00:42:10.870 align:middle line:84% this theorem, if you think about it, 00:42:10.870 --> 00:42:12.790 align:middle line:84% it's not telling me which of them I have, 00:42:12.790 --> 00:42:16.570 align:middle line:84% whether I have a long increasing or a long decreasing sequence. 00:42:16.570 --> 00:42:18.090 align:middle line:84% So in some sense, what I want to do 00:42:18.090 --> 00:42:21.330 align:middle line:84% is I want to keep track of my progress on both of them. 00:42:21.330 --> 00:42:24.230 align:middle line:84% How long is my largest increasing subsequence? 00:42:24.230 --> 00:42:27.130 align:middle line:84% How long is my largest decreasing sequence? 00:42:27.130 --> 00:42:30.570 align:middle line:84% And I want to see how those progress measures grow. 00:42:30.570 --> 00:42:36.340 align:middle line:84% So the proof is going to involve this very careful definition. 00:42:36.340 --> 00:42:40.640 align:middle line:84% Let's consider these integers in order, 00:42:40.640 --> 00:42:48.110 align:middle line:84% and let's consider the S-th element in the sequence. 00:42:48.110 --> 00:42:53.580 align:middle line:90% 00:42:53.580 --> 00:42:56.780 align:middle line:84% And what we're going to do is we're 00:42:56.780 --> 00:42:59.580 align:middle line:84% going to define these helper variables. 00:42:59.580 --> 00:43:09.220 align:middle line:84% So we're going to let I sub S denote the length of the longest 00:43:09.220 --> 00:43:22.230 align:middle line:84% increasing subsequence ending at the S-th location. 00:43:22.230 --> 00:43:24.820 align:middle line:90% 00:43:24.820 --> 00:43:25.420 align:middle line:90% OK? 00:43:25.420 --> 00:43:27.940 align:middle line:90% That's one progress measure. 00:43:27.940 --> 00:43:29.520 align:middle line:90% I is for increasing. 00:43:29.520 --> 00:43:33.790 align:middle line:84% So I've chosen notation that at least has a mnemonic 00:43:33.790 --> 00:43:35.310 align:middle line:90% associated with it. 00:43:35.310 --> 00:43:37.830 align:middle line:84% I have ds is going to be the same thing, 00:43:37.830 --> 00:43:45.750 align:middle line:84% but it'll be the longest decreasing Subsequence. 00:43:45.750 --> 00:43:49.910 align:middle line:84% But again, ending at the S-th location. 00:43:49.910 --> 00:43:52.230 align:middle line:90% OK? 00:43:52.230 --> 00:43:54.570 align:middle line:90% So this is my progress measure. 00:43:54.570 --> 00:43:59.030 align:middle line:84% In fact, this is a little bit abstract, so I'll go about it. 00:43:59.030 --> 00:44:02.950 align:middle line:84% So how are we going to prove this thing? 00:44:02.950 --> 00:44:10.870 align:middle line:84% So first, does anyone know what this term right here means? 00:44:10.870 --> 00:44:13.150 align:middle line:90% WLOG. 00:44:13.150 --> 00:44:14.210 align:middle line:90% Just out of curiosity. 00:44:14.210 --> 00:44:15.030 align:middle line:90% Yeah? 00:44:15.030 --> 00:44:16.410 align:middle line:90% Without loss of generality. 00:44:16.410 --> 00:44:16.910 align:middle line:90% OK. 00:44:16.910 --> 00:44:19.830 align:middle line:84% So I told you that when I was proving things 00:44:19.830 --> 00:44:22.670 align:middle line:84% with the number of people in a party 00:44:22.670 --> 00:44:25.450 align:middle line:84% that I had to do this case analysis, well, 00:44:25.450 --> 00:44:27.170 align:middle line:84% to prevent my proof from being too long, 00:44:27.170 --> 00:44:31.110 align:middle line:84% let me just pretend s is larger than t, because if it was 00:44:31.110 --> 00:44:33.310 align:middle line:84% the other way around, I would just switch what's S 00:44:33.310 --> 00:44:35.390 align:middle line:90% and what's t. 00:44:35.390 --> 00:44:36.150 align:middle line:90% Yeah? 00:44:36.150 --> 00:44:39.050 align:middle line:90% AUDIENCE: [INAUDIBLE]? 00:44:39.050 --> 00:44:42.430 align:middle line:84% Like, what are the-- the longest increasing one 00:44:42.430 --> 00:44:45.557 align:middle line:90% is six and the last [INAUDIBLE]? 00:44:45.557 --> 00:44:46.390 align:middle line:90% ANKUR MOITRA: Sorry. 00:44:46.390 --> 00:44:49.190 align:middle line:90% Which entry in the table here? 00:44:49.190 --> 00:44:50.165 align:middle line:90% Yeah? 00:44:50.165 --> 00:44:50.790 align:middle line:90% AUDIENCE: Yeah. 00:44:50.790 --> 00:44:51.842 align:middle line:90% Three and six. 00:44:51.842 --> 00:44:52.550 align:middle line:90% ANKUR MOITRA: OK. 00:44:52.550 --> 00:44:53.550 align:middle line:90% All right. 00:44:53.550 --> 00:44:54.970 align:middle line:90% So here. 00:44:54.970 --> 00:44:58.470 align:middle line:84% So we want the longest increasing sequence 00:44:58.470 --> 00:45:00.205 align:middle line:90% that ends at three. 00:45:00.205 --> 00:45:02.542 align:middle line:90% AUDIENCE: That ends at six. 00:45:02.542 --> 00:45:04.750 align:middle line:84% ANKUR MOITRA: So do you want this entry in the table? 00:45:04.750 --> 00:45:05.550 align:middle line:90% OK. 00:45:05.550 --> 00:45:08.490 align:middle line:84% So the longest increasing one that ends at six. 00:45:08.490 --> 00:45:09.410 align:middle line:90% So what would it be? 00:45:09.410 --> 00:45:12.578 align:middle line:90% 00:45:12.578 --> 00:45:13.870 align:middle line:90% AUDIENCE: Oh, it's [INAUDIBLE]. 00:45:13.870 --> 00:45:14.662 align:middle line:90% ANKUR MOITRA: Yeah. 00:45:14.662 --> 00:45:15.190 align:middle line:90% Yeah. 00:45:15.190 --> 00:45:17.150 align:middle line:84% So it's not required to be contiguous, 00:45:17.150 --> 00:45:19.750 align:middle line:84% but it has to include that end point. 00:45:19.750 --> 00:45:25.470 align:middle line:84% So that's why these numbers are not monotonically increasing. 00:45:25.470 --> 00:45:28.960 align:middle line:84% So when I have the longest decreasing sequence that 00:45:28.960 --> 00:45:32.500 align:middle line:84% ends at three, that's pretty long, 00:45:32.500 --> 00:45:36.160 align:middle line:84% because I could go 5, 4, 3 and end there. 00:45:36.160 --> 00:45:38.640 align:middle line:84% But as soon as I move over to this next column 00:45:38.640 --> 00:45:41.926 align:middle line:84% right here, if I have to end at 6, 00:45:41.926 --> 00:45:44.240 align:middle line:84% that makes the longest decreasing sequence 00:45:44.240 --> 00:45:45.800 align:middle line:90% only of length 1. 