WEBVTT 00:00:00.000 --> 00:00:12.820 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:12.820 --> 00:00:15.180 align:middle line:90% CASEY RODRIGUEZ: OK. 00:00:15.180 --> 00:00:19.940 align:middle line:84% So I have to admit this is extremely awkward, 00:00:19.940 --> 00:00:21.998 align:middle line:90% lecturing to an empty room. 00:00:21.998 --> 00:00:23.540 align:middle line:84% So I have to imagine there's somebody 00:00:23.540 --> 00:00:26.150 align:middle line:84% on the other end actually listening to me at some point. 00:00:26.150 --> 00:00:29.170 align:middle line:90% 00:00:29.170 --> 00:00:32.380 align:middle line:84% Perhaps this is what YouTube stars have to go through 00:00:32.380 --> 00:00:33.630 align:middle line:90% at some point in their career. 00:00:33.630 --> 00:00:36.280 align:middle line:90% 00:00:36.280 --> 00:00:38.510 align:middle line:84% So what is the purpose of this course? 00:00:38.510 --> 00:00:51.770 align:middle line:84% So this is for 18 100A, Real Analysis. 00:00:51.770 --> 00:00:54.670 align:middle line:84% So the purpose of this course is twofold. 00:00:54.670 --> 00:01:03.790 align:middle line:90% 00:01:03.790 --> 00:01:07.520 align:middle line:84% Really, I think the first primary purpose of this course 00:01:07.520 --> 00:01:16.685 align:middle line:84% is to gain experience with proofs. 00:01:16.685 --> 00:01:21.060 align:middle line:90% 00:01:21.060 --> 00:01:23.730 align:middle line:84% So that means being able to read a proof, 00:01:23.730 --> 00:01:27.180 align:middle line:90% being able to write a proof. 00:01:27.180 --> 00:01:31.750 align:middle line:84% And the second statement, or the second purpose, 00:01:31.750 --> 00:01:37.950 align:middle line:84% which is supposed to be a way to obtain the first purpose, 00:01:37.950 --> 00:01:57.815 align:middle line:84% is to prove statements about real numbers, functions, 00:01:57.815 --> 00:01:58.315 align:middle line:90% and limits. 00:01:58.315 --> 00:02:07.950 align:middle line:90% 00:02:07.950 --> 00:02:08.940 align:middle line:90% OK. 00:02:08.940 --> 00:02:21.770 align:middle line:84% So the second part, this is the analysis part, OK? 00:02:21.770 --> 00:02:23.600 align:middle line:84% So for the first few lectures, we're 00:02:23.600 --> 00:02:30.680 align:middle line:84% going to do what maybe to some will be kind of review. 00:02:30.680 --> 00:02:33.652 align:middle line:84% And for most of you, a lot of this material 00:02:33.652 --> 00:02:35.360 align:middle line:84% in the first few lectures will be review. 00:02:35.360 --> 00:02:40.090 align:middle line:90% 00:02:40.090 --> 00:02:42.730 align:middle line:84% But it's a nice way to ease into the material. 00:02:42.730 --> 00:02:49.340 align:middle line:84% And things will most definitely pick up after a few lectures. 00:02:49.340 --> 00:02:56.580 align:middle line:84% So the first set of objects we're 00:02:56.580 --> 00:03:01.430 align:middle line:84% going to define and try to prove some statements about are sets. 00:03:01.430 --> 00:03:12.500 align:middle line:84% So definition-- and because I use a lot of shorthand, 00:03:12.500 --> 00:03:17.030 align:middle line:84% I will mostly write Dfn from now on instead 00:03:17.030 --> 00:03:18.470 align:middle line:90% of the entire word, definition. 00:03:18.470 --> 00:03:21.270 align:middle line:90% 00:03:21.270 --> 00:03:39.100 align:middle line:84% So a set is a collection of objects 00:03:39.100 --> 00:03:47.200 align:middle line:90% called elements or members, OK? 00:03:47.200 --> 00:03:54.240 align:middle line:84% Now, this course is supposed to be 00:03:54.240 --> 00:03:58.980 align:middle line:84% probably the first really rigorous course in math 00:03:58.980 --> 00:04:01.570 align:middle line:90% that many of you will deal with. 00:04:01.570 --> 00:04:06.220 align:middle line:84% So essentially, everything that we talk about 00:04:06.220 --> 00:04:09.310 align:middle line:84% will be rigorously and unambiguously defined. 00:04:09.310 --> 00:04:11.630 align:middle line:84% But we do have to start somewhere. 00:04:11.630 --> 00:04:16.329 align:middle line:84% And so maybe you think this word, 00:04:16.329 --> 00:04:18.610 align:middle line:84% "collection," is a little ambiguous. 00:04:18.610 --> 00:04:22.990 align:middle line:90% And perhaps you should. 00:04:22.990 --> 00:04:28.270 align:middle line:84% But to actually build up set theory from the ground up 00:04:28.270 --> 00:04:34.660 align:middle line:84% would take us quite beyond the scope of this class and too far 00:04:34.660 --> 00:04:37.810 align:middle line:84% afield of the things that we want to do, 00:04:37.810 --> 00:04:41.290 align:middle line:84% or at least that what I want to do. 00:04:41.290 --> 00:04:44.290 align:middle line:84% OK, so a set is just a collection of objects 00:04:44.290 --> 00:04:45.475 align:middle line:90% called elements, or members. 00:04:45.475 --> 00:04:48.520 align:middle line:90% 00:04:48.520 --> 00:04:55.940 align:middle line:84% There is the simplest set to define: 00:04:55.940 --> 00:05:06.890 align:middle line:84% the empty set is the set with no elements. 00:05:06.890 --> 00:05:16.750 align:middle line:84% And we denote it by this symbol here-- a circle with a dash 00:05:16.750 --> 00:05:17.250 align:middle line:90% through it. 00:05:17.250 --> 00:05:22.590 align:middle line:90% 00:05:22.590 --> 00:05:27.480 align:middle line:84% OK, so with new math typically comes new notation, 00:05:27.480 --> 00:05:29.040 align:middle line:90% new symbols that you use. 00:05:29.040 --> 00:05:33.240 align:middle line:84% So let me introduce a few shorthand notations we'll 00:05:33.240 --> 00:05:37.240 align:middle line:90% use throughout the course. 00:05:37.240 --> 00:05:41.010 align:middle line:84% I mean, quite honestly, this is a little bit 00:05:41.010 --> 00:05:42.570 align:middle line:90% of the fun of doing higher math. 00:05:42.570 --> 00:05:46.080 align:middle line:90% You get all these funny symbols. 00:05:46.080 --> 00:05:48.787 align:middle line:84% And a very accomplished mathematician 00:05:48.787 --> 00:05:50.370 align:middle line:84% at the University of Chicago, one time 00:05:50.370 --> 00:05:54.240 align:middle line:84% said, you're really only interested in the math 00:05:54.240 --> 00:05:56.310 align:middle line:84% where that has the symbols you like 00:05:56.310 --> 00:05:59.810 align:middle line:90% to write over and over again. 00:05:59.810 --> 00:06:04.140 align:middle line:84% So some notation-- a and this symbol 00:06:04.140 --> 00:06:07.930 align:middle line:84% which this symbol, which looks like a e-S 00:06:07.930 --> 00:06:22.400 align:middle line:84% means a is an element of S. A with a dash 00:06:22.400 --> 00:06:27.170 align:middle line:84% through this little e means everything here but. 00:06:27.170 --> 00:06:30.760 align:middle line:90% 00:06:30.760 --> 00:06:36.630 align:middle line:90% So a is not an element of S. 00:06:36.630 --> 00:06:41.640 align:middle line:84% This upside down A means for all. 00:06:41.640 --> 00:06:44.190 align:middle line:90% It's shorthand for all. 00:06:44.190 --> 00:06:51.210 align:middle line:90% Backwards E means there exists. 00:06:51.210 --> 00:06:52.860 align:middle line:90% And a couple of more-- 00:06:52.860 --> 00:06:55.380 align:middle line:90% 00:06:55.380 --> 00:07:05.440 align:middle line:84% if you see an arrow like this, this means implies. 00:07:05.440 --> 00:07:07.150 align:middle line:90% So I've written down one thing. 00:07:07.150 --> 00:07:09.100 align:middle line:90% This implies the next statement. 00:07:09.100 --> 00:07:12.340 align:middle line:90% I'll put an arrow between them. 00:07:12.340 --> 00:07:21.810 align:middle line:84% And an arrow going both ways means if and only 00:07:21.810 --> 00:07:25.740 align:middle line:84% if, meaning if I have a statement P if and only if Q, 00:07:25.740 --> 00:07:31.200 align:middle line:84% that means statement P implies Q and statement Q implies 00:07:31.200 --> 00:07:34.960 align:middle line:90% P, all right? 00:07:34.960 --> 00:07:39.820 align:middle line:84% And if you need a quick refresher on basic logic, 00:07:39.820 --> 00:07:43.910 align:middle line:84% you can find that in the appendix of the textbook. 00:07:43.910 --> 00:07:46.610 align:middle line:90% 00:07:46.610 --> 00:07:54.810 align:middle line:84% OK, so that's the basic definition of set, empty set. 00:07:54.810 --> 00:07:59.020 align:middle line:90% 00:07:59.020 --> 00:08:04.400 align:middle line:84% So set A-- another definition to set 00:08:04.400 --> 00:08:17.020 align:middle line:84% A is subset of B, which we write A, this little symbol 00:08:17.020 --> 00:08:26.650 align:middle line:84% that looks like a C, B If every element in A 00:08:26.650 --> 00:08:30.760 align:middle line:84% is an element in B. Little a's in capital A 00:08:30.760 --> 00:08:39.590 align:middle line:90% means little a is in capital B. 00:08:39.590 --> 00:08:48.440 align:middle line:90% So two sets are equal-- 00:08:48.440 --> 00:08:51.430 align:middle line:90% we write A equals B-- 00:08:51.430 --> 00:09:15.950 align:middle line:84% if A is a subset of B and B is a subset of A. 00:09:15.950 --> 00:09:29.120 align:middle line:84% And A is a proper subset of B if A is a subset of B 00:09:29.120 --> 00:09:36.020 align:middle line:84% and A does not equal B. And we typically 00:09:36.020 --> 00:09:40.700 align:middle line:84% write that by A and with a dash going 00:09:40.700 --> 00:09:45.270 align:middle line:84% through a line underneath the C to signify that it's not equal. 00:09:45.270 --> 00:09:49.610 align:middle line:84% So think of it as not, so less than or equal to, 00:09:49.610 --> 00:09:52.790 align:middle line:84% but not equal to is one way to think about it. 00:09:52.790 --> 00:09:57.410 align:middle line:90% 00:09:57.410 --> 00:10:02.090 align:middle line:84% OK, so let me say something since I'm now 00:10:02.090 --> 00:10:06.260 align:middle line:90% 1, 2, 3 definitions in. 00:10:06.260 --> 00:10:09.230 align:middle line:84% So definitions are a fact of life when it comes to math. 00:10:09.230 --> 00:10:10.910 align:middle line:84% In the beginning of any subject, there's 00:10:10.910 --> 00:10:14.210 align:middle line:84% going to be a lot of definitions because we have to have objects 00:10:14.210 --> 00:10:16.620 align:middle line:90% we want to talk about. 00:10:16.620 --> 00:10:21.570 align:middle line:84% And we have to have these unambiguously defined objects. 