WEBVTT 00:00:00.000 --> 00:00:02.430 align:middle line:90% [SQUEAKING] 00:00:02.430 --> 00:00:04.374 align:middle line:90% [RUSTLING] 00:00:04.374 --> 00:00:15.470 align:middle line:90% [CLICKING] 00:00:15.470 --> 00:00:19.570 align:middle line:84% CASEY RODRIGUEZ: So here we are again. 00:00:19.570 --> 00:00:24.540 align:middle line:84% So I'm going to finish real quick proof of a theorem 00:00:24.540 --> 00:00:26.250 align:middle line:84% that I stated last time due to Cantor. 00:00:26.250 --> 00:00:28.050 align:middle line:90% So let me recall the setting. 00:00:28.050 --> 00:00:35.250 align:middle line:84% So last time, we were finishing up 00:00:35.250 --> 00:00:37.530 align:middle line:84% what I had to say about cardinality, which remember, 00:00:37.530 --> 00:00:44.020 align:middle line:90% is a notion of size of sets. 00:00:44.020 --> 00:00:56.120 align:middle line:84% And at the end, for a given set A, 00:00:56.120 --> 00:01:00.230 align:middle line:84% then we defined the power set of A 00:01:00.230 --> 00:01:07.160 align:middle line:84% to be the set of all subsets of A. 00:01:07.160 --> 00:01:10.040 align:middle line:84% And last time, for example, we did several of them 00:01:10.040 --> 00:01:13.160 align:middle line:84% or looked at a few different power sets. 00:01:13.160 --> 00:01:21.620 align:middle line:84% And simplest example is that the power set of the empty set 00:01:21.620 --> 00:01:25.220 align:middle line:84% is the set containing the empty set. 00:01:25.220 --> 00:01:29.240 align:middle line:84% In particular, the power has one more element 00:01:29.240 --> 00:01:30.050 align:middle line:90% than the empty set. 00:01:30.050 --> 00:01:31.820 align:middle line:90% The empty set has no elements. 00:01:31.820 --> 00:01:35.720 align:middle line:84% The power set of the empty set has one element, namely 00:01:35.720 --> 00:01:36.575 align:middle line:90% the empty set. 00:01:36.575 --> 00:01:39.170 align:middle line:90% 00:01:39.170 --> 00:01:50.130 align:middle line:84% And we are looking at this to answer a question which 00:01:50.130 --> 00:01:52.860 align:middle line:84% I posed at the end of last class, namely, 00:01:52.860 --> 00:01:55.500 align:middle line:84% all the sets we saw from the integers 00:01:55.500 --> 00:02:00.180 align:middle line:84% to even numbers to rational numbers, which 00:02:00.180 --> 00:02:06.020 align:middle line:84% is on the assignment, all have the same cardinality 00:02:06.020 --> 00:02:08.770 align:middle line:90% as the natural numbers. 00:02:08.770 --> 00:02:12.800 align:middle line:84% And that's what we call the countably infinite. 00:02:12.800 --> 00:02:14.660 align:middle line:84% And so a question would be, is there 00:02:14.660 --> 00:02:19.730 align:middle line:84% any set that is bigger in cardinality 00:02:19.730 --> 00:02:21.170 align:middle line:90% then the natural numbers? 00:02:21.170 --> 00:02:23.030 align:middle line:84% Is there any set that's uncountable? 00:02:23.030 --> 00:02:29.090 align:middle line:84% And so this theorem due to Cantor answers that and more. 00:02:29.090 --> 00:02:32.450 align:middle line:90% And it says the following-- 00:02:32.450 --> 00:02:44.810 align:middle line:84% if A is any set, so if A is a set, then the cardinality of A 00:02:44.810 --> 00:02:47.540 align:middle line:84% is strictly smaller than the cardinality of the power 00:02:47.540 --> 00:03:10.320 align:middle line:84% set of A. And as a consequence is that the natural numbers 00:03:10.320 --> 00:03:14.740 align:middle line:84% have smaller size than the power set of the natural numbers, 00:03:14.740 --> 00:03:20.070 align:middle line:84% which has smaller size than the power set of the power 00:03:20.070 --> 00:03:22.890 align:middle line:84% set of the natural numbers, which is smaller in size 00:03:22.890 --> 00:03:24.070 align:middle line:90% than-- and so on. 00:03:24.070 --> 00:03:28.800 align:middle line:84% So there are an infinity of infinities. 00:03:28.800 --> 00:03:31.905 align:middle line:84% There is an infinity of infinite sizes. 00:03:31.905 --> 00:03:47.970 align:middle line:90% 00:03:47.970 --> 00:03:57.830 align:middle line:84% So let's prove this theorem, and it's extremely clever 00:03:57.830 --> 00:04:00.630 align:middle line:90% and a bit mind boggling. 00:04:00.630 --> 00:04:11.030 align:middle line:84% So first, let me prove that A has cardinality less 00:04:11.030 --> 00:04:13.930 align:middle line:84% than or equal to the cardinality of the power set of A. 00:04:13.930 --> 00:04:22.240 align:middle line:90% So let A be a set. 00:04:22.240 --> 00:04:24.850 align:middle line:84% First, we can show that the cardinality of A is less than 00:04:24.850 --> 00:04:27.760 align:middle line:84% or equal to the cardinality of the power set of A. 00:04:27.760 --> 00:04:35.260 align:middle line:84% So we need to find an injection, a one-to-one map from A 00:04:35.260 --> 00:04:38.860 align:middle line:84% into the power set of A. And the simplest one to choose 00:04:38.860 --> 00:04:42.790 align:middle line:84% is one that takes an element of A 00:04:42.790 --> 00:04:46.940 align:middle line:84% to the set containing just that element. 00:04:46.940 --> 00:04:53.260 align:middle line:84% So define f from A into the power set of A 00:04:53.260 --> 00:05:01.480 align:middle line:90% by the function that takes x. 00:05:01.480 --> 00:05:05.860 align:middle line:84% And this should spit out a subset of A. 00:05:05.860 --> 00:05:12.430 align:middle line:84% So this will be the subset that consists solely of x. 00:05:12.430 --> 00:05:17.050 align:middle line:90% And this is clearly one to one. 00:05:17.050 --> 00:05:18.550 align:middle line:90% I'll prove this right now. 00:05:18.550 --> 00:05:22.550 align:middle line:84% Then we need to prove that if f of x equals f of y, 00:05:22.550 --> 00:05:27.220 align:middle line:84% then x equals y, but this is clear from the definition. 00:05:27.220 --> 00:05:36.760 align:middle line:84% Then if f of x equals f of y, this means the set containing 00:05:36.760 --> 00:05:41.410 align:middle line:84% x-- this is by the definition of how we've defined little f-- 00:05:41.410 --> 00:05:44.020 align:middle line:84% means that the set containing x is 00:05:44.020 --> 00:05:46.090 align:middle line:90% equal to the set containing y. 00:05:46.090 --> 00:05:50.020 align:middle line:90% But this means x equals y. 00:05:50.020 --> 00:05:52.090 align:middle line:90% They both contain one element. 00:05:52.090 --> 00:05:56.515 align:middle line:84% Two sets are equal if and only if one side 00:05:56.515 --> 00:05:57.640 align:middle line:90% is an element of the other. 00:05:57.640 --> 00:05:59.830 align:middle line:84% That just means x equals y in the set. 00:05:59.830 --> 00:06:03.280 align:middle line:90% 00:06:03.280 --> 00:06:13.600 align:middle line:84% Thus, f is one to one, which since we 00:06:13.600 --> 00:06:17.620 align:middle line:84% found an injective map from A to the power set of A this means-- 00:06:17.620 --> 00:06:25.200 align:middle line:90% 00:06:25.200 --> 00:06:30.090 align:middle line:84% So now we want to show that they cannot have the same 00:06:30.090 --> 00:06:31.020 align:middle line:90% cardinality. 