00:45:45.800 --> 00:45:48.520 align:middle line:84% So the key is that I'm fixing that my point has 00:45:48.520 --> 00:45:52.000 align:middle line:84% to belong to that subsequence and I'm considering progress 00:45:52.000 --> 00:45:53.820 align:middle line:90% as I go left to right. 00:45:53.820 --> 00:45:54.780 align:middle line:90% So any other questions? 00:45:54.780 --> 00:45:55.863 align:middle line:90% This is a crucial part. 00:45:55.863 --> 00:45:57.280 align:middle line:84% If you don't understand the table, 00:45:57.280 --> 00:45:59.800 align:middle line:90% you won't understand the proof. 00:45:59.800 --> 00:46:01.080 align:middle line:90% Good? 00:46:01.080 --> 00:46:02.560 align:middle line:90% All right. 00:46:02.560 --> 00:46:08.000 align:middle line:84% So without loss of generality, s is larger than t. 00:46:08.000 --> 00:46:10.600 align:middle line:90% And so let's do the following. 00:46:10.600 --> 00:46:21.240 align:middle line:84% So if the element at the S-th location 00:46:21.240 --> 00:46:26.570 align:middle line:84% is strictly larger than the element 00:46:26.570 --> 00:46:34.170 align:middle line:84% at the t-th location, what I claim 00:46:34.170 --> 00:46:38.830 align:middle line:90% is that is at least it plus 1. 00:46:38.830 --> 00:46:41.690 align:middle line:90% 00:46:41.690 --> 00:46:44.450 align:middle line:84% It's the morning and I'm lazy, so who can prove this for me 00:46:44.450 --> 00:46:46.290 align:middle line:90% and justify this? 00:46:46.290 --> 00:46:47.690 align:middle line:90% Why is this true? 00:46:47.690 --> 00:46:49.570 align:middle line:90% Someone who hasn't answered. 00:46:49.570 --> 00:46:51.400 align:middle line:84% Especially my back row is very quiet. 00:46:51.400 --> 00:47:02.630 align:middle line:90% 00:47:02.630 --> 00:47:03.130 align:middle line:90% Yeah? 00:47:03.130 --> 00:47:08.410 align:middle line:90% 00:47:08.410 --> 00:47:09.930 align:middle line:90% That's exactly right. 00:47:09.930 --> 00:47:11.230 align:middle line:90% So let me say that again. 00:47:11.230 --> 00:47:15.730 align:middle line:84% So we can just use the increasing subsequence that 00:47:15.730 --> 00:47:17.850 align:middle line:90% ended at the t-th location. 00:47:17.850 --> 00:47:20.730 align:middle line:84% And because the element at the S-th location 00:47:20.730 --> 00:47:23.620 align:middle line:84% is after the one that happens at the t-th location 00:47:23.620 --> 00:47:27.220 align:middle line:84% because S is larger than t, and the element itself is bigger 00:47:27.220 --> 00:47:29.980 align:middle line:84% too, I can just append that element 00:47:29.980 --> 00:47:33.340 align:middle line:90% and make my subsequence longer. 00:47:33.340 --> 00:47:34.200 align:middle line:90% All right. 00:47:34.200 --> 00:47:38.060 align:middle line:84% So how does the rest of the proof go? 00:47:38.060 --> 00:47:41.380 align:middle line:84% Who can fill in the rest of the proof for me? 00:47:41.380 --> 00:47:44.220 align:middle line:84% So this is one case in the case analysis. 00:47:44.220 --> 00:47:47.660 align:middle line:84% I've said without loss of generality S is larger than t. 00:47:47.660 --> 00:47:50.780 align:middle line:84% And if the element at the S-th location 00:47:50.780 --> 00:47:53.780 align:middle line:84% is larger than the element at the t-th location, 00:47:53.780 --> 00:47:57.660 align:middle line:90% my is is at least it plus 1. 00:47:57.660 --> 00:47:59.100 align:middle line:90% Yeah? 00:47:59.100 --> 00:48:01.480 align:middle line:84% AUDIENCE: Do the same thing without the generality 00:48:01.480 --> 00:48:02.348 align:middle line:90% S greater than t. 00:48:02.348 --> 00:48:03.140 align:middle line:90% ANKUR MOITRA: Yeah. 00:48:03.140 --> 00:48:05.140 align:middle line:84% AUDIENCE: Instead, the element at the S location 00:48:05.140 --> 00:48:07.320 align:middle line:84% is less than the element at the t-th location. 00:48:07.320 --> 00:48:10.783 align:middle line:84% Then you can do the same for ds and dt. 00:48:10.783 --> 00:48:11.700 align:middle line:90% ANKUR MOITRA: Perfect. 00:48:11.700 --> 00:48:14.500 align:middle line:90% That's right. 00:48:14.500 --> 00:48:17.040 align:middle line:84% And it's basically the same reasoning but backwards. 00:48:17.040 --> 00:48:20.460 align:middle line:84% So otherwise, in particular, if the element at the S-th location 00:48:20.460 --> 00:48:24.150 align:middle line:84% is strictly less than the element of the t-th location, 00:48:24.150 --> 00:48:27.110 align:middle line:84% I can just take the decreasing subsequence that 00:48:27.110 --> 00:48:29.990 align:middle line:84% ends at the t-th location, and append what 00:48:29.990 --> 00:48:31.510 align:middle line:90% happens at the S-th location. 00:48:31.510 --> 00:48:34.910 align:middle line:84% That's smaller, so that's a valid decreasing sequence. 00:48:34.910 --> 00:48:36.990 align:middle line:90% And that's the proof. 00:48:36.990 --> 00:48:40.270 align:middle line:84% I just proved for you that these two pairs of integers 00:48:40.270 --> 00:48:43.230 align:middle line:84% are different, because I showed you 00:48:43.230 --> 00:48:45.630 align:middle line:84% that either this one is larger than this, 00:48:45.630 --> 00:48:48.970 align:middle line:90% or this one is larger than this. 00:48:48.970 --> 00:48:51.150 align:middle line:84% And now we're in good shape because, 00:48:51.150 --> 00:48:53.410 align:middle line:90% how does the entire thing work? 00:48:53.410 --> 00:48:56.520 align:middle line:84% So just returning to the proof, putting it all together. 00:48:56.520 --> 00:49:00.470 align:middle line:90% 00:49:00.470 --> 00:49:03.970 align:middle line:84% Even though I said it in words, let's just be precise about it. 00:49:03.970 --> 00:49:07.790 align:middle line:90% 00:49:07.790 --> 00:49:09.020 align:middle line:90% So returning to the proof. 00:49:09.020 --> 00:49:13.790 align:middle line:90% 00:49:13.790 --> 00:49:22.040 align:middle line:84% Let's suppose the theorem is false for the purposes 00:49:22.040 --> 00:49:24.360 align:middle line:90% of contradiction. 