00:10:21.570 --> 00:10:24.960 align:middle line:84% So it may seem like there's going 00:10:24.960 --> 00:10:28.320 align:middle line:84% to be a lot of definitions now, but this will let up. 00:10:28.320 --> 00:10:30.660 align:middle line:84% And we will start proving some theorems, which 00:10:30.660 --> 00:10:34.185 align:middle line:90% are facts, about these objects. 00:10:34.185 --> 00:10:36.060 align:middle line:84% These are the things that we're really after. 00:10:36.060 --> 00:10:39.330 align:middle line:84% We're not really after just making up definitions. 00:10:39.330 --> 00:10:45.600 align:middle line:84% Definitions are meant to be a rigorous way 00:10:45.600 --> 00:10:48.330 align:middle line:84% of defining an object we're interested in studying. 00:10:48.330 --> 00:10:52.146 align:middle line:84% We're interested in proving theorems, facts about them. 00:10:52.146 --> 00:10:55.800 align:middle line:90% 00:10:55.800 --> 00:11:01.055 align:middle line:84% So again, a lot of this is just probably review. 00:11:01.055 --> 00:11:09.260 align:middle line:90% 00:11:09.260 --> 00:11:19.690 align:middle line:84% When we describe sets, we will use these braces 00:11:19.690 --> 00:11:22.570 align:middle line:84% and maybe list the elements in here. 00:11:22.570 --> 00:11:29.050 align:middle line:84% Or we will describe it as x in some set A, 00:11:29.050 --> 00:11:34.750 align:middle line:90% satisfying some property P of x. 00:11:34.750 --> 00:11:37.570 align:middle line:84% Or we won't write this x and A part. 00:11:37.570 --> 00:11:42.400 align:middle line:84% We'll just write all objects x satisfying 00:11:42.400 --> 00:11:47.650 align:middle line:84% x, as being an element of whatever universe we're in, 00:11:47.650 --> 00:11:51.316 align:middle line:84% that satisfy property P of x, OK? 00:11:51.316 --> 00:11:58.650 align:middle line:84% So again, you should read this as all satisfying property 00:11:58.650 --> 00:12:00.960 align:middle line:90% P of x. 00:12:00.960 --> 00:12:03.570 align:middle line:90% So the basic examples-- 00:12:03.570 --> 00:12:07.030 align:middle line:90% 00:12:07.030 --> 00:12:10.000 align:middle line:84% and this is-- you should expect this 00:12:10.000 --> 00:12:14.110 align:middle line:84% after seeing any non-trivial definition. 00:12:14.110 --> 00:12:16.910 align:middle line:84% If you were here, I would ask you to call me out, 00:12:16.910 --> 00:12:19.240 align:middle line:90% so I'll have to police myself. 00:12:19.240 --> 00:12:24.880 align:middle line:84% But after every semi-interesting definition, 00:12:24.880 --> 00:12:26.900 align:middle line:90% you should see examples, OK? 00:12:26.900 --> 00:12:31.240 align:middle line:84% This is how you actually learn about these things, 00:12:31.240 --> 00:12:33.740 align:middle line:84% or at least digest what these things are. 00:12:33.740 --> 00:12:35.350 align:middle line:84% So we have the natural numbers, which 00:12:35.350 --> 00:12:37.930 align:middle line:84% everyone is familiar with since they started to count-- 00:12:37.930 --> 00:12:42.680 align:middle line:90% 1, 2, 3, 4, and so on. 00:12:42.680 --> 00:12:49.360 align:middle line:84% We have the integers, which is 0, 1, minus 1, 2, minus 2. 00:12:49.360 --> 00:12:51.040 align:middle line:84% So all the natural numbers, along 00:12:51.040 --> 00:12:54.040 align:middle line:84% with their additive inverses, along with the 0 element, 00:12:54.040 --> 00:13:01.550 align:middle line:84% an additive identity, we have the rational numbers. 00:13:01.550 --> 00:13:10.240 align:middle line:84% So this is written as m over n such that m and n are integers 00:13:10.240 --> 00:13:12.100 align:middle line:90% and n does not equal to 0. 00:13:12.100 --> 00:13:15.830 align:middle line:90% 00:13:15.830 --> 00:13:29.550 align:middle line:84% And we have R, the real numbers, which I, as of right now, 00:13:29.550 --> 00:13:32.700 align:middle line:84% cannot actually write down what they are in terms 00:13:32.700 --> 00:13:36.510 align:middle line:90% of set-building notation. 00:13:36.510 --> 00:13:39.430 align:middle line:84% In fact, this will be our first goal of the course, 00:13:39.430 --> 00:13:44.160 align:middle line:84% is to give a proper description or definition of what 00:13:44.160 --> 00:13:49.320 align:middle line:90% R actually is. 00:13:49.320 --> 00:13:55.800 align:middle line:84% But you can think of this as you did in calculus, 00:13:55.800 --> 00:14:01.480 align:middle line:90% as Q along with-- 00:14:01.480 --> 00:14:09.650 align:middle line:84% so rationals and irrationals, like pi and 2 and these things. 00:14:09.650 --> 00:14:11.830 align:middle line:84% So this is fine to think about for now. 00:14:11.830 --> 00:14:17.880 align:middle line:90% 00:14:17.880 --> 00:14:27.520 align:middle line:84% So of course, I didn't have to use these. 00:14:27.520 --> 00:14:29.680 align:middle line:84% Maybe I'm interested in odd numbers. 00:14:29.680 --> 00:14:34.180 align:middle line:84% That's a set of numbers of the form 2m minus 1, 00:14:34.180 --> 00:14:38.180 align:middle line:90% where m is a natural number. 00:14:38.180 --> 00:14:44.110 align:middle line:84% So this is just 1, 3, 5, and so on, OK? 00:14:44.110 --> 00:14:49.600 align:middle line:84% And so note that we have the inclusions. 00:14:49.600 --> 00:14:52.600 align:middle line:84% Natural numbers are contained in the integers, which 00:14:52.600 --> 00:14:55.600 align:middle line:84% are contained in the rational numbers, which are contained 00:14:55.600 --> 00:14:57.880 align:middle line:90% in the real numbers, OK? 00:14:57.880 --> 00:15:03.767 align:middle line:84% And if you look at the history of why these things were 00:15:03.767 --> 00:15:05.350 align:middle line:84% thought up in the first place, I mean, 00:15:05.350 --> 00:15:11.040 align:middle line:84% they were thought up to solve polynomial equations 00:15:11.040 --> 00:15:15.350 align:middle line:84% that you couldn't solve in the number system before. 00:15:15.350 --> 00:15:17.630 align:middle line:84% Integers were created because I could not 00:15:17.630 --> 00:15:20.750 align:middle line:84% solve the equation x plus 1 equals 00:15:20.750 --> 00:15:23.450 align:middle line:90% 0 in the natural numbers. 00:15:23.450 --> 00:15:25.790 align:middle line:84% Rationals were thought of because I could not 00:15:25.790 --> 00:15:31.130 align:middle line:84% solve the equation 2x plus 1 equals 0 in the integers. 00:15:31.130 --> 00:15:34.340 align:middle line:84% And the real numbers were thought of because I cannot 00:15:34.340 --> 00:15:37.850 align:middle line:84% solve the equation x squared minus 2 equals 0 00:15:37.850 --> 00:15:39.300 align:middle line:90% in the rational numbers. 00:15:39.300 --> 00:15:43.520 align:middle line:84% Now, I can't solve the equation x squared plus 1 equals 0 00:15:43.520 --> 00:15:48.710 align:middle line:84% in the real numbers, which led to the creation 00:15:48.710 --> 00:15:50.390 align:middle line:90% of complex numbers. 00:15:50.390 --> 00:15:53.570 align:middle line:84% But we will not deal with complex numbers in this class. 00:15:53.570 --> 00:15:55.940 align:middle line:84% Although hopefully, if you keep studying analysis, 00:15:55.940 --> 00:15:58.280 align:middle line:84% you go on to complex analysis, which 00:15:58.280 --> 00:16:07.890 align:middle line:84% is really a beautiful subject of study to this day. 00:16:07.890 --> 00:16:08.615 align:middle line:90% So as I said-- 00:16:08.615 --> 00:16:11.720 align:middle line:90% 00:16:11.720 --> 00:16:14.830 align:middle line:90% let me write this here-- 00:16:14.830 --> 00:16:18.662 align:middle line:84% our first goal, real goal of the class-- and this 00:16:18.662 --> 00:16:19.870 align:middle line:90% is something to keep in mind. 00:16:19.870 --> 00:16:22.810 align:middle line:84% We're not going to do it right now. 00:16:22.810 --> 00:16:36.060 align:middle line:84% Our first real goal is to describe what R is, OK? 00:16:36.060 --> 00:16:39.660 align:middle line:84% I mean, if we're going to be proving statements about 00:16:39.660 --> 00:16:42.000 align:middle line:84% the real numbers, functions of real numbers, and limits, 00:16:42.000 --> 00:16:45.240 align:middle line:84% whatever-- those limits that you learned in calculus-- 00:16:45.240 --> 00:16:49.748 align:middle line:84% then we have to be able to really describe 00:16:49.748 --> 00:16:51.540 align:middle line:84% what we're starting with, the real numbers. 00:16:51.540 --> 00:16:55.240 align:middle line:90% 00:16:55.240 --> 00:17:00.385 align:middle line:84% OK, so let's get back to sets, to our review of sets. 00:17:00.385 --> 00:17:05.109 align:middle line:90% 00:17:05.109 --> 00:17:06.829 align:middle line:90% So there were some examples. 00:17:06.829 --> 00:17:09.490 align:middle line:90% We have a few more definitions. 00:17:09.490 --> 00:17:14.030 align:middle line:90% 00:17:14.030 --> 00:17:24.720 align:middle line:84% The union of two sets, A and B, is the set which we write-- 00:17:24.720 --> 00:17:30.090 align:middle line:84% so this is how we denote it, A U B. This 00:17:30.090 --> 00:17:34.671 align:middle line:90% is the set of all elements x. 00:17:34.671 --> 00:17:49.580 align:middle line:90% x is in A or x is in B. 00:17:49.580 --> 00:17:54.140 align:middle line:90% The intersection of A and B-- 00:17:54.140 --> 00:17:56.570 align:middle line:90% so this was defining the union. 00:17:56.570 --> 00:17:58.310 align:middle line:84% This was defining the intersection-- 00:17:58.310 --> 00:18:02.660 align:middle line:90% 00:18:02.660 --> 00:18:10.080 align:middle line:84% is the set A cap B. And this is a set of all 00:18:10.080 --> 00:18:23.280 align:middle line:84% x's so that x is in A and x is in B. 00:18:23.280 --> 00:18:26.460 align:middle line:84% So the union is take all the things from A, 00:18:26.460 --> 00:18:29.340 align:middle line:84% take all the things from B, and put them together 00:18:29.340 --> 00:18:30.660 align:middle line:90% in one big basket. 00:18:30.660 --> 00:18:33.630 align:middle line:84% The intersection is just take the things 00:18:33.630 --> 00:18:35.235 align:middle line:90% that A and B have in common. 