00:06:31.020 --> 00:06:36.420 align:middle line:90% 00:06:36.420 --> 00:06:47.510 align:middle line:84% We now show that A does not have the same cardinality 00:06:47.510 --> 00:06:52.370 align:middle line:84% as the power set of A. And these two statements are 00:06:52.370 --> 00:06:55.970 align:middle line:84% what is meant when we write down, recall the definition, 00:06:55.970 --> 00:06:59.390 align:middle line:84% of the cardinality of A or size of A 00:06:59.390 --> 00:07:01.910 align:middle line:84% being smaller than the cardinality of the power 00:07:01.910 --> 00:07:05.760 align:middle line:90% set of A. 00:07:05.760 --> 00:07:11.180 align:middle line:84% So we're going to do this by a proof by contradiction. 00:07:11.180 --> 00:07:24.080 align:middle line:90% 00:07:24.080 --> 00:07:26.800 align:middle line:84% So that means we're going to assume 00:07:26.800 --> 00:07:34.280 align:middle line:84% that this does not hold and arrive at a false statement. 00:07:34.280 --> 00:07:45.700 align:middle line:84% So I assume that they do have the same cardinality. 00:07:45.700 --> 00:07:47.410 align:middle line:84% So this is our initial assumption. 00:07:47.410 --> 00:07:52.340 align:middle line:84% We're going to derive a false statement from this assumption. 00:07:52.340 --> 00:07:55.910 align:middle line:84% And the only way to arrive at a false statement 00:07:55.910 --> 00:08:00.200 align:middle line:84% from a given assumption in a logically consistent fashion 00:08:00.200 --> 00:08:05.480 align:middle line:84% is that the original statement, namely this, is false. 00:08:05.480 --> 00:08:11.210 align:middle line:84% In other words, the thing we want to show is true. 00:08:11.210 --> 00:08:16.100 align:middle line:84% So let's assume they have the same cardinality. 00:08:16.100 --> 00:08:18.920 align:middle line:90% What does that mean? 00:08:18.920 --> 00:08:32.659 align:middle line:84% Then there exists a bijective function g going from A 00:08:32.659 --> 00:08:36.409 align:middle line:90% to the power set of A. 00:08:36.409 --> 00:08:38.870 align:middle line:84% Remember, a bijective function means 00:08:38.870 --> 00:08:43.610 align:middle line:90% that G is one to one and onto. 00:08:43.610 --> 00:08:45.640 align:middle line:84% One to one meaning different things of A 00:08:45.640 --> 00:08:47.780 align:middle line:84% gets mapped to different things of the power set. 00:08:47.780 --> 00:08:51.680 align:middle line:84% Onto means everything in the power set 00:08:51.680 --> 00:08:55.130 align:middle line:90% gets mapped onto from A. 00:08:55.130 --> 00:08:58.280 align:middle line:84% So really, the fact that it's surjective 00:08:58.280 --> 00:09:01.820 align:middle line:84% is the only thing I'm going to use. 00:09:01.820 --> 00:09:06.210 align:middle line:84% And I'm going to define a weird set. 00:09:06.210 --> 00:09:08.580 align:middle line:84% And we're going to look at this set. 00:09:08.580 --> 00:09:23.260 align:middle line:84% So define a set B. And this is a subset of A. B 00:09:23.260 --> 00:09:35.140 align:middle line:84% is a set of all x's in A such that x is not in f of x. 00:09:35.140 --> 00:09:36.100 align:middle line:90% This should be g. 00:09:36.100 --> 00:09:40.440 align:middle line:90% 00:09:40.440 --> 00:09:44.020 align:middle line:84% So remember, g maps from A to the power set. 00:09:44.020 --> 00:09:49.050 align:middle line:84% So for any given x in A, g of x is a subset of A. 00:09:49.050 --> 00:09:53.970 align:middle line:84% So the condition to be in B is that this element x is not 00:09:53.970 --> 00:10:00.410 align:middle line:90% in the image of itself by G. 00:10:00.410 --> 00:10:03.440 align:middle line:90% And B could be empty. 00:10:03.440 --> 00:10:07.410 align:middle line:84% There could be no x's in A that satisfy this. 00:10:07.410 --> 00:10:14.790 align:middle line:84% But it's just a subset of A, simply by definition. 00:10:14.790 --> 00:10:16.500 align:middle line:84% We're just looking at elements in A that 00:10:16.500 --> 00:10:18.360 align:middle line:90% satisfy a certain condition. 00:10:18.360 --> 00:10:22.800 align:middle line:90% 00:10:22.800 --> 00:10:27.740 align:middle line:84% So since it's a subset of A, this 00:10:27.740 --> 00:10:32.075 align:middle line:84% means that it's a element of the power set of A. 00:10:32.075 --> 00:10:34.160 align:middle line:84% This is just how the power set is defined. 00:10:34.160 --> 00:10:38.230 align:middle line:90% Remember, it's all subsets of A 00:10:38.230 --> 00:10:41.500 align:middle line:84% So B is an element of the power set of A. Therefore, 00:10:41.500 --> 00:10:47.740 align:middle line:84% something gets mapped to B under this map. 00:10:47.740 --> 00:11:06.020 align:middle line:84% So since g is surjective, there exists b in capital A 00:11:06.020 --> 00:11:18.230 align:middle line:84% such that g of little b equals capital B. So this just 00:11:18.230 --> 00:11:20.570 align:middle line:84% follows from the fact that g is a surjection, 00:11:20.570 --> 00:11:23.150 align:middle line:84% meaning everything in here must get 00:11:23.150 --> 00:11:30.850 align:middle line:90% mapped to by some element in A. 00:11:30.850 --> 00:11:36.210 align:middle line:84% But now, let's take a look at this B guy. 00:11:36.210 --> 00:11:37.470 align:middle line:90% There are two cases-- 00:11:37.470 --> 00:11:43.870 align:middle line:84% either b is in g of b or b is not in g of b. 00:11:43.870 --> 00:11:46.750 align:middle line:90% 00:11:46.750 --> 00:11:57.640 align:middle line:90% So b is in g of b. 00:11:57.640 --> 00:12:03.040 align:middle line:90% 00:12:03.040 --> 00:12:07.870 align:middle line:84% If b is in g of b, remember this is equal to capital B, 00:12:07.870 --> 00:12:09.250 align:middle line:90% by definition. 00:12:09.250 --> 00:12:14.950 align:middle line:84% And to be in capital B means that x is not in g of x. 00:12:14.950 --> 00:12:19.920 align:middle line:90% 00:12:19.920 --> 00:12:22.800 align:middle line:90% So let me just write this again. 00:12:22.800 --> 00:12:30.260 align:middle line:84% This means b is in B, which means b is not in g of b. 00:12:30.260 --> 00:12:32.880 align:middle line:90% 00:12:32.880 --> 00:12:36.510 align:middle line:84% We started off with b and g of b, 00:12:36.510 --> 00:12:39.060 align:middle line:84% and ended with b is not in g of b. 00:12:39.060 --> 00:12:42.090 align:middle line:90% 00:12:42.090 --> 00:12:48.340 align:middle line:84% So we arrive at a false statement in this case. 00:12:48.340 --> 00:12:49.680 align:middle line:90% Then there's one other case-- 00:12:49.680 --> 00:12:51.135 align:middle line:90% b is not in g of b. 00:12:51.135 --> 00:12:56.640 align:middle line:90% 00:12:56.640 --> 00:13:04.100 align:middle line:84% Then if b is not in g of b, this immediately implies, 00:13:04.100 --> 00:13:12.270 align:middle line:84% by the definition of little b, meaning g of b 00:13:12.270 --> 00:13:25.840 align:middle line:84% is in capital B. So let me write it this way. 00:13:25.840 --> 00:13:31.410 align:middle line:84% So if b is not in g of b, then from the definition of capital 00:13:31.410 --> 00:13:46.510 align:middle line:84% B, this implies that b is in capital B. That's just 00:13:46.510 --> 00:13:49.030 align:middle line:84% from the definition of capital B. 00:13:49.030 --> 00:13:58.048 align:middle line:84% But now capital B is equal to g of little b, 00:13:58.048 --> 00:13:59.340 align:middle line:90% which is another contradiction. 