00:49:24.360 --> 00:49:39.040 align:middle line:84% Then what I claim is that there are at most nm possible values 00:49:39.040 --> 00:49:44.880 align:middle line:84% for these pairs is, ds, because that's what it means 00:49:44.880 --> 00:49:46.560 align:middle line:90% for the theorem to be false. 00:49:46.560 --> 00:49:51.040 align:middle line:84% There's no n plus 1 length increasing subsequence. 00:49:51.040 --> 00:49:54.920 align:middle line:84% There's no length m plus 1 decreasing subsequence. 00:49:54.920 --> 00:49:56.820 align:middle line:84% When I look at my pairs of integers, 00:49:56.820 --> 00:50:00.760 align:middle line:84% is never becomes larger than n, and ds 00:50:00.760 --> 00:50:03.760 align:middle line:90% never becomes larger than m. 00:50:03.760 --> 00:50:08.600 align:middle line:84% And these are my holes is because each one 00:50:08.600 --> 00:50:12.240 align:middle line:84% of the possible positions is going to get mapped just 00:50:12.240 --> 00:50:16.120 align:middle line:84% by this definition that I gave you right here of is, ds 00:50:16.120 --> 00:50:20.010 align:middle line:84% to the signature of these pair of integers that just tells me 00:50:20.010 --> 00:50:21.970 align:middle line:90% the name of my hole. 00:50:21.970 --> 00:50:25.770 align:middle line:90% But how long is my sequence? 00:50:25.770 --> 00:50:31.190 align:middle line:84% There are nm plus 1 values in the sequence. 00:50:31.190 --> 00:50:37.410 align:middle line:84% These are all my pigeons, which, by the pigeonhole principle, 00:50:37.410 --> 00:50:48.090 align:middle line:84% means there is S not equal to t that collide in some hole with 00:50:48.090 --> 00:50:53.270 align:middle line:90% is, ds equal to it, dt. 00:50:53.270 --> 00:50:56.150 align:middle line:84% That's what it means for them to collide to the same hole, 00:50:56.150 --> 00:50:58.130 align:middle line:84% because they get mapped to the same thing. 00:50:58.130 --> 00:51:00.850 align:middle line:84% And we know that that's impossible. 00:51:00.850 --> 00:51:04.370 align:middle line:90% But this is impossible. 00:51:04.370 --> 00:51:07.010 align:middle line:84% So that contradicts the assumption 00:51:07.010 --> 00:51:09.930 align:middle line:84% we made at the beginning, which was just that the theorem was 00:51:09.930 --> 00:51:10.430 align:middle line:90% false. 00:51:10.430 --> 00:51:11.530 align:middle line:90% Yes? 00:51:11.530 --> 00:51:13.170 align:middle line:84% AUDIENCE: Without loss of generality, 00:51:13.170 --> 00:51:15.900 align:middle line:84% why are we comparing is to i of t plus 1 00:51:15.900 --> 00:51:17.927 align:middle line:90% instead of directly to it? 00:51:17.927 --> 00:51:18.760 align:middle line:90% ANKUR MOITRA: Sorry. 00:51:18.760 --> 00:51:19.302 align:middle line:90% Say it again. 00:51:19.302 --> 00:51:21.532 align:middle line:90% So why are we comparing? 00:51:21.532 --> 00:51:26.592 align:middle line:84% AUDIENCE: We compare that to i of t plus 1 instead of to it. 00:51:26.592 --> 00:51:27.300 align:middle line:90% ANKUR MOITRA: OK. 00:51:27.300 --> 00:51:30.700 align:middle line:84% So just to be careful right here, this is i sub t 00:51:30.700 --> 00:51:32.980 align:middle line:90% and then plus 1 is outside. 00:51:32.980 --> 00:51:33.740 align:middle line:90% Yeah. 00:51:33.740 --> 00:51:35.740 align:middle line:84% So this is really just saying that whatever 00:51:35.740 --> 00:51:37.820 align:middle line:84% my length of my increasing sequence 00:51:37.820 --> 00:51:42.940 align:middle line:84% is right here, this value right here, I can add 1 to the length. 00:51:42.940 --> 00:51:45.420 align:middle line:84% So the plus 1 is not in the subscript. 00:51:45.420 --> 00:51:47.300 align:middle line:90% Does that make sense? 00:51:47.300 --> 00:51:49.660 align:middle line:90% AUDIENCE: Why not? 00:51:49.660 --> 00:51:52.980 align:middle line:84% Would it hold true that I ask is strictly greater than it or no? 00:51:52.980 --> 00:51:55.300 align:middle line:84% ANKUR MOITRA: is strictly greater-- yes, definitely. 00:51:55.300 --> 00:51:57.820 align:middle line:84% is definitely strictly greater than it 00:51:57.820 --> 00:51:59.500 align:middle line:90% because these are integers. 00:51:59.500 --> 00:52:00.220 align:middle line:90% Yeah. 00:52:00.220 --> 00:52:04.740 align:middle line:84% In fact, is not necessarily equal to it plus 1 00:52:04.740 --> 00:52:07.080 align:middle line:84% because there might be some totally different 00:52:07.080 --> 00:52:09.940 align:middle line:84% other subsequence that's way longer that 00:52:09.940 --> 00:52:12.940 align:middle line:84% just doesn't involve what happens in the t-th location. 00:52:12.940 --> 00:52:14.050 align:middle line:90% But all right. 00:52:14.050 --> 00:52:15.390 align:middle line:90% Any other questions? 00:52:15.390 --> 00:52:17.000 align:middle line:84% This was our hardest proof for today. 00:52:17.000 --> 00:52:17.500 align:middle line:90% Yeah? 00:52:17.500 --> 00:52:19.438 align:middle line:84% AUDIENCE: What are we doing in this case? 00:52:19.438 --> 00:52:20.230 align:middle line:90% ANKUR MOITRA: Yeah. 00:52:20.230 --> 00:52:20.935 align:middle line:90% So the pigeons. 00:52:20.935 --> 00:52:22.310 align:middle line:84% This is why I told you have to be 00:52:22.310 --> 00:52:23.910 align:middle line:90% a little bit careful with this. 00:52:23.910 --> 00:52:27.270 align:middle line:84% The pigeons are actually the positions. 00:52:27.270 --> 00:52:29.510 align:middle line:84% So what's happening here is each one 00:52:29.510 --> 00:52:32.770 align:middle line:84% of these positions, the first element, the second element, 00:52:32.770 --> 00:52:35.590 align:middle line:84% the third element, the fourth element, the fifth, the sixth. 00:52:35.590 --> 00:52:37.350 align:middle line:90% These are my pigeons. 00:52:37.350 --> 00:52:39.190 align:middle line:84% How are they getting mapped to a hole 00:52:39.190 --> 00:52:43.310 align:middle line:84% is I map them to the hole whose name is this integer, comma, 00:52:43.310 --> 00:52:43.900 align:middle line:90% this integer. 00:52:43.900 --> 00:52:47.030 align:middle line:90% 00:52:47.030 --> 00:52:49.190 align:middle line:90% It's just the pairs right here. 00:52:49.190 --> 00:52:50.310 align:middle line:90% It's the pairs. 