00:18:35.235 --> 00:18:39.050 align:middle line:90% 00:18:39.050 --> 00:18:53.640 align:middle line:84% The set difference of A with respect to B 00:18:53.640 --> 00:19:05.620 align:middle line:84% is the set A backslash B. This is 00:19:05.620 --> 00:19:08.200 align:middle line:84% the set of all elements in A such 00:19:08.200 --> 00:19:28.590 align:middle line:84% that x is not in B. The complement of A is the set A-- 00:19:28.590 --> 00:19:31.860 align:middle line:84% so this is how I'm denoting the set. 00:19:31.860 --> 00:19:34.800 align:middle line:84% The next part is how I'm defining the set. 00:19:34.800 --> 00:19:44.601 align:middle line:84% This is a set of all elements in our universe that is not in A. 00:19:44.601 --> 00:19:50.990 align:middle line:84% And when I say universe, I don't mean this universe necessarily. 00:19:50.990 --> 00:19:54.020 align:middle line:84% I mean, if we're looking at subsets of R, 00:19:54.020 --> 00:20:00.760 align:middle line:84% the complement is generally with respect to R. 00:20:00.760 --> 00:20:03.085 align:middle line:84% Or if all of our sets are subsets of Q, 00:20:03.085 --> 00:20:06.070 align:middle line:84% then our universe would be Q, the rationals. 00:20:06.070 --> 00:20:08.035 align:middle line:84% And we're taking the complement in there. 00:20:08.035 --> 00:20:12.565 align:middle line:90% 00:20:12.565 --> 00:20:28.805 align:middle line:84% Two sets are disjoint if their intersection is empty, OK? 00:20:28.805 --> 00:20:32.280 align:middle line:90% 00:20:32.280 --> 00:20:35.730 align:middle line:84% So it took me quite a long time to figure out 00:20:35.730 --> 00:20:40.585 align:middle line:84% this compliment has an E in the middle as opposed to an I, 00:20:40.585 --> 00:20:42.460 align:middle line:84% as in the compliment you would give a friend. 00:20:42.460 --> 00:20:45.240 align:middle line:90% 00:20:45.240 --> 00:20:47.760 align:middle line:84% I had to do a lot of spell-checking in my thesis 00:20:47.760 --> 00:20:50.800 align:middle line:84% when my advisor pointed that out. 00:20:50.800 --> 00:20:52.590 align:middle line:84% So this is just something to keep in mind. 00:20:52.590 --> 00:20:54.780 align:middle line:84% This complement has an E in the middle of it. 00:20:54.780 --> 00:20:57.544 align:middle line:90% 00:20:57.544 --> 00:21:05.860 align:middle line:84% OK, so let me just draw a quick picture. 00:21:05.860 --> 00:21:10.530 align:middle line:84% So this blob over here is A. This 00:21:10.530 --> 00:21:15.120 align:middle line:84% is a set B. This is a set C. In fact, let's 00:21:15.120 --> 00:21:16.590 align:middle line:90% make this a little more-- 00:21:16.590 --> 00:21:18.810 align:middle line:90% OK, let's keep C there. 00:21:18.810 --> 00:21:29.100 align:middle line:84% Then what I have here, that's A intersect B. This bit 00:21:29.100 --> 00:21:31.620 align:middle line:84% over here, with the lines going this way 00:21:31.620 --> 00:21:38.280 align:middle line:84% but not including this, this is A take away B, A backslash B. 00:21:38.280 --> 00:21:45.150 align:middle line:84% And OK, so that was not meant to be 00:21:45.150 --> 00:21:49.770 align:middle line:84% along the same direction as this one, so let's go vertical. 00:21:49.770 --> 00:21:59.540 align:middle line:84% And everything with a vertical line is A intersect B, OK? 00:21:59.540 --> 00:22:04.850 align:middle line:84% So A backslash has the lines going this direction. 00:22:04.850 --> 00:22:07.730 align:middle line:84% A intersect has the lines going this direction. 00:22:07.730 --> 00:22:10.410 align:middle line:90% 00:22:10.410 --> 00:22:14.430 align:middle line:84% A union B has the lines going vertical. 00:22:14.430 --> 00:22:18.450 align:middle line:84% And C is way over here not touching any of A and B. 00:22:18.450 --> 00:22:33.080 align:middle line:84% So A and C are disjoint and B and C are disjoint. 00:22:33.080 --> 00:22:37.487 align:middle line:90% 00:22:37.487 --> 00:22:38.820 align:middle line:90% OK, they have nothing in common. 00:22:38.820 --> 00:22:44.470 align:middle line:90% 00:22:44.470 --> 00:22:46.270 align:middle line:84% OK, so this was a lot of definitions. 00:22:46.270 --> 00:22:54.510 align:middle line:90% 00:22:54.510 --> 00:22:56.940 align:middle line:84% We have not proven a single statement yet, 00:22:56.940 --> 00:22:59.370 align:middle line:90% so it's about time we do. 00:22:59.370 --> 00:23:03.300 align:middle line:84% This is probably one of the most basic theorems 00:23:03.300 --> 00:23:06.060 align:middle line:84% one can prove at the start of a Real Analysis class 00:23:06.060 --> 00:23:09.030 align:middle line:90% or any class about proofs. 00:23:09.030 --> 00:23:14.310 align:middle line:84% This is analogous to when you write your first Hello World 00:23:14.310 --> 00:23:17.860 align:middle line:90% program in a programming class. 00:23:17.860 --> 00:23:30.040 align:middle line:84% So let me state the theorem, which is DeMorgan's Laws. 00:23:30.040 --> 00:23:32.760 align:middle line:90% 00:23:32.760 --> 00:23:34.890 align:middle line:84% And the statement is the following. 00:23:34.890 --> 00:23:46.455 align:middle line:84% So if A, B, C are sets, then I have several things I can say. 00:23:46.455 --> 00:23:51.770 align:middle line:90% 00:23:51.770 --> 00:23:55.520 align:middle line:84% The union of B and C, taking their complement, 00:23:55.520 --> 00:24:00.410 align:middle line:84% this is the intersection of the complements. 00:24:00.410 --> 00:24:03.080 align:middle line:84% So the complement of the union is the intersection 00:24:03.080 --> 00:24:06.320 align:middle line:90% of the compliments. 00:24:06.320 --> 00:24:10.070 align:middle line:84% If I take their intersection and take the complement, 00:24:10.070 --> 00:24:14.510 align:middle line:84% this is the union of the complements. 00:24:14.510 --> 00:24:16.670 align:middle line:84% So the complement of the intersection 00:24:16.670 --> 00:24:21.170 align:middle line:90% is the union of the complements. 00:24:21.170 --> 00:24:24.680 align:middle line:84% Now, these are complements, meaning I am, in some sense, 00:24:24.680 --> 00:24:27.890 align:middle line:84% taking a set difference with respect to the entire universe. 00:24:27.890 --> 00:24:31.670 align:middle line:84% But I can make these things relative to some set A. 00:24:31.670 --> 00:24:36.020 align:middle line:84% So A take away B union C, this is the same 00:24:36.020 --> 00:24:45.840 align:middle line:84% as A take away B intersect A take away C. Really, again, 00:24:45.840 --> 00:24:48.465 align:middle line:84% you should think of this as a special case of one. 00:24:48.465 --> 00:24:51.360 align:middle line:90% 00:24:51.360 --> 00:24:54.580 align:middle line:84% Or at least if you were to write the proof-- 00:24:54.580 --> 00:24:56.413 align:middle line:84% I'm not going to because it's all 00:24:56.413 --> 00:24:58.080 align:middle line:84% going to be contained in the first two-- 00:24:58.080 --> 00:25:01.200 align:middle line:84% then you would see it's really just a proof of this guy. 00:25:01.200 --> 00:25:03.820 align:middle line:90% 00:25:03.820 --> 00:25:12.210 align:middle line:84% A take away B intersects C equals A take away B union A 00:25:12.210 --> 00:25:15.690 align:middle line:90% take away C, OK? 00:25:15.690 --> 00:25:22.730 align:middle line:84% So again, for a quick refresher about logic 00:25:22.730 --> 00:25:26.570 align:middle line:84% I would look at the appendix of the textbook. 00:25:26.570 --> 00:25:30.380 align:middle line:84% In general, so let me make a few remarks 00:25:30.380 --> 00:25:33.260 align:middle line:84% before we move on to the proof about typically 00:25:33.260 --> 00:25:35.630 align:middle line:90% how this is going to look. 00:25:35.630 --> 00:25:38.480 align:middle line:90% 00:25:38.480 --> 00:25:40.940 align:middle line:90% So this is some remarks. 00:25:40.940 --> 00:25:47.090 align:middle line:84% Typically, a theorem is a statement 00:25:47.090 --> 00:25:51.770 align:middle line:84% of the type P implies Q. Let me write this out in English. 00:25:51.770 --> 00:25:56.900 align:middle line:84% If some statement P holds, then Q-- 00:25:56.900 --> 00:26:00.200 align:middle line:84% for us, it's if I have any three sets, 00:26:00.200 --> 00:26:03.330 align:middle line:84% then I have these equalities between these operations 00:26:03.330 --> 00:26:03.830 align:middle line:90% of sets. 00:26:03.830 --> 00:26:08.350 align:middle line:90% 00:26:08.350 --> 00:26:10.405 align:middle line:84% So the general structure you'll see of the class 00:26:10.405 --> 00:26:13.450 align:middle line:84% is I have objects which I define unambiguously. 00:26:13.450 --> 00:26:16.450 align:middle line:84% I want to prove theorems now, meaning true statements 00:26:16.450 --> 00:26:17.830 align:middle line:90% about these objects. 00:26:17.830 --> 00:26:21.800 align:middle line:84% And the real meat is the proof part. 00:26:21.800 --> 00:26:28.180 align:middle line:84% So what is in this mysterious guy, the proof? 00:26:28.180 --> 00:26:29.575 align:middle line:90% It's quite simple. 00:26:29.575 --> 00:26:32.110 align:middle line:90% 00:26:32.110 --> 00:26:43.490 align:middle line:84% You start with-- you assume P, meaning what you were given, 00:26:43.490 --> 00:26:49.660 align:middle line:84% the hypotheses, the hypothesis, P, and-- 00:26:49.660 --> 00:27:00.620 align:middle line:84% I'm going to put dots here-- through logic and most 00:27:00.620 --> 00:27:05.030 align:middle line:84% definitely, most of the time some calculations, 00:27:05.030 --> 00:27:12.910 align:middle line:90% you arrive at Q is true. 00:27:12.910 --> 00:27:17.380 align:middle line:84% And most proofs are ended with this little box here, OK? 00:27:17.380 --> 00:27:22.260 align:middle line:84% So most proofs have this structure. 00:27:22.260 --> 00:27:25.380 align:middle line:90% I take my hypotheses. 00:27:25.380 --> 00:27:30.030 align:middle line:84% And these hypotheses mean something 00:27:30.030 --> 00:27:33.360 align:middle line:84% in terms of the definitions I have given. 00:27:33.360 --> 00:27:38.850 align:middle line:84% And now, I need to use these unambiguous definitions, 00:27:38.850 --> 00:27:41.700 align:middle line:84% along with logic and maybe some calculations, 00:27:41.700 --> 00:27:45.270 align:middle line:84% to conclude that statement Q is true. 00:27:45.270 --> 00:27:47.520 align:middle line:90% That is the essence of a proof. 00:27:47.520 --> 00:27:54.020 align:middle line:90% That is all there is to it. 00:27:54.020 --> 00:27:55.960 align:middle line:84% Now, that doesn't mean it's a simple thing 00:27:55.960 --> 00:27:57.550 align:middle line:90% to learn how to do. 00:27:57.550 --> 00:28:00.010 align:middle line:90% That's the point of this course. 