00:13:59.340 --> 00:14:04.810 align:middle line:84% Because we started with this and ended with this. 00:14:04.810 --> 00:14:07.290 align:middle line:90% So what I've really proven is-- 00:14:07.290 --> 00:14:11.050 align:middle line:90% 00:14:11.050 --> 00:14:14.620 align:middle line:84% so let's ignore this contradiction mark here-- 00:14:14.620 --> 00:14:25.530 align:middle line:84% thus, we have shown that b in g of b implies b not in g of b. 00:14:25.530 --> 00:14:28.740 align:middle line:84% b not in g of b implies b is in g of b. 00:14:28.740 --> 00:14:35.480 align:middle line:84% So we've proven the statement that b is in g of b if 00:14:35.480 --> 00:14:38.900 align:middle line:90% and only if b is not in g of b. 00:14:38.900 --> 00:14:46.010 align:middle line:84% And this is a very false statement-- 00:14:46.010 --> 00:14:51.320 align:middle line:84% cannot have some object being in the set if and only if it's not 00:14:51.320 --> 00:14:54.120 align:middle line:90% in the set. 00:14:54.120 --> 00:14:57.150 align:middle line:84% So we've arrived at a false statement. 00:14:57.150 --> 00:15:01.060 align:middle line:84% And therefore, our initial assumption 00:15:01.060 --> 00:15:06.310 align:middle line:84% which led us there, namely that the cardinality of the set A 00:15:06.310 --> 00:15:08.950 align:middle line:84% has the same cardinality as the power set of A, 00:15:08.950 --> 00:15:09.925 align:middle line:90% must have been false. 00:15:09.925 --> 00:15:24.770 align:middle line:90% 00:15:24.770 --> 00:15:31.170 align:middle line:84% So it feels almost like you're being hoodwinked a little bit 00:15:31.170 --> 00:15:34.200 align:middle line:90% with this proof. 00:15:34.200 --> 00:15:52.420 align:middle line:84% But to give you some sort of cling or connection 00:15:52.420 --> 00:15:58.030 align:middle line:84% back to reality, what's underlying this argument 00:15:58.030 --> 00:16:05.660 align:middle line:84% or what's one way of understanding this argument is 00:16:05.660 --> 00:16:07.170 align:middle line:90% to think about-- 00:16:07.170 --> 00:16:08.990 align:middle line:84% so in some sense, what this argument does 00:16:08.990 --> 00:16:16.210 align:middle line:84% is make you talk about being in B 00:16:16.210 --> 00:16:20.810 align:middle line:84% while having to also reference B itself. 00:16:20.810 --> 00:16:28.920 align:middle line:84% And maybe this seems a bit wild because this is a math course, 00:16:28.920 --> 00:16:32.010 align:middle line:84% but you can do the statement just in the English language. 00:16:32.010 --> 00:16:38.290 align:middle line:84% If I am to tell you that I am a liar, 00:16:38.290 --> 00:16:42.070 align:middle line:84% then if that statement is true, then what I just said, 00:16:42.070 --> 00:16:44.360 align:middle line:84% namely that I am a liar, is false, 00:16:44.360 --> 00:16:46.540 align:middle line:90% which means I'm not a liar. 00:16:46.540 --> 00:16:51.040 align:middle line:84% Therefore, "I am a liar" implies I'm not a liar. 00:16:51.040 --> 00:16:57.520 align:middle line:84% And vice versa-- if I say, "I am a liar," 00:16:57.520 --> 00:17:02.910 align:middle line:84% and that statement is false, meaning 00:17:02.910 --> 00:17:05.829 align:middle line:84% I'm not a liar, then that statement is true, 00:17:05.829 --> 00:17:08.349 align:middle line:90% which implies I am a liar. 00:17:08.349 --> 00:17:11.109 align:middle line:84% So of course, I would not lie to you. 00:17:11.109 --> 00:17:14.800 align:middle line:84% I would just give you alternative facts. 00:17:14.800 --> 00:17:19.390 align:middle line:84% But that statement is kind of what's 00:17:19.390 --> 00:17:22.210 align:middle line:90% underlying this argument. 00:17:22.210 --> 00:17:41.060 align:middle line:84% So I'm going to put that there, true or false. 00:17:41.060 --> 00:17:46.980 align:middle line:90% 00:17:46.980 --> 00:17:59.583 align:middle line:84% That's the connection to these two things, loosely, 00:17:59.583 --> 00:18:01.000 align:middle line:84% or at least the logic is contained 00:18:01.000 --> 00:18:03.610 align:middle line:84% in trying to verify this statement. 00:18:03.610 --> 00:18:07.440 align:middle line:90% 00:18:07.440 --> 00:18:11.580 align:middle line:84% So that's all I'm going to say about cardinality. 00:18:11.580 --> 00:18:14.370 align:middle line:84% We're going to move on to the real numbers now. 00:18:14.370 --> 00:18:27.090 align:middle line:90% 00:18:27.090 --> 00:18:33.320 align:middle line:84% And so this is really, like I said in the previous lecture, 00:18:33.320 --> 00:18:49.360 align:middle line:84% our first real goal of the class is to describe R. namely, 00:18:49.360 --> 00:18:53.530 align:middle line:84% what exactly is the set of real numbers? 00:18:53.530 --> 00:18:55.480 align:middle line:84% What characterizes the set of real numbers? 00:18:55.480 --> 00:18:59.440 align:middle line:90% 00:18:59.440 --> 00:19:02.100 align:middle line:84% And so let me state this as a theorem 00:19:02.100 --> 00:19:06.330 align:middle line:84% just completely ahead of time, so that it's there for us. 00:19:06.330 --> 00:19:10.930 align:middle line:84% And this is our goal, although we're not going to prove it. 00:19:10.930 --> 00:19:13.230 align:middle line:84% Our goal is just to understand what this theorem says. 00:19:13.230 --> 00:19:23.720 align:middle line:84% So this is a complete description of what R is. 00:19:23.720 --> 00:19:47.440 align:middle line:84% There exists a unique ordered field containing Q 00:19:47.440 --> 00:20:05.560 align:middle line:84% with the least upper bound property, which we denote 00:20:05.560 --> 00:20:12.110 align:middle line:90% by R, instead of real numbers. 00:20:12.110 --> 00:20:18.230 align:middle line:84% So as you sit there and listen to this, you shouldn't have-- 00:20:18.230 --> 00:20:19.910 align:middle line:84% maybe you do, but you really shouldn't 00:20:19.910 --> 00:20:22.550 align:middle line:84% expect to have any idea about what this theorem says, 00:20:22.550 --> 00:20:26.330 align:middle line:90% what all these words mean. 00:20:26.330 --> 00:20:34.205 align:middle line:84% So our goal for this lecture is to make sense of these words. 00:20:34.205 --> 00:20:39.290 align:middle line:90% 00:20:39.290 --> 00:20:43.080 align:middle line:84% So the rest you should understand. 00:20:43.080 --> 00:20:48.890 align:middle line:84% So our goal for this lecture is to make sense of these words, 00:20:48.890 --> 00:20:51.830 align:middle line:84% "ordered field," "least upper bound property." 00:20:51.830 --> 00:21:00.460 align:middle line:84% Because these are the two defining characteristics of R. 00:21:00.460 --> 00:21:03.190 align:middle line:84% And I'm not going to prove this theorem. 00:21:03.190 --> 00:21:06.610 align:middle line:84% We're just going to take it as a given. 00:21:06.610 --> 00:21:11.520 align:middle line:84% We could prove this theorem but it takes quite a bit of time. 00:21:11.520 --> 00:21:15.030 align:middle line:84% And I'd rather start studying properties of R 00:21:15.030 --> 00:21:23.880 align:middle line:84% than building R. This is not so uncommon in math, 00:21:23.880 --> 00:21:30.590 align:middle line:84% that one is not especially interested in the proof 00:21:30.590 --> 00:21:35.340 align:middle line:84% of actually why certain things exist, 00:21:35.340 --> 00:21:38.760 align:middle line:84% but we are definitely interested in the properties of that thing 00:21:38.760 --> 00:21:39.840 align:middle line:90% once we know it exists. 