00:52:50.310 --> 00:52:53.870 align:middle line:84% So my holes-- that's why I drew the little hole right there 00:52:53.870 --> 00:52:56.990 align:middle line:84% is that this is the name of the hole. 00:52:56.990 --> 00:53:00.150 align:middle line:84% And by assumption, if the theorem is false, 00:53:00.150 --> 00:53:06.070 align:middle line:84% is doesn't go larger than n, ds doesn't go larger than m. 00:53:06.070 --> 00:53:11.800 align:middle line:84% So there are at most n times m possibilities for this. 00:53:11.800 --> 00:53:13.360 align:middle line:90% Yeah? 00:53:13.360 --> 00:53:14.800 align:middle line:84% AUDIENCE: So here, maybe you could 00:53:14.800 --> 00:53:17.600 align:middle line:84% have guessed the kind of thing, because the problem is 00:53:17.600 --> 00:53:18.420 align:middle line:90% like a product. 00:53:18.420 --> 00:53:19.212 align:middle line:90% ANKUR MOITRA: Yeah. 00:53:19.212 --> 00:53:20.580 align:middle line:90% AUDIENCE: But if you're-- 00:53:20.580 --> 00:53:25.900 align:middle line:84% I guess if it didn't have this nice [INAUDIBLE], 00:53:25.900 --> 00:53:27.920 align:middle line:90% how would you know? 00:53:27.920 --> 00:53:30.560 align:middle line:90% ANKUR MOITRA: Math is hard. 00:53:30.560 --> 00:53:32.240 align:middle line:84% I mean, we're teaching you things 00:53:32.240 --> 00:53:35.720 align:middle line:84% which we're not going to give you this on a test 00:53:35.720 --> 00:53:38.320 align:middle line:84% and ask you to miraculously guess it. 00:53:38.320 --> 00:53:40.160 align:middle line:84% But, for example, we will ask you 00:53:40.160 --> 00:53:42.520 align:middle line:84% on the set to do a version of this 00:53:42.520 --> 00:53:45.327 align:middle line:84% with three types of subsequences, 00:53:45.327 --> 00:53:47.160 align:middle line:84% and you're going to have this as a blueprint 00:53:47.160 --> 00:53:51.080 align:middle line:84% to go off of to try and understand what's going on. 00:53:51.080 --> 00:53:53.640 align:middle line:84% So this would be extraordinarily clever if you just 00:53:53.640 --> 00:53:55.320 align:middle line:90% came up with this on the spot. 00:53:55.320 --> 00:53:57.360 align:middle line:84% There are also twists to very simple problems 00:53:57.360 --> 00:54:00.280 align:middle line:84% that will make things very difficult and open. 00:54:00.280 --> 00:54:03.200 align:middle line:90% But yeah, let's go with it. 00:54:03.200 --> 00:54:03.700 align:middle line:90% All right. 00:54:03.700 --> 00:54:06.240 align:middle line:90% Any other questions? 00:54:06.240 --> 00:54:06.960 align:middle line:90% Good? 00:54:06.960 --> 00:54:07.580 align:middle line:90% All right. 00:54:07.580 --> 00:54:08.080 align:middle line:90% Yeah. 00:54:08.080 --> 00:54:08.997 align:middle line:90% AUDIENCE: [INAUDIBLE]? 00:54:08.997 --> 00:54:13.970 align:middle line:90% 00:54:13.970 --> 00:54:17.590 align:middle line:84% ANKUR MOITRA: Well, OK, that's maybe a bit of a digression, 00:54:17.590 --> 00:54:20.170 align:middle line:90% but basically if I-- 00:54:20.170 --> 00:54:22.090 align:middle line:84% this really mimics, if you've taken 00:54:22.090 --> 00:54:26.010 align:middle line:84% algorithms classes, a dynamic programming type of solution. 00:54:26.010 --> 00:54:29.530 align:middle line:84% Because if I want to know what is the longest subsequence that 00:54:29.530 --> 00:54:32.010 align:middle line:84% ends here, I really want to piece it together 00:54:32.010 --> 00:54:34.570 align:middle line:84% by answers the subproblems before. 00:54:34.570 --> 00:54:36.570 align:middle line:84% And this often has the structure is 00:54:36.570 --> 00:54:40.050 align:middle line:84% that once I tell you that the subsequence includes 00:54:40.050 --> 00:54:42.290 align:middle line:84% this element and it's increasing, 00:54:42.290 --> 00:54:44.810 align:middle line:84% I don't actually have to know anything else about which 00:54:44.810 --> 00:54:46.430 align:middle line:90% elements belong to it. 00:54:46.430 --> 00:54:48.790 align:middle line:84% I just have to know it's increasing and its length, 00:54:48.790 --> 00:54:51.050 align:middle line:84% and the fact that it belongs to it in order 00:54:51.050 --> 00:54:54.090 align:middle line:84% to be able to piece together what happens to the rest. 00:54:54.090 --> 00:54:56.710 align:middle line:84% So this, a lot of these things in mathematics, 00:54:56.710 --> 00:54:59.210 align:middle line:84% they really mimic what we do algorithmically 00:54:59.210 --> 00:55:01.680 align:middle line:84% in terms of how we would go about finding it. 00:55:01.680 --> 00:55:03.430 align:middle line:84% And we'll see that more and more in class. 00:55:03.430 --> 00:55:06.370 align:middle line:84% That's true even when it comes to writing proof and writing 00:55:06.370 --> 00:55:08.180 align:middle line:90% code. 00:55:08.180 --> 00:55:09.180 align:middle line:90% OK. 00:55:09.180 --> 00:55:10.020 align:middle line:90% Good? 00:55:10.020 --> 00:55:11.260 align:middle line:90% All right. 00:55:11.260 --> 00:55:16.972 align:middle line:84% So let's do one of my favorite examples 00:55:16.972 --> 00:55:19.860 align:middle line:90% of the pigeonhole principle. 00:55:19.860 --> 00:55:24.300 align:middle line:84% Does anyone want to be my victim? 00:55:24.300 --> 00:55:24.840 align:middle line:90% Oh, boy. 00:55:24.840 --> 00:55:27.820 align:middle line:84% You guys volunteered very quickly. 00:55:27.820 --> 00:55:28.580 align:middle line:90% All right. 00:55:28.580 --> 00:55:29.100 align:middle line:90% Sure. 00:55:29.100 --> 00:55:31.180 align:middle line:84% Can you tell me about-- you can stay there. 00:55:31.180 --> 00:55:33.780 align:middle line:84% Can you tell me something about yourself, where 00:55:33.780 --> 00:55:35.200 align:middle line:90% you're from, what your name is. 00:55:35.200 --> 00:55:37.700 align:middle line:90% Introduce yourself to the class. 00:55:37.700 --> 00:55:38.820 align:middle line:90% AUDIENCE: I'm Anthony. 00:55:38.820 --> 00:55:40.320 align:middle line:90% I'm from Saint Louis, Missouri. 