00:28:00.010 --> 00:28:06.245 align:middle line:84% But distilled down, that's what a proof is, OK? 00:28:06.245 --> 00:28:09.230 align:middle line:90% 00:28:09.230 --> 00:28:11.930 align:middle line:84% And Q-- so I said P usually means something in terms 00:28:11.930 --> 00:28:13.220 align:middle line:90% of the definitions we have. 00:28:13.220 --> 00:28:16.640 align:middle line:84% But also, Q will usually mean something in the definitions 00:28:16.640 --> 00:28:17.660 align:middle line:90% that we've given. 00:28:17.660 --> 00:28:22.970 align:middle line:84% And so our job is to verify Q. So let's 00:28:22.970 --> 00:28:28.370 align:middle line:90% go with proving this theorem. 00:28:28.370 --> 00:28:31.040 align:middle line:84% And in fact, I'm only going to prove property 1. 00:28:31.040 --> 00:28:37.330 align:middle line:84% Property 2, 3, and 4 I'll likely put on the homework. 00:28:37.330 --> 00:28:44.330 align:middle line:90% So let B and C be sets. 00:28:44.330 --> 00:28:46.580 align:middle line:84% So I mean, this is the only hypothesis I get. 00:28:46.580 --> 00:28:50.120 align:middle line:84% I'm trying to prove that B union C 00:28:50.120 --> 00:28:52.730 align:middle line:84% complement equals the intersection 00:28:52.730 --> 00:28:54.350 align:middle line:90% of the complements. 00:28:54.350 --> 00:28:55.680 align:middle line:90% So what does that mean? 00:28:55.680 --> 00:28:56.720 align:middle line:90% So we want to prove. 00:28:56.720 --> 00:29:04.483 align:middle line:84% So this is-- it's quite helpful, especially 00:29:04.483 --> 00:29:06.150 align:middle line:84% when you're first starting to do proofs, 00:29:06.150 --> 00:29:09.100 align:middle line:84% to write down what you're actually trying to prove. 00:29:09.100 --> 00:29:13.600 align:middle line:84% So even though I have this statement here, 00:29:13.600 --> 00:29:16.150 align:middle line:84% it's an equality between two sets. 00:29:16.150 --> 00:29:20.650 align:middle line:84% Equality between two sets means something specifically, right? 00:29:20.650 --> 00:29:23.710 align:middle line:90% We have that in our definition-- 00:29:23.710 --> 00:29:28.840 align:middle line:84% where is it-- over there that two sets 00:29:28.840 --> 00:29:33.230 align:middle line:84% are equal if one is a subset of the other and vice versa. 00:29:33.230 --> 00:29:34.690 align:middle line:90% So that's what we have to prove. 00:29:34.690 --> 00:29:39.460 align:middle line:84% We have to prove that the left side, B union C complement, 00:29:39.460 --> 00:29:42.940 align:middle line:84% is a subset of B complement intersect C complement and vice 00:29:42.940 --> 00:29:43.790 align:middle line:90% versa. 00:29:43.790 --> 00:29:56.685 align:middle line:84% So we want to prove that is a subset of B and-- 00:29:56.685 --> 00:30:05.030 align:middle line:90% 00:30:05.030 --> 00:30:05.530 align:middle line:90% OK? 00:30:05.530 --> 00:30:09.780 align:middle line:90% 00:30:09.780 --> 00:30:11.512 align:middle line:84% So that's what the equality means. 00:30:11.512 --> 00:30:12.720 align:middle line:90% That's what we have to prove. 00:30:12.720 --> 00:30:15.720 align:middle line:84% We have to prove those two statements now, OK? 00:30:15.720 --> 00:30:18.510 align:middle line:84% And that's as distilled down as far as we can go. 00:30:18.510 --> 00:30:21.130 align:middle line:90% 00:30:21.130 --> 00:30:22.200 align:middle line:90% So let's prove this. 00:30:22.200 --> 00:30:27.960 align:middle line:84% Now, we'll prove this using, again, logic and what 00:30:27.960 --> 00:30:31.040 align:middle line:90% these things actually are. 00:30:31.040 --> 00:30:35.790 align:middle line:84% So let's prove this first statement here. 00:30:35.790 --> 00:30:40.370 align:middle line:84% So I have to show that every element in this set 00:30:40.370 --> 00:30:42.860 align:middle line:90% is an element of this set. 00:30:42.860 --> 00:30:49.680 align:middle line:84% So I'll even write this down as WTS. 00:30:49.680 --> 00:30:50.680 align:middle line:90% That means Want To Show. 00:30:50.680 --> 00:30:54.250 align:middle line:90% 00:30:54.250 --> 00:30:56.740 align:middle line:84% This is the first thing we'll show. 00:30:56.740 --> 00:30:58.390 align:middle line:84% As we go on, I'm not going to write 00:30:58.390 --> 00:31:00.650 align:middle line:90% as much as I'm doing right now. 00:31:00.650 --> 00:31:03.070 align:middle line:84% But this is the first theorem and proof you're seeing, 00:31:03.070 --> 00:31:06.470 align:middle line:84% so I should write down quite a bit. 00:31:06.470 --> 00:31:08.770 align:middle line:84% So the first thing we want to show 00:31:08.770 --> 00:31:11.410 align:middle line:90% is we have this inclusion, OK? 00:31:11.410 --> 00:31:13.940 align:middle line:84% That means every element here is an element here. 00:31:13.940 --> 00:31:22.430 align:middle line:84% So let x be in B union C complement. 00:31:22.430 --> 00:31:25.280 align:middle line:84% And now, we're going to trace what this means. 00:31:25.280 --> 00:31:28.200 align:middle line:84% And we'll eventually arrive at x as in this. 00:31:28.200 --> 00:31:36.000 align:middle line:84% So then x is not in B union C. That's just 00:31:36.000 --> 00:31:39.180 align:middle line:84% the definition of the complement. 00:31:39.180 --> 00:31:51.810 align:middle line:84% Now, x is not in B union C means x is not in B and x is not in C 00:31:51.810 --> 00:31:53.160 align:middle line:90% because the union is-- 00:31:53.160 --> 00:31:56.160 align:middle line:84% something's in the union if it's in B or C. 00:31:56.160 --> 00:32:02.400 align:middle line:84% So something's not in the union if it's not in B and not in C. 00:32:02.400 --> 00:32:07.120 align:middle line:84% Now, this implies, simply again by the definition 00:32:07.120 --> 00:32:09.880 align:middle line:84% of what it means to be in the complement, 00:32:09.880 --> 00:32:15.520 align:middle line:84% x is in B complement and x is in C complement. 00:32:15.520 --> 00:32:20.410 align:middle line:90% 00:32:20.410 --> 00:32:23.380 align:middle line:84% But this, again, is simply the definition 00:32:23.380 --> 00:32:31.000 align:middle line:84% of x being in B complement intersect C complement, OK? 00:32:31.000 --> 00:32:35.500 align:middle line:84% So you see, we started off with an element in this guy 00:32:35.500 --> 00:32:41.430 align:middle line:84% and we showed that it's also an element of the right-hand side. 00:32:41.430 --> 00:32:51.210 align:middle line:84% So thus, B union C complement is contained in B complement 00:32:51.210 --> 00:32:54.140 align:middle line:90% intersect C complement. 00:32:54.140 --> 00:32:56.945 align:middle line:84% Now, we want to do this other inclusion here. 00:32:56.945 --> 00:33:10.976 align:middle line:90% 00:33:10.976 --> 00:33:16.700 align:middle line:84% Now, this is one of those rare situations where 00:33:16.700 --> 00:33:19.968 align:middle line:84% you get to essentially reverse the entire argument 00:33:19.968 --> 00:33:20.885 align:middle line:90% and get what you want. 00:33:20.885 --> 00:33:25.050 align:middle line:90% 00:33:25.050 --> 00:33:28.965 align:middle line:84% But let's just go through it in a linear fashion. 00:33:28.965 --> 00:33:33.220 align:middle line:90% 00:33:33.220 --> 00:33:38.730 align:middle line:84% Let's take something from here and show it's in here. 00:33:38.730 --> 00:33:43.340 align:middle line:84% So let x be in the intersection of the complements. 00:33:43.340 --> 00:33:48.770 align:middle line:84% Then that means x is in B complement 00:33:48.770 --> 00:33:53.750 align:middle line:90% and x is in C complement. 00:33:53.750 --> 00:33:58.790 align:middle line:90% That means x is not in B-- 00:33:58.790 --> 00:34:00.410 align:middle line:90% so that's this statement. 00:34:00.410 --> 00:34:03.770 align:middle line:84% That's the definition of being in the complement-- 00:34:03.770 --> 00:34:07.650 align:middle line:84% and x is not in C. That's, again, 00:34:07.650 --> 00:34:11.810 align:middle line:84% the definition of being in the complement. 00:34:11.810 --> 00:34:14.460 align:middle line:84% Now, just like we used here in this step, 00:34:14.460 --> 00:34:16.489 align:middle line:90% this is equivalent to-- 00:34:16.489 --> 00:34:17.929 align:middle line:90% so really, I should-- 00:34:17.929 --> 00:34:21.139 align:middle line:84% in this statement, I should have written 00:34:21.139 --> 00:34:23.570 align:middle line:84% this statement is equivalent to this statement, 00:34:23.570 --> 00:34:26.520 align:middle line:90% but we'll remove that. 00:34:26.520 --> 00:34:28.770 align:middle line:84% So x is not in B and x is not in C. 00:34:28.770 --> 00:34:36.030 align:middle line:84% This means x is not in their union, which 00:34:36.030 --> 00:34:46.429 align:middle line:84% implies that x is in the complement of the union, OK? 00:34:46.429 --> 00:35:00.520 align:middle line:84% So thus, we've proven is a subset of B. 00:35:00.520 --> 00:35:05.600 align:middle line:84% And since we've shown both sets are 00:35:05.600 --> 00:35:07.400 align:middle line:84% a subset of each other, that means, 00:35:07.400 --> 00:35:12.440 align:middle line:84% by the definition of two sets being equal, they are equal. 00:35:12.440 --> 00:35:21.910 align:middle line:90% 00:35:21.910 --> 00:35:24.240 align:middle line:84% Again, this box means really nothing. 00:35:24.240 --> 00:35:25.990 align:middle line:84% It just means that's the end of the proof. 00:35:25.990 --> 00:35:40.730 align:middle line:90% 00:35:40.730 --> 00:35:47.540 align:middle line:90% All right, let's move over here. 00:35:47.540 --> 00:35:54.142 align:middle line:90% 00:35:54.142 --> 00:35:54.850 align:middle line:90% This is terrible. 00:35:54.850 --> 00:36:10.550 align:middle line:90% 00:36:10.550 --> 00:36:13.175 align:middle line:84% And not everybody uses that little box to finish a proof. 00:36:13.175 --> 00:36:15.910 align:middle line:90% 00:36:15.910 --> 00:36:17.950 align:middle line:90% Some people don't put anything. 00:36:17.950 --> 00:36:20.350 align:middle line:84% When I was in graduate school, I was 00:36:20.350 --> 00:36:25.110 align:middle line:84% a TA for this guy named Paul Sally who 00:36:25.110 --> 00:36:32.840 align:middle line:84% was a fantastic teacher and really loved math, 00:36:32.840 --> 00:36:35.630 align:middle line:90% who would end-- 00:36:35.630 --> 00:36:40.580 align:middle line:84% So amazing story about this guy is, when I was his TA, 00:36:40.580 --> 00:36:43.890 align:middle line:90% he was in his 70s, I think. 