00:21:39.840 --> 00:21:44.800 align:middle line:90% 00:21:44.800 --> 00:21:46.927 align:middle line:84% So let's get started on making sense 00:21:46.927 --> 00:21:48.010 align:middle line:90% of what this theorem says. 00:21:48.010 --> 00:21:50.540 align:middle line:90% 00:21:50.540 --> 00:21:54.710 align:middle line:84% Let's start off with ordered stuff. 00:21:54.710 --> 00:21:57.290 align:middle line:84% This has ordered field, least upper bound property. 00:21:57.290 --> 00:22:03.450 align:middle line:84% Let's start off with what do I mean by order. 00:22:03.450 --> 00:22:05.863 align:middle line:84% We've seen that just a little bit already 00:22:05.863 --> 00:22:07.530 align:middle line:84% when we talked about the natural numbers 00:22:07.530 --> 00:22:13.090 align:middle line:84% and this well-ordered property of the natural numbers. 00:22:13.090 --> 00:22:16.972 align:middle line:84% But I was just using words to label a certain property 00:22:16.972 --> 00:22:17.930 align:middle line:90% of the natural numbers. 00:22:17.930 --> 00:22:20.650 align:middle line:84% I didn't say they meant anything specifically. 00:22:20.650 --> 00:22:22.240 align:middle line:90% It was just a label for that. 00:22:22.240 --> 00:22:26.320 align:middle line:84% But when I say "order" now, this will definitely mean something. 00:22:26.320 --> 00:22:45.630 align:middle line:84% So ordered sets and fields definition-- 00:22:45.630 --> 00:22:52.300 align:middle line:90% 00:22:52.300 --> 00:23:08.020 align:middle line:84% an ordered set is a set S with a relation, 00:23:08.020 --> 00:23:12.310 align:middle line:84% which we label with this less than symbol . 00:23:12.310 --> 00:23:17.840 align:middle line:84% And this relation satisfies two properties. 00:23:17.840 --> 00:23:22.660 align:middle line:84% One is you can always check whether two elements are bigger 00:23:22.660 --> 00:23:24.980 align:middle line:90% than each other or equal. 00:23:24.980 --> 00:23:33.970 align:middle line:84% So for all x, y in S, either x equals y, x is less than y, 00:23:33.970 --> 00:23:39.790 align:middle line:84% or that's just restating, y is less than x. 00:23:39.790 --> 00:23:44.618 align:middle line:90% 00:23:44.618 --> 00:23:56.980 align:middle line:84% And if x is less than y, and y is less than z, 00:23:56.980 --> 00:24:01.030 align:middle line:90% then x is less than z. 00:24:01.030 --> 00:24:09.160 align:middle line:84% So an ordered set is just a set with some relation which 00:24:09.160 --> 00:24:11.320 align:middle line:84% has two properties, namely that for any two 00:24:11.320 --> 00:24:13.840 align:middle line:84% elements in the set, I can compare the two. 00:24:13.840 --> 00:24:15.370 align:middle line:90% That's basically what this says. 00:24:15.370 --> 00:24:18.610 align:middle line:84% And I have this transitive property 00:24:18.610 --> 00:24:21.190 align:middle line:84% that if x is less than y, and y is less than z, then 00:24:21.190 --> 00:24:24.610 align:middle line:90% x is less than z. 00:24:24.610 --> 00:24:28.630 align:middle line:84% And so as I said before, whenever 00:24:28.630 --> 00:24:37.040 align:middle line:84% you have some sort of mildly interesting definition, 00:24:37.040 --> 00:24:39.320 align:middle line:84% you should definitely try to come up 00:24:39.320 --> 00:24:41.010 align:middle line:90% with examples and non-examples. 00:24:41.010 --> 00:24:45.485 align:middle line:84% So what's the simplest example of an ordered set? 00:24:45.485 --> 00:24:48.242 align:middle line:90% 00:24:48.242 --> 00:24:49.700 align:middle line:84% Well, the natural numbers, which we 00:24:49.700 --> 00:24:53.420 align:middle line:84% discussed earlier, but also the integers, 00:24:53.420 --> 00:24:56.900 align:middle line:84% where I have to define what this relation means. 00:24:56.900 --> 00:25:08.000 align:middle line:84% We say m is less than n, this m is less than n, 00:25:08.000 --> 00:25:14.960 align:middle line:84% if n minus m is a natural number. 00:25:14.960 --> 00:25:17.720 align:middle line:84% So this is how we define our order. 00:25:17.720 --> 00:25:22.385 align:middle line:84% So n is bigger than m if n minus m is a natural number. 00:25:22.385 --> 00:25:27.660 align:middle line:90% 00:25:27.660 --> 00:25:31.530 align:middle line:84% And you can check that this is just the usual ordering on z. 00:25:31.530 --> 00:25:35.980 align:middle line:84% And that satisfies these two properties. 00:25:35.980 --> 00:25:44.932 align:middle line:84% The standard order on Q, namely that we'll say q 00:25:44.932 --> 00:25:53.220 align:middle line:84% is less than r if there exist natural numbers, m, n, 00:25:53.220 --> 00:25:59.923 align:middle line:84% such that r minus q equals m over n. 00:25:59.923 --> 00:26:07.730 align:middle line:90% 00:26:07.730 --> 00:26:11.240 align:middle line:84% So these are just the usual orders 00:26:11.240 --> 00:26:14.382 align:middle line:84% of orderings on the integers and rational numbers 00:26:14.382 --> 00:26:15.965 align:middle line:84% that you're used to and that you know. 00:26:15.965 --> 00:26:21.050 align:middle line:84% I'm just writing out exactly how one would define this order. 00:26:21.050 --> 00:26:24.800 align:middle line:84% And you can check just using this definition of these orders 00:26:24.800 --> 00:26:29.590 align:middle line:84% that these orders satisfy 1 and 2. 00:26:29.590 --> 00:26:33.590 align:middle line:90% 00:26:33.590 --> 00:26:50.270 align:middle line:84% Now, a simple non-example, which I guess you could call it 00:26:50.270 --> 00:26:55.520 align:middle line:84% a relation which satisfies 2 but not 1, is the following-- 00:26:55.520 --> 00:27:01.190 align:middle line:90% so let's take our set S to be-- 00:27:01.190 --> 00:27:03.335 align:middle line:84% and create the power set of the natural numbers. 00:27:03.335 --> 00:27:06.000 align:middle line:90% 00:27:06.000 --> 00:27:23.450 align:middle line:84% And we define a relation A less than B by A is less than B 00:27:23.450 --> 00:27:31.050 align:middle line:90% if A is a subset of B. 00:27:31.050 --> 00:27:34.830 align:middle line:84% So I'm just defining a relation on the power set-- 00:27:34.830 --> 00:27:38.040 align:middle line:84% A less than B if A is a subset of B. 00:27:38.040 --> 00:27:40.110 align:middle line:84% And maybe I shouldn't even use less than, 00:27:40.110 --> 00:27:42.780 align:middle line:84% because this makes you think that it's automatically 00:27:42.780 --> 00:27:43.420 align:middle line:90% an order. 00:27:43.420 --> 00:27:46.620 align:middle line:84% So let's make it a script-looking less than. 00:27:46.620 --> 00:27:49.500 align:middle line:90% 00:27:49.500 --> 00:27:51.870 align:middle line:84% This relation is not an order, though. 00:27:51.870 --> 00:27:55.990 align:middle line:90% 00:27:55.990 --> 00:27:58.720 align:middle line:84% I keep saying "order," so we usually refer 00:27:58.720 --> 00:28:05.230 align:middle line:90% to this relation as an order. 00:28:05.230 --> 00:28:11.040 align:middle line:90% 00:28:11.040 --> 00:28:14.370 align:middle line:84% So it's clear that it satisfies 2. 