00:55:40.320 --> 00:55:44.540 align:middle line:90% 00:55:44.540 --> 00:55:45.160 align:middle line:90% I don't know. 00:55:45.160 --> 00:55:46.743 align:middle line:84% ANKUR MOITRA: Do you have any hobbies? 00:55:46.743 --> 00:55:50.300 align:middle line:90% 00:55:50.300 --> 00:55:52.140 align:middle line:90% AUDIENCE: Oh. 00:55:52.140 --> 00:55:54.100 align:middle line:90% I have a two letter curve. 00:55:54.100 --> 00:55:56.340 align:middle line:90% ANKUR MOITRA: OK. 00:55:56.340 --> 00:55:56.860 align:middle line:90% Wow. 00:55:56.860 --> 00:55:57.520 align:middle line:90% All right. 00:55:57.520 --> 00:55:59.923 align:middle line:90% 00:55:59.923 --> 00:56:01.840 align:middle line:84% There are not many possibilities for that one. 00:56:01.840 --> 00:56:02.620 align:middle line:90% All right. 00:56:02.620 --> 00:56:07.630 align:middle line:84% Now, I can, just based on that, using my MIT superpowers, 00:56:07.630 --> 00:56:10.150 align:middle line:84% I can tell you something deeply personal 00:56:10.150 --> 00:56:13.310 align:middle line:90% about your life history. 00:56:13.310 --> 00:56:16.790 align:middle line:84% In your family tree, it's not a tree. 00:56:16.790 --> 00:56:18.910 align:middle line:90% And I know that for a fact. 00:56:18.910 --> 00:56:21.430 align:middle line:84% So we're going to prove that using the pigeonhole principle. 00:56:21.430 --> 00:56:25.070 align:middle line:84% I should confess that I once used this in 042 and I asked 00:56:25.070 --> 00:56:26.770 align:middle line:90% for a volunteer or a victim. 00:56:26.770 --> 00:56:28.870 align:middle line:84% That's why I call it a victim now. 00:56:28.870 --> 00:56:32.370 align:middle line:84% And the girl raised her hand and told me a bit about herself, 00:56:32.370 --> 00:56:35.870 align:middle line:84% and I told her, your family tree is not a family tree. 00:56:35.870 --> 00:56:40.910 align:middle line:84% And then she got very red and she said, I know. 00:56:40.910 --> 00:56:43.490 align:middle line:84% And I've been a little bit scared to use this since then, 00:56:43.490 --> 00:56:45.090 align:middle line:84% so that's why I call it a victim. 00:56:45.090 --> 00:56:46.723 align:middle line:90% So you signed up for it. 00:56:46.723 --> 00:56:48.390 align:middle line:84% But it's not really just true about you. 00:56:48.390 --> 00:56:50.930 align:middle line:84% It's true about everyone in class. 00:56:50.930 --> 00:56:52.590 align:middle line:84% It's just a mathematical fact that 00:56:52.590 --> 00:56:56.070 align:middle line:84% follows from some very basic axioms of biology, 00:56:56.070 --> 00:57:00.370 align:middle line:84% things which are you really can't poke holes in. 00:57:00.370 --> 00:57:02.270 align:middle line:90% And it's true for all of us. 00:57:02.270 --> 00:57:04.240 align:middle line:84% So let me tell you about this fun application 00:57:04.240 --> 00:57:05.460 align:middle line:90% of the pigeonhole principle. 00:57:05.460 --> 00:57:08.600 align:middle line:84% First, let me state the lemma, and then we'll 00:57:08.600 --> 00:57:10.920 align:middle line:84% figure out, as usual, how to model it 00:57:10.920 --> 00:57:14.800 align:middle line:84% and how exactly to think about it as pigeons and holes. 00:57:14.800 --> 00:57:28.799 align:middle line:84% So what I claim is that everyone's family tree 00:57:28.799 --> 00:57:36.860 align:middle line:84% contains someone whose parents are blood relatives. 00:57:36.860 --> 00:57:39.440 align:middle line:90% 00:57:39.440 --> 00:57:40.470 align:middle line:90% It's just true. 00:57:40.470 --> 00:57:47.600 align:middle line:90% 00:57:47.600 --> 00:57:48.600 align:middle line:90% OK. 00:57:48.600 --> 00:57:53.520 align:middle line:84% So now I'm just going to assert some very incontrovertible 00:57:53.520 --> 00:57:56.180 align:middle line:84% things biologically that are pretty safe. 00:57:56.180 --> 00:58:00.600 align:middle line:84% But this will be enough to be able to prove this fact. 00:58:00.600 --> 00:58:03.890 align:middle line:90% So let's do the setup for this. 00:58:03.890 --> 00:58:06.250 align:middle line:84% So what are the axioms we're using? 00:58:06.250 --> 00:58:14.840 align:middle line:84% So first of all, everyone has two biological parents. 00:58:14.840 --> 00:58:18.932 align:middle line:90% 00:58:18.932 --> 00:58:20.390 align:middle line:84% Of course, if I removed biological, 00:58:20.390 --> 00:58:21.790 align:middle line:90% it would be more complicated. 00:58:21.790 --> 00:58:23.750 align:middle line:84% But I'm just saying this biologically. 00:58:23.750 --> 00:58:27.330 align:middle line:84% Everyone has two biological parents. 00:58:27.330 --> 00:58:36.130 align:middle line:90% No one has children after 100. 00:58:36.130 --> 00:58:37.080 align:middle line:90% Seems pretty safe. 00:58:37.080 --> 00:58:39.730 align:middle line:90% 00:58:39.730 --> 00:58:51.490 align:middle line:84% The human race is at least 4,000 years old. 00:58:51.490 --> 00:58:54.330 align:middle line:90% That's a very safe assumption. 00:58:54.330 --> 00:58:58.630 align:middle line:84% And this is something that can be checked by a biologist, 00:58:58.630 --> 00:59:03.020 align:middle line:84% is that we have upper bounds for the number of total humans 00:59:03.020 --> 00:59:04.460 align:middle line:90% who've ever lived. 00:59:04.460 --> 00:59:06.580 align:middle line:84% What I claim is that there are at most 10 00:59:06.580 --> 00:59:11.570 align:middle line:84% to the 12 humans who've lived all throughout history. 00:59:11.570 --> 00:59:14.460 align:middle line:90% 00:59:14.460 --> 00:59:19.260 align:middle line:84% And just from these axioms, I claim that we can prove this. 00:59:19.260 --> 00:59:21.700 align:middle line:90% So let's prove it. 00:59:21.700 --> 00:59:24.980 align:middle line:84% And we have to figure out the right mathematical abstraction 00:59:24.980 --> 00:59:26.720 align:middle line:90% for how to think about this. 00:59:26.720 --> 00:59:31.540 align:middle line:84% It's the same way as the friends at a party where 00:59:31.540 --> 00:59:37.440 align:middle line:84% we had this way of thinking about it as degrees in a graph. 