00:36:43.890 --> 00:36:47.170 align:middle line:90% But he had also had diabetes. 00:36:47.170 --> 00:36:52.590 align:middle line:84% So he had lost both of his legs beneath his knees. 00:36:52.590 --> 00:36:55.170 align:middle line:90% He was also legally blind. 00:36:55.170 --> 00:36:58.260 align:middle line:90% And he had a patch over one eye. 00:36:58.260 --> 00:37:02.040 align:middle line:84% So he himself often referred to himself 00:37:02.040 --> 00:37:06.820 align:middle line:90% as the a pirate mathematician. 00:37:06.820 --> 00:37:11.300 align:middle line:84% But he would end his proofs with-- 00:37:11.300 --> 00:37:13.160 align:middle line:84% at least in his textbook-- he didn't 00:37:13.160 --> 00:37:15.260 align:middle line:84% ask me to do this on the board, thankfully. 00:37:15.260 --> 00:37:22.240 align:middle line:84% He would end his proofs with a picture of himself 00:37:22.240 --> 00:37:27.800 align:middle line:84% with this cob pipe that he had, very much 00:37:27.800 --> 00:37:31.510 align:middle line:90% in the pirate fashion. 00:37:31.510 --> 00:37:35.360 align:middle line:84% Anyways, OK, moving on from things that end proofs, 00:37:35.360 --> 00:37:45.070 align:middle line:84% let's go on to a next subject, induction. 00:37:45.070 --> 00:37:53.500 align:middle line:84% So induction is a way to prove theorems about natural numbers, 00:37:53.500 --> 00:37:55.480 align:middle line:90% OK? 00:37:55.480 --> 00:38:01.800 align:middle line:84% The theorem itself is more of a tool rather 00:38:01.800 --> 00:38:07.260 align:middle line:84% than an interesting fact on its own, OK? 00:38:07.260 --> 00:38:10.088 align:middle line:90% So let me state the theorem. 00:38:10.088 --> 00:38:11.880 align:middle line:84% And then we'll go over a couple of examples 00:38:11.880 --> 00:38:13.530 align:middle line:90% on how to use induction. 00:38:13.530 --> 00:38:20.940 align:middle line:90% So let me recall from-- 00:38:20.940 --> 00:38:22.650 align:middle line:90% I think I just erased it. 00:38:22.650 --> 00:38:24.000 align:middle line:90% N is the natural numbers. 00:38:24.000 --> 00:38:27.490 align:middle line:90% 00:38:27.490 --> 00:38:33.140 align:middle line:84% And it has an ordering, meaning-- 00:38:33.140 --> 00:38:36.290 align:middle line:84% so we'll precisely define what ordering means. 00:38:36.290 --> 00:38:39.020 align:middle line:84% But just in your head, this means the usual thing-- 00:38:39.020 --> 00:38:42.215 align:middle line:84% 1 is less than 2 is less than 3 is less than 4. 00:38:42.215 --> 00:38:47.910 align:middle line:90% 00:38:47.910 --> 00:38:51.010 align:middle line:84% So a property of the natural numbers, 00:38:51.010 --> 00:38:56.550 align:middle line:84% which will take as an axiom, is the well-ordering property. 00:38:56.550 --> 00:39:07.270 align:middle line:90% 00:39:07.270 --> 00:39:09.360 align:middle line:84% So an axiom is not something you prove. 00:39:09.360 --> 00:39:13.830 align:middle line:84% You assume this about the objects 00:39:13.830 --> 00:39:19.620 align:middle line:84% that you've defined or are studying up to this point. 00:39:19.620 --> 00:39:26.120 align:middle line:84% And so the statement is if I take 00:39:26.120 --> 00:39:33.930 align:middle line:84% a subset of natural numbers, which is non-empty, 00:39:33.930 --> 00:39:44.100 align:middle line:84% then S has a least element or smallest element. 00:39:44.100 --> 00:39:45.490 align:middle line:90% Now, what does this mean? 00:39:45.490 --> 00:39:50.610 align:middle line:84% Let me write this last statement out. 00:39:50.610 --> 00:39:51.360 align:middle line:90% i.e. 00:39:51.360 --> 00:39:56.502 align:middle line:90% There exists an x in S-- 00:39:56.502 --> 00:40:01.470 align:middle line:84% "st" I will often write, meaning such that or so that-- 00:40:01.470 --> 00:40:11.820 align:middle line:84% such that x is less than or equal to y for all y in S, OK? 00:40:11.820 --> 00:40:16.350 align:middle line:84% So every non-empty subset of the natural numbers 00:40:16.350 --> 00:40:19.720 align:middle line:90% has a smallest element, OK? 00:40:19.720 --> 00:40:22.030 align:middle line:84% We're going to take that as an axiom, 00:40:22.030 --> 00:40:25.240 align:middle line:84% as just a property of the natural numbers, which 00:40:25.240 --> 00:40:28.150 align:middle line:90% we'll assume. 00:40:28.150 --> 00:40:32.930 align:middle line:84% Now, using this axiom, we're going to prove-- 00:40:32.930 --> 00:40:36.200 align:middle line:84% it's not really often you hear it called 00:40:36.200 --> 00:40:38.780 align:middle line:84% as a principle of mathematical induction, 00:40:38.780 --> 00:40:41.870 align:middle line:84% but this will state it as a theorem instead of a principle, 00:40:41.870 --> 00:40:43.700 align:middle line:84% whatever a principle is supposed to be. 00:40:43.700 --> 00:41:00.880 align:middle line:84% So induction, so this is due to Pascal. 00:41:00.880 --> 00:41:05.660 align:middle line:84% Or at least in its first rigorous formulation 00:41:05.660 --> 00:41:26.940 align:middle line:84% is let Pn be a statement depending on natural number n. 00:41:26.940 --> 00:41:28.810 align:middle line:84% OK, so maybe we have some equality 00:41:28.810 --> 00:41:32.750 align:middle line:84% between two quantities that involves a natural number n, 00:41:32.750 --> 00:41:33.250 align:middle line:90% OK? 00:41:33.250 --> 00:41:37.020 align:middle line:84% That could be our statement P of n. 00:41:37.020 --> 00:41:40.890 align:middle line:90% Now, we're going to assume-- 00:41:40.890 --> 00:41:44.550 align:middle line:84% so what are our hypotheses about this statement? 00:41:44.550 --> 00:41:46.470 align:middle line:90% What's our if? 00:41:46.470 --> 00:41:51.510 align:middle line:84% Assume that this statement satisfies two properties. 00:41:51.510 --> 00:41:55.665 align:middle line:84% This first property is usually referred to as a base case. 00:41:55.665 --> 00:42:01.200 align:middle line:90% 00:42:01.200 --> 00:42:05.640 align:middle line:90% That is that P of 1 is true. 00:42:05.640 --> 00:42:10.845 align:middle line:84% And the second property is called the inductive step. 00:42:10.845 --> 00:42:18.350 align:middle line:90% 00:42:18.350 --> 00:42:24.700 align:middle line:84% So this statement satisfies the following property 00:42:24.700 --> 00:42:31.280 align:middle line:84% that if you assume P of m is true, 00:42:31.280 --> 00:42:35.950 align:middle line:84% then you can prove that P of m plus 1 is true. 00:42:35.950 --> 00:42:39.140 align:middle line:90% 00:42:39.140 --> 00:42:40.750 align:middle line:84% So I have a statement which satisfies 00:42:40.750 --> 00:42:43.630 align:middle line:90% both of these properties, OK? 00:42:43.630 --> 00:42:47.145 align:middle line:84% In particular, since I'm assuming P of 1 is true, 00:42:47.145 --> 00:42:50.500 align:middle line:84% by the second property, P of 2 is true. 00:42:50.500 --> 00:42:53.350 align:middle line:84% And then again by the second property, P of 3 is true. 00:42:53.350 --> 00:42:55.840 align:middle line:84% And then P of 4, and then P of 5. 00:42:55.840 --> 00:42:59.560 align:middle line:84% And so if you followed that last line of reasoning, 00:42:59.560 --> 00:43:02.740 align:middle line:84% this means you should be able to guess 00:43:02.740 --> 00:43:06.010 align:middle line:84% what the conclusion of this theorem is. 00:43:06.010 --> 00:43:21.790 align:middle line:84% Then P n is true for all natural numbers, OK? 00:43:21.790 --> 00:43:26.540 align:middle line:84% All right, so we're going to use the well-ordering property 00:43:26.540 --> 00:43:31.940 align:middle line:84% of the natural numbers to prove this theorem 00:43:31.940 --> 00:43:35.710 align:middle line:90% about the induction. 00:43:35.710 --> 00:43:41.920 align:middle line:90% OK, so we have our assumptions. 00:43:41.920 --> 00:43:44.190 align:middle line:84% I'm not going to-- although, I said over 00:43:44.190 --> 00:43:46.110 align:middle line:84% there let B, C be sets, I'm not going 00:43:46.110 --> 00:43:48.900 align:middle line:84% to rewrite the assumptions that we have about our statement 00:43:48.900 --> 00:43:53.520 align:middle line:84% P. We're just going to start trying to prove P of n 00:43:53.520 --> 00:43:54.870 align:middle line:90% is true for all n. 00:43:54.870 --> 00:43:59.640 align:middle line:84% So let me write our conclusion slightly differently. 00:43:59.640 --> 00:44:02.200 align:middle line:90% 00:44:02.200 --> 00:44:15.490 align:middle line:84% Let S be the set of all natural numbers such that P of n 00:44:15.490 --> 00:44:17.310 align:middle line:90% is not true. 00:44:17.310 --> 00:44:20.510 align:middle line:84% So what I want to show is that P of n is true for all n. 00:44:20.510 --> 00:44:26.490 align:middle line:84% So that's equivalent to saying we want 00:44:26.490 --> 00:44:34.960 align:middle line:90% to show that S is empty, OK? 00:44:34.960 --> 00:44:37.940 align:middle line:84% The set of natural numbers where P of n is not true, 00:44:37.940 --> 00:44:38.710 align:middle line:90% this is empty. 00:44:38.710 --> 00:44:42.820 align:middle line:84% This is equivalent to saying P of n is true for all n. 00:44:42.820 --> 00:44:45.370 align:middle line:90% 00:44:45.370 --> 00:44:46.960 align:middle line:84% And the way we're going to do this 00:44:46.960 --> 00:44:57.600 align:middle line:84% is another staple of mathematical proofs 00:44:57.600 --> 00:45:12.880 align:middle line:84% is trying to prove this by contradiction, OK? 00:45:12.880 --> 00:45:15.830 align:middle line:90% So what does that mean? 00:45:15.830 --> 00:45:21.280 align:middle line:84% Let me make a few comments about what that means, 00:45:21.280 --> 00:45:22.690 align:middle line:90% proof by a contradiction. 00:45:22.690 --> 00:46:01.760 align:middle line:90% 00:46:01.760 --> 00:46:05.390 align:middle line:84% OK, so in a proof by contradiction-- 00:46:05.390 --> 00:46:07.420 align:middle line:84% so this is-- what I'm about to write down 00:46:07.420 --> 00:46:09.980 align:middle line:90% is not part of the proof. 00:46:09.980 --> 00:46:13.735 align:middle line:84% This is commentary not to be included in the proof. 00:46:13.735 --> 00:46:16.430 align:middle line:90% 00:46:16.430 --> 00:46:18.290 align:middle line:84% What does it mean to say we're going 00:46:18.290 --> 00:46:23.180 align:middle line:84% to prove S is equal to the empty set by contradiction? 