00:28:14.370 --> 00:28:21.510 align:middle line:90% 00:28:21.510 --> 00:28:22.010 align:middle line:90% Why? 00:28:22.010 --> 00:28:31.580 align:middle line:84% Because A is a subset of B. B is a subset of C. 00:28:31.580 --> 00:28:41.630 align:middle line:84% Then A is a subset of C, i.e., A is less than C. 00:28:41.630 --> 00:28:43.795 align:middle line:84% It just follows from the definition 00:28:43.795 --> 00:28:45.670 align:middle line:84% of what it means to be a subset of the other. 00:28:45.670 --> 00:28:49.150 align:middle line:84% If A is a subset of B, that means every element of A 00:28:49.150 --> 00:28:51.340 align:middle line:84% is an element of B. If B's an element of C, that 00:28:51.340 --> 00:28:54.460 align:middle line:84% means every element of B is an element of C. 00:28:54.460 --> 00:28:59.590 align:middle line:84% And therefore, every element of A is an element of C. 00:28:59.590 --> 00:29:05.170 align:middle line:84% So this relation satisfies the second transitive property, 00:29:05.170 --> 00:29:07.570 align:middle line:84% but it doesn't satisfy the first. 00:29:07.570 --> 00:29:14.110 align:middle line:90% 00:29:14.110 --> 00:29:25.970 align:middle line:84% I cannot always measure if one thing is bigger than the other. 00:29:25.970 --> 00:29:26.600 align:middle line:90% Why is this? 00:29:26.600 --> 00:29:31.710 align:middle line:90% 00:29:31.710 --> 00:29:34.530 align:middle line:84% For example, the set containing 0 00:29:34.530 --> 00:29:37.905 align:middle line:84% does not equal the set containing 1. 00:29:37.905 --> 00:29:42.300 align:middle line:90% 00:29:42.300 --> 00:29:54.730 align:middle line:84% But neither of these things hold. 00:29:54.730 --> 00:29:59.500 align:middle line:84% So for something to be an order, or a set to be an ordered set, 00:29:59.500 --> 00:30:03.400 align:middle line:84% it has to have this property that I can always compare 00:30:03.400 --> 00:30:06.200 align:middle line:90% two elements of the set. 00:30:06.200 --> 00:30:10.490 align:middle line:84% And for this relation, which seems like it could be 00:30:10.490 --> 00:30:12.980 align:middle line:84% an order-- it satisfies the second property-- 00:30:12.980 --> 00:30:16.820 align:middle line:84% it does not satisfy the first property because I cannot 00:30:16.820 --> 00:30:21.770 align:middle line:84% always compare two subsets of the natural numbers by saying 00:30:21.770 --> 00:30:24.920 align:middle line:90% one is a subset of the other. 00:30:24.920 --> 00:30:26.210 align:middle line:90% So we just saw that. 00:30:26.210 --> 00:30:29.960 align:middle line:84% We have two sets which are not equal, 00:30:29.960 --> 00:30:32.630 align:middle line:84% but one is not bigger than the other. 00:30:32.630 --> 00:30:52.280 align:middle line:90% 00:30:52.280 --> 00:30:58.610 align:middle line:84% And so one more example, which again, I 00:30:58.610 --> 00:31:02.780 align:middle line:84% will leave for you to check that it does 00:31:02.780 --> 00:31:08.820 align:middle line:84% satisfy conditions 1 and 2 just based on how it's defined. 00:31:08.820 --> 00:31:17.490 align:middle line:84% For example, this is the dictionary 00:31:17.490 --> 00:31:24.767 align:middle line:84% ordering of Q Cartesian product Q. I think I wrote this down. 00:31:24.767 --> 00:31:26.850 align:middle line:84% I mean, you should know what the Cartesian product 00:31:26.850 --> 00:31:27.820 align:middle line:90% is of two things. 00:31:27.820 --> 00:31:33.150 align:middle line:84% But let me just recall that I have two sets. 00:31:33.150 --> 00:31:35.190 align:middle line:84% The Cartesian product of A and B, 00:31:35.190 --> 00:31:43.530 align:middle line:84% this is the set of all ordered pairs of elements 00:31:43.530 --> 00:31:44.325 align:middle line:90% from those sets. 00:31:44.325 --> 00:31:47.550 align:middle line:90% 00:31:47.550 --> 00:31:51.990 align:middle line:84% So the dictionary ordering of Q is what? 00:31:51.990 --> 00:31:55.710 align:middle line:84% So I need to define this relation which 00:31:55.710 --> 00:31:57.880 align:middle line:90% I claim is an order. 00:31:57.880 --> 00:32:12.680 align:middle line:84% So we'll say that a, b is less than q, r if one of two things 00:32:12.680 --> 00:32:13.180 align:middle line:90% happens. 00:32:13.180 --> 00:32:17.710 align:middle line:90% 00:32:17.710 --> 00:32:26.670 align:middle line:84% If either a is less than q or a equals 00:32:26.670 --> 00:32:31.150 align:middle line:90% q, and b is less than r-- 00:32:31.150 --> 00:32:33.830 align:middle line:90% 00:32:33.830 --> 00:32:37.170 align:middle line:84% so dictionary ordering or alphabetical ordering of Q. 00:32:37.170 --> 00:32:40.760 align:middle line:84% You can just check and see which is smaller first. 00:32:40.760 --> 00:32:43.910 align:middle line:84% If they're both equal, then you check the next letter, 00:32:43.910 --> 00:32:47.000 align:middle line:84% and see which one's smaller there. 00:32:47.000 --> 00:32:55.430 align:middle line:84% So then this relation that I've defined here 00:32:55.430 --> 00:33:05.850 align:middle line:84% is in order on Q cross Q, making it an ordered set. 00:33:05.850 --> 00:33:20.320 align:middle line:90% 00:33:20.320 --> 00:33:22.300 align:middle line:84% So this is what an ordered set is. 00:33:22.300 --> 00:33:25.690 align:middle line:84% We'll get to an ordered field in just a second. 00:33:25.690 --> 00:33:30.790 align:middle line:84% But now, let me define what I mean by this least upper 00:33:30.790 --> 00:33:31.910 align:middle line:90% bound property. 00:33:31.910 --> 00:33:36.670 align:middle line:84% And this is really what sets R apart from Q. 00:33:36.670 --> 00:33:42.130 align:middle line:84% So we'll see in a minute that both R and Q are ordered 00:33:42.130 --> 00:33:44.980 align:middle line:84% fields, so that's not what separates R from Q, 00:33:44.980 --> 00:33:46.330 align:middle line:90% the rational numbers. 00:33:46.330 --> 00:33:49.878 align:middle line:84% But what does separate R from Q is the second property, 00:33:49.878 --> 00:33:51.170 align:middle line:90% the least upper bound property. 00:33:51.170 --> 00:33:57.670 align:middle line:84% So if I removed that property, and just 00:33:57.670 --> 00:34:00.220 align:middle line:84% said there exists a unique ordered field containing Q, 00:34:00.220 --> 00:34:05.020 align:middle line:84% that would just be Q. We don't need to add anything to it. 00:34:05.020 --> 00:34:08.860 align:middle line:84% But it's the second property, the least upper bound property, 00:34:08.860 --> 00:34:18.030 align:middle line:90% that really separates R from Q. 00:34:18.030 --> 00:34:23.219 align:middle line:84% So to define this, I need to define 00:34:23.219 --> 00:34:26.370 align:middle line:90% what a least upper bound is. 00:34:26.370 --> 00:34:30.090 align:middle line:84% And this is all in the setting of an ordered set. 00:34:30.090 --> 00:34:44.800 align:middle line:84% Let S be an ordered set, and let E be a subset of S. 00:34:44.800 --> 00:34:46.989 align:middle line:84% So I'm going to make a series of definitions here. 00:34:46.989 --> 00:34:49.830 align:middle line:90% 00:34:49.830 --> 00:34:54.270 align:middle line:84% First, if there exists an element of S-- 00:34:54.270 --> 00:34:58.463 align:middle line:84% so not necessarily the set I'm looking at, the subset 00:34:58.463 --> 00:34:59.130 align:middle line:90% I'm looking at-- 00:34:59.130 --> 00:35:03.640 align:middle line:90% 00:35:03.640 --> 00:35:11.700 align:middle line:84% such that for all x in E, x is less than or equal to b. 00:35:11.700 --> 00:35:13.750 align:middle line:84% So here, I have this order less than. 00:35:13.750 --> 00:35:17.100 align:middle line:84% Less than or equal to means just what it means in English, 00:35:17.100 --> 00:35:20.220 align:middle line:84% either x is less than b or x is equal to b, 00:35:20.220 --> 00:35:26.370 align:middle line:84% and the same thing with bigger than or equal to, and so on. 00:35:26.370 --> 00:35:33.590 align:middle line:84% But just keep in mind that this ordered 00:35:33.590 --> 00:35:35.510 align:middle line:90% set is a general ordered set. 00:35:35.510 --> 00:35:38.570 align:middle line:84% You could think of it as the dictionary ordering 00:35:38.570 --> 00:35:41.015 align:middle line:90% on Q cross Q. 00:35:41.015 --> 00:35:44.060 align:middle line:84% So if there exists a b such that for all x in E, 00:35:44.060 --> 00:35:50.980 align:middle line:84% x is less than or equal to b, then 00:35:50.980 --> 00:35:57.280 align:middle line:90% we say that E is bounded above. 