00:59:37.440 --> 00:59:40.200 align:middle line:84% And what we can do here is we can do the same kind of thing, 00:59:40.200 --> 00:59:42.230 align:middle line:84% but we can trace back our family tree. 00:59:42.230 --> 00:59:50.605 align:middle line:90% 00:59:50.605 --> 00:59:52.980 align:middle line:84% In essence, what I'm doing when I trace back 00:59:52.980 --> 00:59:56.340 align:middle line:84% this family tree is I'm defining a whole bunch of nodes 00:59:56.340 --> 01:00:00.390 align:middle line:84% who have jobs or roles in your family tree. 01:00:00.390 --> 01:00:02.870 align:middle line:90% So there's you at the start. 01:00:02.870 --> 01:00:06.420 align:middle line:84% Then if I trace it back, you have your parents. 01:00:06.420 --> 01:00:08.950 align:middle line:90% 01:00:08.950 --> 01:00:15.490 align:middle line:84% If I trace it back further, I have your grandparents. 01:00:15.490 --> 01:00:22.030 align:middle line:90% 01:00:22.030 --> 01:00:28.790 align:middle line:84% And when I do this, I'm going to trace this back at least 40 01:00:28.790 --> 01:00:33.790 align:middle line:84% generations, because human history is at least 4,000 years 01:00:33.790 --> 01:00:38.030 align:middle line:84% old and no one has children after 100. 01:00:38.030 --> 01:00:43.390 align:middle line:84% So I get at least that my depth is at least 40. 01:00:43.390 --> 01:00:51.230 align:middle line:84% So what this means is that there are at least 2 01:00:51.230 --> 01:00:55.470 align:middle line:84% to the 40 nodes in this abstract family tree which 01:00:55.470 --> 01:00:58.040 align:middle line:90% are just job descriptions. 01:00:58.040 --> 01:01:01.700 align:middle line:90% Your maternal father of-- 01:01:01.700 --> 01:01:06.520 align:middle line:84% I mean, all the way back of tracing all the possibilities. 01:01:06.520 --> 01:01:13.320 align:middle line:84% And the trouble is that, but 2 to the 40 is larger, 01:01:13.320 --> 01:01:19.020 align:middle line:84% is strictly larger than 10 cubed raised to the fourth power, 01:01:19.020 --> 01:01:23.080 align:middle line:84% because 2 to the 10 is larger than 1,000. 01:01:23.080 --> 01:01:25.120 align:middle line:84% And so this is larger than the number 01:01:25.120 --> 01:01:27.920 align:middle line:84% of nodes, all of the possibilities, 01:01:27.920 --> 01:01:30.600 align:middle line:84% all the possible humans in human history. 01:01:30.600 --> 01:01:37.280 align:middle line:84% So by the pigeonhole principle, what we get is that at least two 01:01:37.280 --> 01:01:42.370 align:middle line:84% nodes, two jobs get mapped to the same person. 01:01:42.370 --> 01:01:48.680 align:middle line:90% 01:01:48.680 --> 01:01:50.100 align:middle line:90% So that's pretty cool. 01:01:50.100 --> 01:01:51.760 align:middle line:84% This is just the mathematical fact 01:01:51.760 --> 01:01:56.080 align:middle line:84% that you can use on your friends to annoy them. 01:01:56.080 --> 01:01:58.300 align:middle line:90% Please don't do that. 01:01:58.300 --> 01:01:59.800 align:middle line:84% So this is what I wanted to tell you 01:01:59.800 --> 01:02:01.203 align:middle line:90% about the pigeonhole principle. 01:02:01.203 --> 01:02:03.120 align:middle line:84% Now, as I told you, we're going to be covering 01:02:03.120 --> 01:02:06.220 align:middle line:84% a bunch of introductory topics in the first few lectures, 01:02:06.220 --> 01:02:10.702 align:middle line:84% so pigeonhole principle will get more practice on the p sets. 01:02:10.702 --> 01:02:12.160 align:middle line:84% We're actually going to be spending 01:02:12.160 --> 01:02:15.880 align:middle line:84% the next couple of lectures on probability and then counting. 01:02:15.880 --> 01:02:17.480 align:middle line:84% So we'll also be talking a little bit 01:02:17.480 --> 01:02:20.340 align:middle line:84% about elementary probability in the first recitation, 01:02:20.340 --> 01:02:21.800 align:middle line:90% the first homework. 01:02:21.800 --> 01:02:23.818 align:middle line:84% So let me tell you at least about the formalisms 01:02:23.818 --> 01:02:25.360 align:middle line:84% for probability, and then we're going 01:02:25.360 --> 01:02:28.260 align:middle line:84% to go into a lot more detail over the next couple 01:02:28.260 --> 01:02:31.080 align:middle line:84% lectures that I'll be explaining. 01:02:31.080 --> 01:02:33.480 align:middle line:84% So let me tell you about probability just 01:02:33.480 --> 01:02:37.295 align:middle line:90% as an intro for later lectures. 01:02:37.295 --> 01:02:38.920 align:middle line:84% And we're going to connect all of this, 01:02:38.920 --> 01:02:41.180 align:middle line:90% obviously, to counting as well. 01:02:41.180 --> 01:02:44.120 align:middle line:90% 01:02:44.120 --> 01:02:47.140 align:middle line:84% So basically in class, especially for our later topics, 01:02:47.140 --> 01:02:51.320 align:middle line:84% everything is going to be built on counting one way or another. 01:02:51.320 --> 01:02:53.580 align:middle line:84% We're going to use it for all kinds of fun things. 01:02:53.580 --> 01:02:56.930 align:middle line:90% 01:02:56.930 --> 01:02:58.970 align:middle line:84% But let me give you a little bit of motivation 01:02:58.970 --> 01:03:00.650 align:middle line:90% before we get there. 01:03:00.650 --> 01:03:04.790 align:middle line:84% So we're going to be focusing mostly on discrete probability. 01:03:04.790 --> 01:03:07.970 align:middle line:90% 01:03:07.970 --> 01:03:10.610 align:middle line:84% So why study discrete probability? 01:03:10.610 --> 01:03:16.250 align:middle line:84% Well, it's very useful for things like counting. 01:03:16.250 --> 01:03:20.970 align:middle line:84% You want to understand basic setups like, 01:03:20.970 --> 01:03:24.050 align:middle line:84% what are the chances of creating certain kinds of objects 01:03:24.050 --> 01:03:27.690 align:middle line:84% that are nice when you choose an object at random? 01:03:27.690 --> 01:03:29.450 align:middle line:84% That's very related to counting, and it's 01:03:29.450 --> 01:03:33.530 align:middle line:84% intertwined with basic questions about probability. 01:03:33.530 --> 01:03:35.710 align:middle line:84% Later when we do more advanced topics, 01:03:35.710 --> 01:03:39.250 align:middle line:84% we're going to use probability in really cool ways 01:03:39.250 --> 01:03:43.810 align:middle line:84% to show the existence of very strange objects 01:03:43.810 --> 01:03:46.075 align:middle line:84% that are otherwise hard to construct. 