00:46:23.180 --> 00:46:28.490 align:middle line:84% We're going to assume that the statement we want to prove 00:46:28.490 --> 00:46:29.600 align:middle line:90% is false. 00:46:29.600 --> 00:46:31.880 align:middle line:84% Or not false, but we want to assume 00:46:31.880 --> 00:46:37.050 align:middle line:84% that the negation of the statement we want to prove 00:46:37.050 --> 00:46:43.170 align:middle line:84% is true and then arrive at a false statement, OK? 00:46:43.170 --> 00:46:46.740 align:middle line:90% So we want to assume-- 00:46:46.740 --> 00:46:48.593 align:middle line:90% this is what we're going to do. 00:46:48.593 --> 00:46:50.010 align:middle line:84% We're going to assume the negation 00:46:50.010 --> 00:46:51.840 align:middle line:84% of the statement we want to prove-- namely, 00:46:51.840 --> 00:46:58.850 align:middle line:90% S is non-empty, OK? 00:46:58.850 --> 00:47:14.030 align:middle line:84% And from this, we want to derive a false statement, OK? 00:47:14.030 --> 00:47:17.710 align:middle line:90% 00:47:17.710 --> 00:47:22.810 align:middle line:84% And so if we are to do-- if we were able to do that, then-- 00:47:22.810 --> 00:47:27.490 align:middle line:84% let me just say, again, you can check in the appendix 00:47:27.490 --> 00:47:34.210 align:middle line:84% or you can just believe me that the rules of logic 00:47:34.210 --> 00:47:48.340 align:middle line:84% then say that our initial assumption, that S was not 00:47:48.340 --> 00:47:52.680 align:middle line:84% empty, is false to begin with, OK? 00:47:52.680 --> 00:47:56.400 align:middle line:84% So rules of logic, meaning I cannot start from a true 00:47:56.400 --> 00:48:02.130 align:middle line:84% assumption and derive, in a logically consistent way, 00:48:02.130 --> 00:48:06.630 align:middle line:90% a false statement, OK? 00:48:06.630 --> 00:48:09.630 align:middle line:84% That is, if we believe that the rules of logic we're using 00:48:09.630 --> 00:48:19.070 align:middle line:84% are consistent, which that's a little bit hairy to talk about. 00:48:19.070 --> 00:48:24.630 align:middle line:84% But for our purposes of our class, 00:48:24.630 --> 00:48:28.340 align:middle line:84% you can believe me that the rules of logic we use-- 00:48:28.340 --> 00:48:30.590 align:middle line:84% or at least accept that the rules 00:48:30.590 --> 00:48:36.980 align:middle line:84% of logic we're going to use are consistent and sound. 00:48:36.980 --> 00:48:40.380 align:middle line:84% OK, so back to the proof at hand. 00:48:40.380 --> 00:48:42.770 align:middle line:84% We have this set of natural numbers 00:48:42.770 --> 00:48:44.520 align:middle line:90% where the statement is not true. 00:48:44.520 --> 00:48:46.103 align:middle line:90% We want to show it is empty. 00:48:46.103 --> 00:48:48.020 align:middle line:84% We're going to do it by contradiction, meaning 00:48:48.020 --> 00:48:50.450 align:middle line:84% we're going to assume the negation of the statement 00:48:50.450 --> 00:48:51.350 align:middle line:90% we want to prove-- 00:48:51.350 --> 00:48:53.000 align:middle line:90% namely, S is non-empty. 00:48:53.000 --> 00:48:55.910 align:middle line:84% And we're going to derive a false statement 00:48:55.910 --> 00:48:58.790 align:middle line:90% from that assumption, OK? 00:48:58.790 --> 00:49:00.170 align:middle line:90% And by the rules of logic-- 00:49:00.170 --> 00:49:02.360 align:middle line:84% that means that our initial assumption-- 00:49:02.360 --> 00:49:04.970 align:middle line:90% that S is non-empty-- 00:49:04.970 --> 00:49:09.360 align:middle line:90% is, in fact, false, OK? 00:49:09.360 --> 00:49:16.450 align:middle line:84% All right, so towards a contradiction, 00:49:16.450 --> 00:49:21.220 align:middle line:90% suppose that S is non-empty, OK? 00:49:21.220 --> 00:49:23.770 align:middle line:84% Now, we're going to use the well-ordering property 00:49:23.770 --> 00:49:26.540 align:middle line:90% of the natural numbers. 00:49:26.540 --> 00:49:37.870 align:middle line:84% By the well-ordering property of the natural numbers, 00:49:37.870 --> 00:49:48.190 align:middle line:90% S has a least element, x, OK? 00:49:48.190 --> 00:49:52.100 align:middle line:90% 00:49:52.100 --> 00:49:58.120 align:middle line:90% Now, what do we know about x? 00:49:58.120 --> 00:50:01.510 align:middle line:90% So first off, x cannot be 1, OK? 00:50:01.510 --> 00:50:03.970 align:middle line:84% S is a set where this property does not hold. 00:50:03.970 --> 00:50:06.460 align:middle line:90% x cannot be 1 because-- 00:50:06.460 --> 00:50:13.560 align:middle line:84% let me again rewrite this fact that S has the least element. 00:50:13.560 --> 00:50:18.900 align:middle line:84% Let me just reiterate that S has a least element in the set, OK? 00:50:18.900 --> 00:50:22.770 align:middle line:84% Now, x cannot be 1 because we're assuming the base case, 00:50:22.770 --> 00:50:24.700 align:middle line:90% meaning P of 1 is true. 00:50:24.700 --> 00:50:34.670 align:middle line:84% So since P of 1 is true, that means 1 is not 00:50:34.670 --> 00:50:39.750 align:middle line:90% an S, which means x is not 1. 00:50:39.750 --> 00:50:45.052 align:middle line:90% 00:50:45.052 --> 00:50:47.580 align:middle line:84% In particular, x must be bigger than 1. 00:50:47.580 --> 00:50:51.630 align:middle line:84% So x is some magical natural number out there bigger than 1 00:50:51.630 --> 00:50:56.990 align:middle line:84% that's the least element of this set S. 00:50:56.990 --> 00:51:09.700 align:middle line:84% OK, since x is the least element of S-- 00:51:09.700 --> 00:51:11.140 align:middle line:90% so let me draw. 00:51:11.140 --> 00:51:15.290 align:middle line:90% 00:51:15.290 --> 00:51:18.520 align:middle line:84% On the number line, we have 1, 2, 3, 4. 00:51:18.520 --> 00:51:21.250 align:middle line:84% Out there is some magic point x, which 00:51:21.250 --> 00:51:25.600 align:middle line:84% is the least element of S. And the rest of the subset S 00:51:25.600 --> 00:51:30.790 align:middle line:84% lies to the right of this number x, right? 00:51:30.790 --> 00:51:34.210 align:middle line:84% Because it's the least element of S. 00:51:34.210 --> 00:51:41.330 align:middle line:84% And therefore, x minus 1 cannot be in S. 00:51:41.330 --> 00:51:49.870 align:middle line:84% So since x is the least element of S and x minus 1 is less than 00:51:49.870 --> 00:51:56.350 align:middle line:84% x, this means that x minus 1 is not in S. Otherwise, 00:51:56.350 --> 00:52:05.220 align:middle line:84% it would be a smaller element than x in S. So thus, 00:52:05.220 --> 00:52:08.640 align:middle line:84% what does it mean to not be in S? 00:52:08.640 --> 00:52:11.370 align:middle line:84% It means that P of x minus 1 is true. 00:52:11.370 --> 00:52:15.296 align:middle line:90% 00:52:15.296 --> 00:52:20.400 align:middle line:84% By the definition of S, this means P of x minus 1 00:52:20.400 --> 00:52:25.040 align:middle line:90% is true, OK? 00:52:25.040 --> 00:52:27.560 align:middle line:90% 00:52:27.560 --> 00:52:36.260 align:middle line:84% But by the second property we're assuming about our statement P, 00:52:36.260 --> 00:52:44.120 align:middle line:84% this means that the next guy in line, x minus 1 plus 1, 00:52:44.120 --> 00:52:58.650 align:middle line:84% is true, which is just x, which means that x is not in S, OK? 00:52:58.650 --> 00:53:04.350 align:middle line:84% So from the assumption-- so let me just recap. 00:53:04.350 --> 00:53:06.390 align:middle line:84% From the assumption that S is non-empty, 00:53:06.390 --> 00:53:08.640 align:middle line:90% we've derived two facts. 00:53:08.640 --> 00:53:16.380 align:middle line:84% 1, x has the least element in S. And that element is also not 00:53:16.380 --> 00:53:17.670 align:middle line:90% in S. 00:53:17.670 --> 00:53:34.710 align:middle line:84% So written out, we've concluded there 00:53:34.710 --> 00:53:41.840 align:middle line:84% exists a natural number which is both in S and not in S. 00:53:41.840 --> 00:53:44.030 align:middle line:90% And this is a false statement. 00:53:44.030 --> 00:53:48.950 align:middle line:84% You cannot have an object and member that's both in the set 00:53:48.950 --> 00:53:53.070 align:middle line:90% and not in the set, OK? 00:53:53.070 --> 00:53:56.040 align:middle line:84% And at the end of contradiction arguments, 00:53:56.040 --> 00:53:58.840 align:middle line:84% I'll usually put two arrows hitting each other. 00:53:58.840 --> 00:54:00.370 align:middle line:90% So that's a contradiction. 00:54:00.370 --> 00:54:03.660 align:middle line:84% Therefore, our initial assumption that S is non-empty 00:54:03.660 --> 00:54:04.455 align:middle line:90% has to be false. 00:54:04.455 --> 00:54:10.480 align:middle line:90% 00:54:10.480 --> 00:54:16.530 align:middle line:84% And therefore, S is the empty set, OK? 00:54:16.530 --> 00:54:20.430 align:middle line:84% So I encourage you to go through that proof 00:54:20.430 --> 00:54:24.540 align:middle line:84% a little slowly because maybe you got turned around 00:54:24.540 --> 00:54:30.940 align:middle line:84% by taking the complements or the general scheme of how 00:54:30.940 --> 00:54:33.810 align:middle line:90% a proof by contradiction works. 00:54:33.810 --> 00:54:40.020 align:middle line:84% But don't spend too much time on it because, as I've said, 00:54:40.020 --> 00:54:43.860 align:middle line:84% this theorem itself and its proof are not the thing we're 00:54:43.860 --> 00:54:44.820 align:middle line:90% really interested in. 00:54:44.820 --> 00:54:47.850 align:middle line:84% Or at least, it's not the most interesting. 00:54:47.850 --> 00:54:49.740 align:middle line:84% It's more of a tool that we'll use to prove 00:54:49.740 --> 00:54:53.620 align:middle line:90% more interesting statements. 00:54:53.620 --> 00:54:57.580 align:middle line:84% OK, so how do we actually use this theorem, induction, 00:54:57.580 --> 00:54:59.210 align:middle line:90% to prove other statements? 00:54:59.210 --> 00:55:04.030 align:middle line:84% So I guess I should include this here. 00:55:04.030 --> 00:55:06.130 align:middle line:84% This falls under the umbrella of logic, 00:55:06.130 --> 00:55:08.227 align:middle line:84% meaning we're going to approve previous-- 00:55:08.227 --> 00:55:10.060 align:middle line:84% we're going to use previous statements we've 00:55:10.060 --> 00:55:13.060 align:middle line:90% proven to prove new statements. 