00:35:57.280 --> 00:36:03.480 align:middle line:90% 00:36:03.480 --> 00:36:24.340 align:middle line:84% And this element of b is an upper bound for E. 00:36:24.340 --> 00:36:28.540 align:middle line:84% So if I can find some element of my set bigger than everything 00:36:28.540 --> 00:36:32.260 align:middle line:84% in this set E, I say E's bounded above, 00:36:32.260 --> 00:36:34.660 align:middle line:90% and will be an upper bound. 00:36:34.660 --> 00:36:38.270 align:middle line:90% I also have lower bounds. 00:36:38.270 --> 00:36:45.870 align:middle line:84% If there exists b in S such that for all x in E, 00:36:45.870 --> 00:36:50.510 align:middle line:84% b is smaller than or equal to everything in E, 00:36:50.510 --> 00:37:11.450 align:middle line:84% so b is less than or equal to x, then we say E is bounded below, 00:37:11.450 --> 00:37:41.960 align:middle line:84% and b is a lower bound for E. So b sits below everything in E. 00:37:41.960 --> 00:38:08.100 align:middle line:84% Now, we call an element in S the least upper bound for E 00:38:08.100 --> 00:38:10.020 align:middle line:90% if it satisfies two conditions. 00:38:10.020 --> 00:38:15.540 align:middle line:90% 00:38:15.540 --> 00:38:18.663 align:middle line:84% One is, if there's a least upper bound, 00:38:18.663 --> 00:38:20.330 align:middle line:84% there should at least be an upper bound. 00:38:20.330 --> 00:38:29.570 align:middle line:90% 00:38:29.570 --> 00:38:32.630 align:middle line:84% b is an upper bound for E, and it 00:38:32.630 --> 00:38:36.480 align:middle line:84% should be, in some sense, the least of all upper bounds. 00:38:36.480 --> 00:38:41.270 align:middle line:84% So if I take any other upper bound, 00:38:41.270 --> 00:38:43.550 align:middle line:84% b sub 0 should sit below that one. 00:38:43.550 --> 00:39:00.940 align:middle line:90% 00:39:00.940 --> 00:39:07.430 align:middle line:84% So an element is the least upper bound for a set 00:39:07.430 --> 00:39:09.920 align:middle line:84% E if it's an upper bound and it's 00:39:09.920 --> 00:39:12.050 align:middle line:84% the least among all other upper bounds. 00:39:12.050 --> 00:39:16.500 align:middle line:84% It sits below every other upper bound. 00:39:16.500 --> 00:39:34.120 align:middle line:84% And in this case, we also say b0 is the supremum of E. 00:39:34.120 --> 00:39:45.860 align:middle line:84% And we write b0 equals sup E. Now, 00:39:45.860 --> 00:39:50.520 align:middle line:84% this was having to deal with upper bounds. 00:39:50.520 --> 00:39:51.960 align:middle line:84% We can also deal with lower bounds 00:39:51.960 --> 00:39:54.540 align:middle line:84% or we also have a definition corresponding 00:39:54.540 --> 00:39:57.940 align:middle line:84% to lower bounds for what would be the greatest lower bound. 00:39:57.940 --> 00:40:16.640 align:middle line:84% So we call an element of S the greatest lower bound 00:40:16.640 --> 00:40:22.640 align:middle line:84% for E if two conditions hold, which 00:40:22.640 --> 00:40:24.612 align:middle line:84% is kind of similar to the conditions we 00:40:24.612 --> 00:40:25.820 align:middle line:90% had for at least upper bound. 00:40:25.820 --> 00:40:37.820 align:middle line:84% Except now for lower bounds, b0 is a lower bound for E, 00:40:37.820 --> 00:40:42.300 align:middle line:84% and it is the greatest of all lower bounds. 00:40:42.300 --> 00:40:59.863 align:middle line:84% If b is any lower bound for E, then b 00:40:59.863 --> 00:41:01.030 align:middle line:90% is less than or equal to b0. 00:41:01.030 --> 00:41:08.200 align:middle line:90% 00:41:08.200 --> 00:41:14.330 align:middle line:84% And so there's some Latin name attached to least upper bound, 00:41:14.330 --> 00:41:18.000 align:middle line:84% so there's a Latin name attached to the greatest lower bound, 00:41:18.000 --> 00:41:21.380 align:middle line:90% which we call the infimum. 00:41:21.380 --> 00:41:35.110 align:middle line:84% We also call b0 the infimum of E. And we write b0 00:41:35.110 --> 00:41:40.210 align:middle line:90% equals inf E. 00:41:40.210 --> 00:41:43.600 align:middle line:84% So we have this mildly interesting and complex 00:41:43.600 --> 00:41:52.180 align:middle line:84% definition, which means we should look at some examples 00:41:52.180 --> 00:41:54.670 align:middle line:90% to get a feel for it. 00:41:54.670 --> 00:42:00.260 align:middle line:84% So let's look at some simple examples. 00:42:00.260 --> 00:42:03.790 align:middle line:84% So let's take our big ordered set to be Z. 00:42:03.790 --> 00:42:11.350 align:middle line:84% And let's take our set E to be minus 1, 0, and 2. 00:42:11.350 --> 00:42:25.720 align:middle line:90% 00:42:25.720 --> 00:42:29.990 align:middle line:90% So what about this guy? 00:42:29.990 --> 00:42:31.180 align:middle line:90% What is the supremum? 00:42:31.180 --> 00:42:32.132 align:middle line:90% What is the infimum? 00:42:32.132 --> 00:42:34.690 align:middle line:90% 00:42:34.690 --> 00:42:38.920 align:middle line:84% So really, if I'm going to prove or if I'm 00:42:38.920 --> 00:42:41.800 align:middle line:84% going to make a statement something is 00:42:41.800 --> 00:42:43.960 align:middle line:84% equal to the supremum or something is 00:42:43.960 --> 00:42:47.830 align:middle line:84% equal to the infimum, I should actually give a proof of that, 00:42:47.830 --> 00:42:50.470 align:middle line:84% meaning if I say something is an infimum or something is 00:42:50.470 --> 00:42:52.600 align:middle line:84% a supremum, then I need to prove that it 00:42:52.600 --> 00:42:55.900 align:middle line:84% satisfies these two conditions and these two conditions. 00:42:55.900 --> 00:43:00.370 align:middle line:84% But I'm just going over examples, 00:43:00.370 --> 00:43:05.220 align:middle line:84% so I will not give a full proof of that. 00:43:05.220 --> 00:43:08.080 align:middle line:84% This is just to get some intuition going. 00:43:08.080 --> 00:43:18.280 align:middle line:84% So now, first off, what would be an upper bound for E? 00:43:18.280 --> 00:43:20.470 align:middle line:90% Well, 3, 4, 5-- 00:43:20.470 --> 00:43:23.890 align:middle line:84% 2 is an upper bound because 2 is bigger than or equal to 00:43:23.890 --> 00:43:27.170 align:middle line:90% everything in the set. 00:43:27.170 --> 00:43:34.990 align:middle line:90% So 2, 3, 4, 5-- 00:43:34.990 --> 00:43:41.840 align:middle line:84% these are all upper bounds for this set E. 00:43:41.840 --> 00:43:45.440 align:middle line:84% But what is the supremum, meaning the least upper bound? 