01:03:46.075 --> 01:03:48.450 align:middle line:84% And we're going to be doing this through something called 01:03:48.450 --> 01:03:51.580 align:middle line:90% the probabilistic method. 01:03:51.580 --> 01:03:53.340 align:middle line:84% It's another example where you can 01:03:53.340 --> 01:03:55.340 align:middle line:84% use the language of probability to show 01:03:55.340 --> 01:03:57.800 align:middle line:84% that these very strange objects exist, 01:03:57.800 --> 01:04:01.140 align:middle line:84% even though no one knows how to find any particular one. 01:04:01.140 --> 01:04:02.700 align:middle line:84% So it's kind of like the way that we 01:04:02.700 --> 01:04:08.180 align:middle line:84% reason about those subsets of sums of 20 four digit numbers, 01:04:08.180 --> 01:04:10.060 align:middle line:84% that we knew that there were subsets 01:04:10.060 --> 01:04:15.060 align:middle line:84% whose sum was the same without actually knowing which ones. 01:04:15.060 --> 01:04:16.780 align:middle line:84% Later, when we talk a little bit more 01:04:16.780 --> 01:04:27.100 align:middle line:84% about some topics in algorithms, things like primality testing 01:04:27.100 --> 01:04:31.900 align:middle line:84% using some of the elementary number theory that we do, 01:04:31.900 --> 01:04:35.740 align:middle line:84% deciding whether a given huge integer is prime or not, 01:04:35.740 --> 01:04:38.020 align:middle line:84% it'll turn out that, fundamentally, these algorithms 01:04:38.020 --> 01:04:38.800 align:middle line:90% are randomized. 01:04:38.800 --> 01:04:40.300 align:middle line:84% So really, to reason about it, we'll 01:04:40.300 --> 01:04:44.060 align:middle line:84% have to have the language of probability to begin with. 01:04:44.060 --> 01:04:47.100 align:middle line:84% And much later in class, when we talk 01:04:47.100 --> 01:04:50.190 align:middle line:84% about things like coding theory, we're 01:04:50.190 --> 01:04:53.470 align:middle line:84% going to use probability intrinsically to model 01:04:53.470 --> 01:04:56.310 align:middle line:90% types of communication. 01:04:56.310 --> 01:04:58.370 align:middle line:84% For example, when there's noise between us. 01:04:58.370 --> 01:05:00.670 align:middle line:84% Like when a satellite is sending images back 01:05:00.670 --> 01:05:04.030 align:middle line:84% to Earth, what are ways to reliably communicate 01:05:04.030 --> 01:05:07.030 align:middle line:84% in the face of errors that are introduced 01:05:07.030 --> 01:05:09.855 align:middle line:84% when one party is communicating with the other? 01:05:09.855 --> 01:05:11.230 align:middle line:84% So we're going to use probability 01:05:11.230 --> 01:05:13.630 align:middle line:84% in all of these different contexts. 01:05:13.630 --> 01:05:17.730 align:middle line:84% I want to give you a little bit of the setup for probability 01:05:17.730 --> 01:05:22.590 align:middle line:84% before we get into a lot of the key rules and definitions 01:05:22.590 --> 01:05:24.830 align:middle line:90% next time. 01:05:24.830 --> 01:05:30.470 align:middle line:84% So probability, even though it's familiar to all of us, 01:05:30.470 --> 01:05:34.550 align:middle line:84% it uses some very powerful abstractions and definitions. 01:05:34.550 --> 01:05:36.030 align:middle line:84% So the most important of which is 01:05:36.030 --> 01:05:39.150 align:middle line:90% what's called the sample space. 01:05:39.150 --> 01:05:42.190 align:middle line:84% So in a basic probabilistic setup, the way to think about it 01:05:42.190 --> 01:05:45.590 align:middle line:84% is that points in this sample space. 01:05:45.590 --> 01:05:49.880 align:middle line:90% 01:05:49.880 --> 01:05:54.440 align:middle line:84% We associate them with possible outcomes, 01:05:54.440 --> 01:06:01.280 align:middle line:84% things that can happen out of the range of all possibilities. 01:06:01.280 --> 01:06:03.720 align:middle line:84% Just to do a familiar example, we 01:06:03.720 --> 01:06:06.220 align:middle line:84% could do something like roll a die. 01:06:06.220 --> 01:06:09.520 align:middle line:90% 01:06:09.520 --> 01:06:14.740 align:middle line:84% And if it's a normal die, we have six possibilities. 01:06:14.740 --> 01:06:17.480 align:middle line:84% That's our sample space, because that describes 01:06:17.480 --> 01:06:19.580 align:middle line:84% all of the possible things that could happen, 01:06:19.580 --> 01:06:21.800 align:middle line:90% what number shows up. 01:06:21.800 --> 01:06:23.560 align:middle line:84% Now the important thing is not just 01:06:23.560 --> 01:06:25.420 align:middle line:84% what's the range of possibilities, 01:06:25.420 --> 01:06:27.600 align:middle line:84% but really, what's the likelihood of each 01:06:27.600 --> 01:06:30.600 align:middle line:90% of these different outcomes. 01:06:30.600 --> 01:06:37.720 align:middle line:84% So to each point x in this sample space, 01:06:37.720 --> 01:06:44.090 align:middle line:84% we're going to assign some probability p of x which 01:06:44.090 --> 01:06:46.330 align:middle line:90% has the following properties. 01:06:46.330 --> 01:06:47.810 align:middle line:84% These are all very intuitive, so we 01:06:47.810 --> 01:06:50.610 align:middle line:84% need that the probability is non-negative, 01:06:50.610 --> 01:06:54.690 align:middle line:84% and this should be true for all points in the sample space. 01:06:54.690 --> 01:06:56.970 align:middle line:84% And we want that, in total, the sum 01:06:56.970 --> 01:07:03.970 align:middle line:84% of all the probabilities over the entire sample space is 1. 01:07:03.970 --> 01:07:06.910 align:middle line:84% So these are the things that define the setup 01:07:06.910 --> 01:07:07.850 align:middle line:90% for basic probability. 01:07:07.850 --> 01:07:11.050 align:middle line:84% We have a sample space and we have this probability associated 01:07:11.050 --> 01:07:12.890 align:middle line:90% with each one of the outcomes. 