00:55:13.060 --> 00:55:16.075 align:middle line:84% But anyway, so how do we use induction in practice? 00:55:16.075 --> 00:55:27.340 align:middle line:90% 00:55:27.340 --> 00:55:34.250 align:middle line:84% So if we want to prove some statement-- 00:55:34.250 --> 00:55:39.480 align:middle line:90% 00:55:39.480 --> 00:55:44.610 align:middle line:90% for all n, Pn is true-- 00:55:44.610 --> 00:55:47.730 align:middle line:84% in the print then, this theorem about induction-- 00:55:47.730 --> 00:55:49.710 align:middle line:84% this theorem of induction-- tells us 00:55:49.710 --> 00:56:02.280 align:middle line:84% we just have to do two things, OK? 00:56:02.280 --> 00:56:08.230 align:middle line:90% We have to prove the base case. 00:56:08.230 --> 00:56:09.370 align:middle line:90% And this is usually easy. 00:56:09.370 --> 00:56:12.530 align:middle line:90% 00:56:12.530 --> 00:56:14.560 align:middle line:84% You just stick the number 1 into the statement 00:56:14.560 --> 00:56:16.120 align:middle line:90% that you want to prove. 00:56:16.120 --> 00:56:21.010 align:middle line:90% And that's the end of the story. 00:56:21.010 --> 00:56:27.600 align:middle line:90% 00:56:27.600 --> 00:56:28.950 align:middle line:90% And the second step is usually-- 00:56:28.950 --> 00:56:30.450 align:middle line:84% or the second thing we have to prove 00:56:30.450 --> 00:56:33.450 align:middle line:84% is the more involved part, which is 00:56:33.450 --> 00:56:50.400 align:middle line:84% we have to prove that the statement that if PM is true, 00:56:50.400 --> 00:56:56.580 align:middle line:90% then P of m plus 1 is true, OK? 00:56:56.580 --> 00:56:59.040 align:middle line:84% If we want to do a proof by induction, 00:56:59.040 --> 00:57:01.540 align:middle line:84% there's two smaller proofs we have to do. 00:57:01.540 --> 00:57:04.170 align:middle line:84% First, we have to prove P of 1 is true. 00:57:04.170 --> 00:57:06.270 align:middle line:84% And then we have to prove this statement. 00:57:06.270 --> 00:57:09.540 align:middle line:90% If P of m, then P of m plus 1. 00:57:09.540 --> 00:57:12.340 align:middle line:90% 00:57:12.340 --> 00:57:15.050 align:middle line:84% So again, this is usually referred to as the base case. 00:57:15.050 --> 00:57:16.580 align:middle line:90% This is the inductive step. 00:57:16.580 --> 00:57:19.135 align:middle line:84% So let's try and actually do this. 00:57:19.135 --> 00:57:29.850 align:middle line:90% 00:57:29.850 --> 00:57:39.970 align:middle line:84% All right, so-- so another question I get at the beginning 00:57:39.970 --> 00:57:42.460 align:middle line:84% of a course, especially about proofs-- 00:57:42.460 --> 00:57:45.387 align:middle line:84% because there's a lot of uncertainty about what you 00:57:45.387 --> 00:57:47.470 align:middle line:84% can assume is true, what can you use, what can you 00:57:47.470 --> 00:57:51.820 align:middle line:84% not use, right now, at this point, you 00:57:51.820 --> 00:57:57.040 align:middle line:84% can use whatever you know about any 00:57:57.040 --> 00:58:00.460 align:middle line:84% of the algebraic properties, you know about the real numbers, 00:58:00.460 --> 00:58:02.860 align:middle line:90% the rational numbers-- 00:58:02.860 --> 00:58:08.680 align:middle line:84% and by algebraic properties, I mean if a plus b equals 00:58:08.680 --> 00:58:13.510 align:middle line:84% c, then a plus b times d equals c times d-- 00:58:13.510 --> 00:58:16.640 align:middle line:84% and what you know about inequalities. 00:58:16.640 --> 00:58:20.110 align:middle line:84% So we're going to go much more in depth into ordering, 00:58:20.110 --> 00:58:27.090 align:middle line:84% which is what inequality is a part of. 00:58:27.090 --> 00:58:30.030 align:middle line:84% But you can use all the properties 00:58:30.030 --> 00:58:33.420 align:middle line:84% you know about solving inequalities or manipulating 00:58:33.420 --> 00:58:37.530 align:middle line:84% inequalities, meaning if I have one number is less than 00:58:37.530 --> 00:58:39.780 align:middle line:84% or equal to another number, then when I multiply 00:58:39.780 --> 00:58:42.300 align:middle line:84% both sides by a positive number, that 00:58:42.300 --> 00:58:43.600 align:middle line:90% doesn't change the inequality. 00:58:43.600 --> 00:58:48.240 align:middle line:84% So you can use all of these algebraic properties 00:58:48.240 --> 00:58:55.890 align:middle line:84% of rationals and real numbers from here on out. 00:58:55.890 --> 00:58:58.950 align:middle line:84% I mean, so we're going to be proving things about calculus. 00:58:58.950 --> 00:59:03.270 align:middle line:84% So you certainly cannot use anything about continuity, 00:59:03.270 --> 00:59:07.060 align:middle line:84% differentiation, or anything like that. 00:59:07.060 --> 00:59:12.380 align:middle line:84% But for now, you can use all the algebraic properties you know. 00:59:12.380 --> 00:59:16.190 align:middle line:84% So the first statement we're going to try and prove 00:59:16.190 --> 00:59:24.740 align:middle line:84% using induction is the statement that for all c not equal 00:59:24.740 --> 00:59:30.650 align:middle line:84% to 1, for all n, a natural number, 1 00:59:30.650 --> 00:59:37.820 align:middle line:84% plus c plus c squared plus c to the n equals 1 minus c to the n 00:59:37.820 --> 00:59:41.510 align:middle line:90% plus 1 over 1 minus c, OK? 00:59:41.510 --> 00:59:49.790 align:middle line:84% So this here is our statement P of n. 00:59:49.790 --> 00:59:53.036 align:middle line:84% It depends on the natural number n, OK? 00:59:53.036 --> 00:59:57.630 align:middle line:84% So we're going to do this by induction, which means we're 00:59:57.630 --> 00:59:59.618 align:middle line:90% going to do those two things. 00:59:59.618 --> 01:00:01.410 align:middle line:84% We're going to prove the base case, P of 1, 01:00:01.410 --> 01:00:02.710 align:middle line:90% which I said is easy. 01:00:02.710 --> 01:00:04.820 align:middle line:84% And then we're going to prove the second case, 01:00:04.820 --> 01:00:07.470 align:middle line:84% the second property, the inductive step, which 01:00:07.470 --> 01:00:12.150 align:middle line:84% is a little more involved, but not so much involved, 01:00:12.150 --> 01:00:13.860 align:middle line:90% at least in the beginning. 01:00:13.860 --> 01:00:19.890 align:middle line:84% So let me call this inequality star. 01:00:19.890 --> 01:00:27.140 align:middle line:90% 01:00:27.140 --> 01:00:31.350 align:middle line:84% We're going to prove star by induction. 01:00:31.350 --> 01:00:35.150 align:middle line:84% So first, we will do the base case. 01:00:35.150 --> 01:00:41.120 align:middle line:90% 01:00:41.120 --> 01:00:46.370 align:middle line:84% And like I said, the base case is usually you 01:00:46.370 --> 01:00:50.460 align:middle line:84% just plug in n equals 1 and verify that P of n is true. 01:00:50.460 --> 01:00:53.570 align:middle line:90% And that's what we do. 01:00:53.570 --> 01:00:57.980 align:middle line:84% 1 plus c to the 1, which is the left-hand side, does, in fact, 01:00:57.980 --> 01:01:05.150 align:middle line:84% equal 1 minus c to the 1 plus 1 over 1 minus c because this 01:01:05.150 --> 01:01:05.885 align:middle line:90% right-hand side-- 01:01:05.885 --> 01:01:11.390 align:middle line:90% 01:01:11.390 --> 01:01:14.480 align:middle line:84% 1 minus c squared is 1 minus c times 1 plus c-- 01:01:14.480 --> 01:01:17.880 align:middle line:90% 01:01:17.880 --> 01:01:19.050 align:middle line:90% the 1 minus c's cancel. 01:01:19.050 --> 01:01:22.870 align:middle line:90% 01:01:22.870 --> 01:01:26.050 align:middle line:84% And so the base case is proven, all right? 01:01:26.050 --> 01:01:31.120 align:middle line:90% 01:01:31.120 --> 01:01:41.670 align:middle line:84% Now, we do the inductive step, OK? 01:01:41.670 --> 01:01:47.050 align:middle line:84% So we're going to assume that star holds for n equals m 01:01:47.050 --> 01:01:48.430 align:middle line:90% So we're going to assume P of m. 01:01:48.430 --> 01:01:57.090 align:middle line:84% So assume that 1 plus c plus c to the m 01:01:57.090 --> 01:02:04.580 align:middle line:84% equals 1 minus c over 1 minus c, OK? 01:02:04.580 --> 01:02:07.880 align:middle line:90% Now, we want to show-- 01:02:07.880 --> 01:02:11.330 align:middle line:84% again, let's write out what we want to do, what's our plan. 01:02:11.330 --> 01:02:15.004 align:middle line:90% 01:02:15.004 --> 01:02:22.360 align:middle line:84% We want to prove that this equality, that this line 01:02:22.360 --> 01:02:33.690 align:middle line:84% star holds for n equals m plus 1, OK? 01:02:33.690 --> 01:02:37.540 align:middle line:84% So again, what I wrote here, this 01:02:37.540 --> 01:02:47.370 align:middle line:84% is basically star for n equals m, OK? 01:02:47.370 --> 01:02:51.630 align:middle line:84% And let me call this second inequality-- 01:02:51.630 --> 01:02:53.310 align:middle line:90% the second equality 2 star. 01:02:53.310 --> 01:02:57.780 align:middle line:84% So this is my assumption for m, n equals m. 01:02:57.780 --> 01:03:00.690 align:middle line:90% 01:03:00.690 --> 01:03:09.280 align:middle line:84% OK, so let's take the left side for n equals m plus 1 and see 01:03:09.280 --> 01:03:15.630 align:middle line:84% if we cannot massage it to get the right-hand side for-- 01:03:15.630 --> 01:03:18.360 align:middle line:84% I should say the right-hand side for n equals m plus 1. 01:03:18.360 --> 01:03:23.700 align:middle line:90% 01:03:23.700 --> 01:03:25.630 align:middle line:90% So here is the calculation part. 01:03:25.630 --> 01:03:33.300 align:middle line:84% So we have 1 plus c plus c to the m plus c to the m plus 1. 01:03:33.300 --> 01:03:38.610 align:middle line:84% This is the m plus 1 case of the left-hand side of star, 01:03:38.610 --> 01:03:40.590 align:middle line:84% which we want to show is to the-- 01:03:40.590 --> 01:03:46.510 align:middle line:84% is equal to the n equals n plus 1 case of the right-hand side. 01:03:46.510 --> 01:03:48.970 align:middle line:90% Now, this is equal to-- 01:03:48.970 --> 01:03:53.620 align:middle line:84% now, we already know what this is equal to by our assumption. 01:03:53.620 --> 01:03:57.670 align:middle line:84% This is by the second star there, which 01:03:57.670 --> 01:03:59.680 align:middle line:90% is what we're assuming is true. 01:03:59.680 --> 01:04:04.150 align:middle line:84% This is equal to 1 minus c n plus 1 over 1 01:04:04.150 --> 01:04:12.340 align:middle line:90% minus c plus cm plus 1. 