00:43:45.440 --> 00:43:47.520 align:middle line:90% That would be 2. 00:43:47.520 --> 00:43:49.720 align:middle line:84% If I take anything less than 2 that's not an upper 00:43:49.720 --> 00:43:52.220 align:middle line:84% bound because there's something in the set bigger than that. 00:43:52.220 --> 00:43:55.520 align:middle line:90% 00:43:55.520 --> 00:43:59.890 align:middle line:84% And if I take anything bigger than that, 00:43:59.890 --> 00:44:02.140 align:middle line:84% then this is going to be bigger than 2, 00:44:02.140 --> 00:44:03.970 align:middle line:90% but 2 is still an upper bound. 00:44:03.970 --> 00:44:06.520 align:middle line:90% So 2 is the least upper bound. 00:44:06.520 --> 00:44:11.150 align:middle line:90% 00:44:11.150 --> 00:44:12.490 align:middle line:90% Now what about lower bounds? 00:44:12.490 --> 00:44:17.710 align:middle line:84% A lower bound would be minus 1, because minus 1 is less than 00:44:17.710 --> 00:44:19.540 align:middle line:84% or equal to everything in the set. 00:44:19.540 --> 00:44:27.390 align:middle line:84% But then so would be minus 2, minus 3, and so on. 00:44:27.390 --> 00:44:32.900 align:middle line:84% But the greatest lower bound would be minus 1. 00:44:32.900 --> 00:44:45.912 align:middle line:90% 00:44:45.912 --> 00:44:47.120 align:middle line:90% Now let's do another example. 00:44:47.120 --> 00:44:55.600 align:middle line:90% 00:44:55.600 --> 00:45:03.990 align:middle line:84% Let's now look at S, the rational numbers, 00:45:03.990 --> 00:45:11.280 align:middle line:84% and let's take E to be a set of rational numbers such 00:45:11.280 --> 00:45:19.770 align:middle line:84% that 0 is bigger than or equal to-- 00:45:19.770 --> 00:45:25.120 align:middle line:90% Q is between 0 and 1 inclusive. 00:45:25.120 --> 00:45:26.800 align:middle line:84% So then what are some upper bounds? 00:45:26.800 --> 00:45:30.880 align:middle line:90% 00:45:30.880 --> 00:45:34.420 align:middle line:84% Everything in E is less than or equal to 1. 00:45:34.420 --> 00:45:47.288 align:middle line:84% So 1 is also a perfectly good upper bound, 3/2, 5/4, 6/5, 00:45:47.288 --> 00:45:47.788 align:middle line:90% 7/6. 00:45:47.788 --> 00:45:51.820 align:middle line:90% 00:45:51.820 --> 00:45:54.340 align:middle line:84% Anything bigger than or equal to 1 00:45:54.340 --> 00:45:57.070 align:middle line:84% is also an upper bound for this set. 00:45:57.070 --> 00:46:02.780 align:middle line:84% And the least upper bound would be 1. 00:46:02.780 --> 00:46:07.800 align:middle line:90% 00:46:07.800 --> 00:46:14.520 align:middle line:84% So if I were to try to draw Q for this set-- 00:46:14.520 --> 00:46:19.900 align:middle line:90% 00:46:19.900 --> 00:46:20.800 align:middle line:90% let's not do that-- 00:46:20.800 --> 00:46:24.750 align:middle line:90% 00:46:24.750 --> 00:46:29.110 align:middle line:84% I usually draw this line is for the real number line, 00:46:29.110 --> 00:46:34.720 align:middle line:84% but let's imagine it's Q. So everything 00:46:34.720 --> 00:46:39.370 align:middle line:84% bigger than or equal to 1 sits above everything in E. 00:46:39.370 --> 00:46:42.730 align:middle line:84% So everything bigger than or equal to 1 is an upper bound, 00:46:42.730 --> 00:46:45.700 align:middle line:90% and the least upper bound is 1. 00:46:45.700 --> 00:46:46.885 align:middle line:90% What are some lower bounds? 00:46:46.885 --> 00:46:49.870 align:middle line:90% 00:46:49.870 --> 00:46:53.740 align:middle line:84% Everything in the set E is bigger than or equal to 0. 00:46:53.740 --> 00:46:56.420 align:middle line:90% So 0 is a lower bound. 00:46:56.420 --> 00:47:04.990 align:middle line:84% So is minus 1/2, minus 1/3, minus 1/4, minus 1/5, and so 00:47:04.990 --> 00:47:06.880 align:middle line:90% on. 00:47:06.880 --> 00:47:11.010 align:middle line:84% But anything less than or equal to 0 is a lower bound. 00:47:11.010 --> 00:47:13.620 align:middle line:84% Nothing bigger than 0 can be a lower bound 00:47:13.620 --> 00:47:16.350 align:middle line:90% because I can always find some-- 00:47:16.350 --> 00:47:20.460 align:middle line:84% so this is not a proof, but this is some explanation. 00:47:20.460 --> 00:47:24.663 align:middle line:90% 00:47:24.663 --> 00:47:26.080 align:middle line:84% Maybe I should have also said this 00:47:26.080 --> 00:47:33.990 align:middle line:84% for the upper bound statement, but anything bigger than 0 00:47:33.990 --> 00:47:37.530 align:middle line:90% is not a lower bound. 00:47:37.530 --> 00:47:41.580 align:middle line:90% 00:47:41.580 --> 00:47:46.260 align:middle line:84% Because what if I take some number, call it r, 00:47:46.260 --> 00:47:53.580 align:middle line:84% bigger than 0, and less than 1, then r cannot be a lower bound 00:47:53.580 --> 00:47:59.730 align:middle line:84% because I can find something in E less than r, namely r over 2. 00:47:59.730 --> 00:48:04.200 align:middle line:90% 00:48:04.200 --> 00:48:08.970 align:middle line:84% So therefore-- and we're going to do some proofs where 00:48:08.970 --> 00:48:11.280 align:middle line:84% we actually have to get our hands dirty 00:48:11.280 --> 00:48:14.640 align:middle line:84% and prove something is an infimum or the supremum. 00:48:14.640 --> 00:48:16.770 align:middle line:84% So you'll see how that works, but for now, let's 00:48:16.770 --> 00:48:20.040 align:middle line:90% just go off of intuition. 00:48:20.040 --> 00:48:23.490 align:middle line:90% The infimum of this set is 0. 00:48:23.490 --> 00:48:28.710 align:middle line:90% 00:48:28.710 --> 00:48:32.370 align:middle line:84% Now, both of these examples so far 00:48:32.370 --> 00:48:37.140 align:middle line:84% have this property that both the sup and the inf 00:48:37.140 --> 00:48:41.140 align:middle line:84% belong to the set E that I'm looking at. 00:48:41.140 --> 00:48:43.930 align:middle line:84% But this is not necessarily always the case. 00:48:43.930 --> 00:48:50.370 align:middle line:84% So I could change this slightly so our universe, 00:48:50.370 --> 00:48:55.500 align:middle line:84% the ordered set we're looking in, is still Q, 00:48:55.500 --> 00:49:08.860 align:middle line:84% and the subset E is now, let's say, q is bigger than 0 00:49:08.860 --> 00:49:11.140 align:middle line:90% and less than 1. 00:49:11.140 --> 00:49:18.600 align:middle line:84% Then sup of E is still going to be 1, 00:49:18.600 --> 00:49:22.020 align:middle line:84% but this is not an element of this set E. 00:49:22.020 --> 00:49:27.130 align:middle line:84% It's of course an element of S, this universe we're in, 00:49:27.130 --> 00:49:37.140 align:middle line:84% but it's not an element of the set E. 00:49:37.140 --> 00:49:39.300 align:middle line:84% And likewise, the infimum is still 0, 00:49:39.300 --> 00:49:42.330 align:middle line:90% but it's not an element of E. 00:49:42.330 --> 00:49:44.280 align:middle line:84% So there are situations where you 00:49:44.280 --> 00:49:47.400 align:middle line:84% have a set, a subset of an ordered set, 00:49:47.400 --> 00:49:50.190 align:middle line:84% which has an infimum and supremum which 00:49:50.190 --> 00:49:53.550 align:middle line:84% do not exist in this smaller set you're looking at. 00:49:53.550 --> 00:49:55.620 align:middle line:84% Whether or not the supremum or infimum 00:49:55.620 --> 00:49:58.350 align:middle line:84% may exist in the universe that you're 00:49:58.350 --> 00:50:01.320 align:middle line:84% looking at in the bigger set is an entirely different issue. 