01:07:12.890 --> 01:07:15.950 align:middle line:84% We can certainly do simple examples like rolling a dice. 01:07:15.950 --> 01:07:18.890 align:middle line:84% If it's a fair die, then each one of these possibilities 01:07:18.890 --> 01:07:20.930 align:middle line:90% has one sixth probability. 01:07:20.930 --> 01:07:23.010 align:middle line:84% We're going to do a lot more exotic examples 01:07:23.010 --> 01:07:26.110 align:middle line:84% in class, things where the sample space might be infinite. 01:07:26.110 --> 01:07:27.270 align:middle line:90% That's perfectly fine. 01:07:27.270 --> 01:07:30.808 align:middle line:84% We can still have these axioms hold. 01:07:30.808 --> 01:07:32.850 align:middle line:84% An example of something we might be interested in 01:07:32.850 --> 01:07:36.050 align:middle line:84% is the number of times we flip a coin before we 01:07:36.050 --> 01:07:38.090 align:middle line:90% see our first heads. 01:07:38.090 --> 01:07:40.550 align:middle line:84% That, in principle, can go on forever. 01:07:40.550 --> 01:07:43.940 align:middle line:84% There's no bound on how large that sample space is, 01:07:43.940 --> 01:07:46.820 align:middle line:84% and we can still talk about all of the basic axioms 01:07:46.820 --> 01:07:51.020 align:middle line:84% of probability and reason about things. 01:07:51.020 --> 01:07:54.220 align:middle line:84% And we'll be particularly interested in what 01:07:54.220 --> 01:07:55.090 align:middle line:90% are called events. 01:07:55.090 --> 01:08:02.780 align:middle line:90% 01:08:02.780 --> 01:08:07.860 align:middle line:84% So events are just subsets of the sample space. 01:08:07.860 --> 01:08:10.500 align:middle line:84% They're just some subset of possibilities. 01:08:10.500 --> 01:08:13.820 align:middle line:84% And in that case, we can think about this probability 01:08:13.820 --> 01:08:17.020 align:middle line:84% of the entire event as just the sum of all 01:08:17.020 --> 01:08:20.939 align:middle line:84% of the individual outcomes that land in this event, 01:08:20.939 --> 01:08:22.919 align:middle line:90% their total probability. 01:08:22.919 --> 01:08:23.419 align:middle line:90% OK? 01:08:23.419 --> 01:08:27.259 align:middle line:90% 01:08:27.259 --> 01:08:29.779 align:middle line:84% So for example, you could think about an event 01:08:29.779 --> 01:08:32.209 align:middle line:90% like rolling an even number. 01:08:32.209 --> 01:08:37.939 align:middle line:90% 01:08:37.939 --> 01:08:41.490 align:middle line:84% That's not just one possible outcome in the sample space, 01:08:41.490 --> 01:08:45.390 align:middle line:90% but it's the set 2, 4, 6. 01:08:45.390 --> 01:08:51.830 align:middle line:84% And the last little baby piece of background 01:08:51.830 --> 01:08:58.270 align:middle line:84% is that we'll also be interested in complements of events. 01:08:58.270 --> 01:09:03.390 align:middle line:84% So for example, I could look at the probability of t bar. 01:09:03.390 --> 01:09:06.430 align:middle line:84% Sometimes you'll see it written as not T as well. 01:09:06.430 --> 01:09:10.510 align:middle line:84% And that's just everything that is in the sample 01:09:10.510 --> 01:09:12.710 align:middle line:90% space outside the set t. 01:09:12.710 --> 01:09:15.870 align:middle line:84% So for example, this is defined as all of the points 01:09:15.870 --> 01:09:20.029 align:middle line:84% in the outcome space that don't occur in this original event. 01:09:20.029 --> 01:09:22.510 align:middle line:84% So we can define this notion of complementary events, which 01:09:22.510 --> 01:09:26.910 align:middle line:84% is everything else besides the events in T. 01:09:26.910 --> 01:09:29.310 align:middle line:84% Basic fact is that a lot of these things 01:09:29.310 --> 01:09:31.550 align:middle line:84% behave the way you'd expect them to. 01:09:31.550 --> 01:09:34.149 align:middle line:84% So the probability of the complement is just 1 01:09:34.149 --> 01:09:37.590 align:middle line:84% minus the probability of the event itself. 01:09:37.590 --> 01:09:41.040 align:middle line:90% It's a very simple proof. 01:09:41.040 --> 01:09:42.640 align:middle line:84% We're going to stop now, but I'll 01:09:42.640 --> 01:09:44.223 align:middle line:84% tell you a little bit about what we're 01:09:44.223 --> 01:09:45.640 align:middle line:90% going to be doing next time. 01:09:45.640 --> 01:09:48.399 align:middle line:84% So now that we know at least the basic definitions 01:09:48.399 --> 01:09:50.060 align:middle line:84% for probability theory, we're going 01:09:50.060 --> 01:09:52.560 align:middle line:84% to be talking about conditional probability, which is always 01:09:52.560 --> 01:09:54.960 align:middle line:84% something that's very counterintuitive. 01:09:54.960 --> 01:09:57.460 align:middle line:84% So we're going to define what conditional probabilities are. 01:09:57.460 --> 01:09:59.085 align:middle line:84% I'll tell you some of the basic results 01:09:59.085 --> 01:10:00.340 align:middle line:90% and conditional probability. 01:10:00.340 --> 01:10:01.840 align:middle line:84% And then I'm going to give you a lot 01:10:01.840 --> 01:10:05.040 align:middle line:84% of fun examples of how people get 01:10:05.040 --> 01:10:06.797 align:middle line:90% conditional probability wrong. 01:10:06.797 --> 01:10:08.380 align:middle line:84% So there are a lot of famous examples, 01:10:08.380 --> 01:10:11.720 align:middle line:84% like the Monty Hall problem that many of you may have seen. 01:10:11.720 --> 01:10:14.460 align:middle line:84% For me, this was always a little bit abstract. 01:10:14.460 --> 01:10:16.720 align:middle line:84% Like, it's a cute mathematical puzzle. 01:10:16.720 --> 01:10:20.320 align:middle line:84% But I'm going to give you some scarier examples of real world 01:10:20.320 --> 01:10:23.440 align:middle line:84% things that happen with doctors where they misinterpret 01:10:23.440 --> 01:10:24.982 align:middle line:90% conditional probabilities. 01:10:24.982 --> 01:10:26.440 align:middle line:84% So we'll do that next time and I'll 01:10:26.440 --> 01:10:28.800 align:middle line:84% tell you some personal stories about that one. 01:10:28.800 --> 01:10:31.110 align:middle line:90% So see you on Thursday. 01:10:31.110 --> 01:10:37.000 align:middle line:90%