01:04:12.340 --> 01:04:15.670 align:middle line:84% And so now, we just do a little bit of algebra. 01:04:15.670 --> 01:04:18.080 align:middle line:84% This is equal to 1 over 1 minus c 01:04:18.080 --> 01:04:27.520 align:middle line:84% m plus 1 plus c m plus 1 minus c m plus 2 over 1 minus c. 01:04:27.520 --> 01:04:31.630 align:middle line:84% Those cancel and I'm left with 1 minus c to the m plus 2. 01:04:31.630 --> 01:04:36.070 align:middle line:84% And I'll write it just so that you can see this is really 01:04:36.070 --> 01:04:39.910 align:middle line:90% the m plus 1 case, all right? 01:04:39.910 --> 01:04:46.100 align:middle line:90% 01:04:46.100 --> 01:04:51.730 align:middle line:84% So again, we arrived at this first step 01:04:51.730 --> 01:04:57.270 align:middle line:84% by our assumption, the second starred equation, OK? 01:04:57.270 --> 01:05:00.660 align:middle line:90% 01:05:00.660 --> 01:05:10.880 align:middle line:84% So thus, star holds for n equals m plus 1. 01:05:10.880 --> 01:05:17.300 align:middle line:84% So by induction-- or really, I should say the theorem 01:05:17.300 --> 01:05:18.635 align:middle line:90% of induction that we proved-- 01:05:18.635 --> 01:05:23.660 align:middle line:90% 01:05:23.660 --> 01:05:29.840 align:middle line:84% our equality between those two objects, or two expressions, 01:05:29.840 --> 01:05:38.910 align:middle line:90% is valid for all n, OK? 01:05:38.910 --> 01:05:58.390 align:middle line:90% 01:05:58.390 --> 01:05:58.890 align:middle line:90% OK. 01:05:58.890 --> 01:06:50.980 align:middle line:90% 01:06:50.980 --> 01:06:59.090 align:middle line:84% OK, so let's do one more example of using induction. 01:06:59.090 --> 01:07:07.900 align:middle line:84% So let's prove if C is a real number bigger 01:07:07.900 --> 01:07:16.135 align:middle line:84% than or equal to minus 1, then for all n, 01:07:16.135 --> 01:07:22.340 align:middle line:84% a natural number, 1 plus c to the n 01:07:22.340 --> 01:07:30.240 align:middle line:84% is bigger than or equal to 1 plus n times c, OK? 01:07:30.240 --> 01:07:32.830 align:middle line:90% 01:07:32.830 --> 01:07:37.780 align:middle line:84% All right, so we're going to do this by induction again. 01:07:37.780 --> 01:07:39.568 align:middle line:84% That means we need to prove the base case 01:07:39.568 --> 01:07:41.110 align:middle line:84% and we need to do the inductive step. 01:07:41.110 --> 01:07:43.710 align:middle line:90% 01:07:43.710 --> 01:07:46.160 align:middle line:90% So base case, as always, will-- 01:07:46.160 --> 01:07:47.960 align:middle line:90% so this is just right here. 01:07:47.960 --> 01:07:52.440 align:middle line:84% We're going to do this by induction. 01:07:52.440 --> 01:08:00.560 align:middle line:84% So as you can see, the base case is, 01:08:00.560 --> 01:08:05.980 align:middle line:84% again, n equals 1 is clear just by looking at it. 01:08:05.980 --> 01:08:12.200 align:middle line:84% 1 plus c to the 1, in fact, equals 1 plus 1 times c. 01:08:12.200 --> 01:08:16.520 align:middle line:84% So it's certainly bigger than or equal to 1 plus n times c. 01:08:16.520 --> 01:08:28.670 align:middle line:84% So I think that's the last stars I'll use for this lecture. 01:08:28.670 --> 01:08:37.640 align:middle line:84% So our statement, our inequality star star star, 01:08:37.640 --> 01:08:40.985 align:middle line:90% holds for n equals 1. 01:08:40.985 --> 01:08:42.859 align:middle line:84% All right, so that's our base case. 01:08:42.859 --> 01:08:46.670 align:middle line:84% Now, we're going to assume that this inequality holds 01:08:46.670 --> 01:08:49.939 align:middle line:84% for n equals m and try to prove that it 01:08:49.939 --> 01:08:51.484 align:middle line:90% holds for n equals m plus 1. 01:08:51.484 --> 01:08:59.390 align:middle line:90% 01:08:59.390 --> 01:09:05.300 align:middle line:84% So we're assuming this when n equals m. 01:09:05.300 --> 01:09:12.950 align:middle line:84% So 1 plus c to the m is bigger than or equal to 1 plus m times 01:09:12.950 --> 01:09:15.270 align:middle line:90% c. 01:09:15.270 --> 01:09:19.350 align:middle line:84% And we want to prove this inequality with n 01:09:19.350 --> 01:09:21.390 align:middle line:90% equal to m plus 1. 01:09:21.390 --> 01:09:26.859 align:middle line:84% And we're just assuming this guy, OK? 01:09:26.859 --> 01:09:31.420 align:middle line:84% So I want to get the statement for n equals m plus 1. 01:09:31.420 --> 01:09:34.760 align:middle line:84% One way to do that is this left side. 01:09:34.760 --> 01:09:36.130 align:middle line:90% I want to get-- 01:09:36.130 --> 01:09:40.180 align:middle line:84% let's look at the n equals m plus 1 side 01:09:40.180 --> 01:09:43.660 align:middle line:90% and see what we can do with it. 01:09:43.660 --> 01:09:47.109 align:middle line:84% So again, this is a calculation part and logic. 01:09:47.109 --> 01:09:51.540 align:middle line:84% So we have 1 plus c to the m plus 1. 01:09:51.540 --> 01:09:55.620 align:middle line:84% So that's the n equals m plus 1 side of this. 01:09:55.620 --> 01:09:58.220 align:middle line:90% 01:09:58.220 --> 01:10:03.920 align:middle line:84% This is equal to 1 plus c times 1 plus c to the m. 01:10:03.920 --> 01:10:06.920 align:middle line:84% Now, we're assuming, again, this inequality. 01:10:06.920 --> 01:10:09.230 align:middle line:90% This is the n equals m case. 01:10:09.230 --> 01:10:10.620 align:middle line:90% So we can use it. 01:10:10.620 --> 01:10:11.810 align:middle line:90% So we're assuming it. 01:10:11.810 --> 01:10:13.820 align:middle line:90% We use it. 01:10:13.820 --> 01:10:17.660 align:middle line:84% And since C is bigger than or equal to minus 1, 1 plus C 01:10:17.660 --> 01:10:18.790 align:middle line:90% is non-negative. 01:10:18.790 --> 01:10:22.710 align:middle line:84% So this thing is bigger than or equal to this thing. 01:10:22.710 --> 01:10:25.340 align:middle line:84% So if I multiply both sides by 1 plus c, 01:10:25.340 --> 01:10:26.820 align:middle line:90% I preserve the inequality. 01:10:26.820 --> 01:10:37.570 align:middle line:84% So this is bigger than or equal to 1 plus mc, OK? 01:10:37.570 --> 01:10:41.200 align:middle line:84% Again, this just follows from essentially the assumption 01:10:41.200 --> 01:10:44.810 align:middle line:84% multiplied through by 1 plus c, OK? 01:10:44.810 --> 01:10:49.800 align:middle line:90% 01:10:49.800 --> 01:10:56.190 align:middle line:84% So now, I'm going to finagle this. 01:10:56.190 --> 01:10:57.690 align:middle line:90% So let me just-- 01:10:57.690 --> 01:11:01.660 align:middle line:90% 01:11:01.660 --> 01:11:03.530 align:middle line:84% I'm not doing anything different here. 01:11:03.530 --> 01:11:05.020 align:middle line:84% I'm just going to rewrite this over 01:11:05.020 --> 01:11:12.700 align:middle line:84% here so that I can have a chain of inequality. 01:11:12.700 --> 01:11:16.140 align:middle line:84% So I have 1 plus c to the m plus 1 is greater than or equal to 1 01:11:16.140 --> 01:11:20.760 align:middle line:90% plus c 1 plus mc. 01:11:20.760 --> 01:11:23.340 align:middle line:84% All right, so now, this is bigger than or equal to this. 01:11:23.340 --> 01:11:27.750 align:middle line:84% And this here-- so when I write equal, 01:11:27.750 --> 01:11:29.940 align:middle line:84% I do not mean that this left side is now 01:11:29.940 --> 01:11:32.310 align:middle line:84% equal to what I'm about to write here. 01:11:32.310 --> 01:11:36.600 align:middle line:84% That means the previous thing on here is equal to what I'm about 01:11:36.600 --> 01:11:46.250 align:middle line:84% to write here, OK/ this is a typical fashion and writing 01:11:46.250 --> 01:11:49.370 align:middle line:84% down inequalities-- or I guess, practice. 01:11:49.370 --> 01:11:51.650 align:middle line:90% So this is equal to 1-- 01:11:51.650 --> 01:11:54.440 align:middle line:90% so just doing the algebra-- 01:11:54.440 --> 01:12:01.760 align:middle line:84% m plus 1 times c plus m times c squared, OK? 01:12:01.760 --> 01:12:07.925 align:middle line:84% Now, this part is exactly the n equals m plus 1 side of this. 01:12:07.925 --> 01:12:10.470 align:middle line:90% 01:12:10.470 --> 01:12:13.470 align:middle line:84% And I have a little room to give because now this 01:12:13.470 --> 01:12:17.130 align:middle line:84% is plus something that's non-negative. 01:12:17.130 --> 01:12:21.930 align:middle line:90% 01:12:21.930 --> 01:12:23.700 align:middle line:84% So let me just rewrite this again. 01:12:23.700 --> 01:12:27.840 align:middle line:84% This means that 1 plus c to the m plus 1 01:12:27.840 --> 01:12:29.880 align:middle line:84% is greater than or equal to 1 plus-- 01:12:29.880 --> 01:12:34.910 align:middle line:90% 01:12:34.910 --> 01:12:37.490 align:middle line:84% so again, I'm kind of writing a lot here. 01:12:37.490 --> 01:12:40.730 align:middle line:84% I will stop writing as much as the course goes on. 01:12:40.730 --> 01:12:44.308 align:middle line:90% 01:12:44.308 --> 01:12:46.350 align:middle line:84% But I encourage you, especially in the beginning, 01:12:46.350 --> 01:12:53.190 align:middle line:84% to write all the steps and logic, OK? 01:12:53.190 --> 01:12:56.530 align:middle line:84% So again, I'm not rewriting anything. 01:12:56.530 --> 01:13:01.090 align:middle line:84% I'm just summarizing what I've done here. 01:13:01.090 --> 01:13:04.030 align:middle line:84% Now, this right-hand side-- so I have this as bigger than 01:13:04.030 --> 01:13:05.080 align:middle line:90% or equal to this. 01:13:05.080 --> 01:13:07.660 align:middle line:84% And this right-hand side, since I 01:13:07.660 --> 01:13:09.820 align:middle line:84% have a number plus something non-negative, 01:13:09.820 --> 01:13:12.550 align:middle line:84% m times c squared-- m's a natural number. 01:13:12.550 --> 01:13:19.650 align:middle line:84% This is bigger than or equal to 1 m plus 1 times c. 01:13:19.650 --> 01:13:22.250 align:middle line:90% 01:13:22.250 --> 01:13:30.300 align:middle line:84% Thus, 1 plus c to the m plus 1 is greater than or equal to, 01:13:30.300 --> 01:13:32.690 align:middle line:84% which is the n equals n plus 1 case. 01:13:32.690 --> 01:13:46.540 align:middle line:90% 01:13:46.540 --> 01:13:58.440 align:middle line:84% So by induction, this inequality triple star holds for all n. 01:13:58.440 --> 01:14:13.720 align:middle line:90% 01:14:13.720 --> 01:14:15.390 align:middle line:90% All right. 01:14:15.390 --> 01:14:22.000 align:middle line:90%