00:50:01.320 --> 00:50:04.690 align:middle line:90% 00:50:04.690 --> 00:50:09.580 align:middle line:84% In fact, that's the next issue we're going to talk about. 00:50:09.580 --> 00:50:14.490 align:middle line:90% 00:50:14.490 --> 00:50:20.170 align:middle line:84% So let me just reiterate what we saw a minute ago. 00:50:20.170 --> 00:50:25.890 align:middle line:84% So we can be in some ordered set Z or Q, 00:50:25.890 --> 00:50:30.810 align:middle line:84% and the smaller set, which we're taking the inf or sup in, 00:50:30.810 --> 00:50:33.600 align:middle line:84% it could belong to the smaller set-- 00:50:33.600 --> 00:50:39.870 align:middle line:84% for example, in this case 1 and 0 were in the set E-- 00:50:39.870 --> 00:50:40.740 align:middle line:90% or not. 00:50:40.740 --> 00:50:44.310 align:middle line:84% There is this case where neither of them were in E. 00:50:44.310 --> 00:50:46.980 align:middle line:84% And of course, if I put less than or equal to here, 00:50:46.980 --> 00:50:51.480 align:middle line:84% then I would had the inf in E and the sup not in E, 00:50:51.480 --> 00:50:52.450 align:middle line:90% and then vice versa. 00:50:52.450 --> 00:50:57.750 align:middle line:84% So inf's and sup's of these sets don't necessarily 00:50:57.750 --> 00:51:02.430 align:middle line:84% need to belong to that set you're looking at. 00:51:02.430 --> 00:51:09.560 align:middle line:84% But they at least existed in the big set Q. 00:51:09.560 --> 00:51:13.240 align:middle line:90% Now, big, ordered sets-- 00:51:13.240 --> 00:51:17.140 align:middle line:84% so this ordered set that has this property that the inf 00:51:17.140 --> 00:51:21.550 align:middle line:84% and sup of bounded above and below sets 00:51:21.550 --> 00:51:26.270 align:middle line:90% exist in the bigger set always-- 00:51:26.270 --> 00:51:29.680 align:middle line:84% is what we call an ordered set with the least upper 00:51:29.680 --> 00:51:30.305 align:middle line:90% bound property. 00:51:30.305 --> 00:51:37.330 align:middle line:90% 00:51:37.330 --> 00:51:52.790 align:middle line:84% So the definition-- ordered set S 00:51:52.790 --> 00:52:20.750 align:middle line:84% has the least property if every subset of S which 00:52:20.750 --> 00:52:52.200 align:middle line:84% is non-empty and bounded above has a supremum in S. 00:52:52.200 --> 00:52:55.440 align:middle line:84% So a set has the least upper bound property 00:52:55.440 --> 00:52:58.345 align:middle line:84% if every non-empty bounded set has a supremum. 00:52:58.345 --> 00:53:01.540 align:middle line:90% 00:53:01.540 --> 00:53:08.480 align:middle line:84% And so we could come up with simple examples 00:53:08.480 --> 00:53:16.190 align:middle line:84% of sets that do have the least upper bound property. 00:53:16.190 --> 00:53:20.390 align:middle line:84% For example, let's take S to be-- 00:53:20.390 --> 00:53:22.600 align:middle line:84% I mean, this is kind of the simplest one-- 00:53:22.600 --> 00:53:26.362 align:middle line:84% S with a single element, let's say 0. 00:53:26.362 --> 00:53:29.050 align:middle line:90% 00:53:29.050 --> 00:53:33.430 align:middle line:84% Here there's no order to put on it. 00:53:33.430 --> 00:53:36.710 align:middle line:84% Every element in here is equal to itself. 00:53:36.710 --> 00:53:41.200 align:middle line:84% And therefore, every non-empty subset of S 00:53:41.200 --> 00:53:43.990 align:middle line:84% is just the whole set, and the supremum 00:53:43.990 --> 00:53:48.070 align:middle line:90% would then be that one element. 00:53:48.070 --> 00:53:52.375 align:middle line:84% So this is kind of the silliest one you could do. 00:53:52.375 --> 00:53:56.410 align:middle line:90% 00:53:56.410 --> 00:54:02.320 align:middle line:84% Let's say you could have two elements. 00:54:02.320 --> 00:54:10.960 align:middle line:84% And then if E is a subset of S, E is one of four guys. 00:54:10.960 --> 00:54:15.940 align:middle line:84% If E is a non-empty subset of S here, the order is 1, 00:54:15.940 --> 00:54:17.830 align:middle line:90% 0 is less than 1. 00:54:17.830 --> 00:54:25.990 align:middle line:84% If E is equal to 0, then sup equals 0. 00:54:25.990 --> 00:54:31.630 align:middle line:84% If E is equal to 1, sup E equals 1. 00:54:31.630 --> 00:54:40.625 align:middle line:84% And if E is equal to 0, 1, this implies sup E equals 1. 00:54:40.625 --> 00:54:45.890 align:middle line:84% So this is in S, this is in S, this is in S. 00:54:45.890 --> 00:54:52.370 align:middle line:84% So every subset of S which is non-empty has a supremum in S. 00:54:52.370 --> 00:54:58.240 align:middle line:84% So these are not the most interesting examples 00:54:58.240 --> 00:55:01.045 align:middle line:84% of sets with this least upper bound property. 00:55:01.045 --> 00:55:04.010 align:middle line:90% 00:55:04.010 --> 00:55:10.120 align:middle line:84% So for example, maybe a more interesting one 00:55:10.120 --> 00:55:19.760 align:middle line:84% would be if I take S to be, let's say, with the dots here, 00:55:19.760 --> 00:55:22.280 align:middle line:90% minus 3, minus 2, minus 1. 00:55:22.280 --> 00:55:24.680 align:middle line:84% So let me just write it this way-- 00:55:24.680 --> 00:55:29.960 align:middle line:84% minus 1, minus 2, minus 3, minus 4, and so on, 00:55:29.960 --> 00:55:33.740 align:middle line:84% with the usual ordering coming from the integers. 00:55:33.740 --> 00:55:39.800 align:middle line:84% So minus 2 is less than minus 1, minus 3 is less than minus 2, 00:55:39.800 --> 00:55:40.970 align:middle line:90% and so on. 00:55:40.970 --> 00:55:45.150 align:middle line:84% I claim this that does have the least upper bound property. 00:55:45.150 --> 00:55:45.650 align:middle line:90% Why? 00:55:45.650 --> 00:55:53.670 align:middle line:84% Because if E is a subset of S, E non-empty. 00:55:53.670 --> 00:55:58.780 align:middle line:90% 00:55:58.780 --> 00:56:03.030 align:middle line:90% In fact, both of these-- 00:56:03.030 --> 00:56:06.240 align:middle line:84% this was always bounded above because there's 00:56:06.240 --> 00:56:07.320 align:middle line:90% only two elements. 00:56:07.320 --> 00:56:09.330 align:middle line:84% And you can choose the biggest one, 1. 00:56:09.330 --> 00:56:13.680 align:middle line:84% Every subset of S is also bounded above by minus 1. 00:56:13.680 --> 00:56:17.760 align:middle line:84% So every subset of S is automatically bounded above. 00:56:17.760 --> 00:56:21.240 align:middle line:84% So I just need to check that every non-empty set has 00:56:21.240 --> 00:56:28.240 align:middle line:84% a supremum in S. If E is a subset of S and is non-empty, 00:56:28.240 --> 00:56:32.550 align:middle line:84% then let me look at the set minus E. This is a label. 00:56:32.550 --> 00:56:36.030 align:middle line:84% This is not meant to mean anything always. 00:56:36.030 --> 00:56:41.520 align:middle line:84% This is a label, which is the set of elements minus x. 00:56:41.520 --> 00:56:44.100 align:middle line:90% 00:56:44.100 --> 00:56:46.380 align:middle line:84% So I take all my elements in E, which 00:56:46.380 --> 00:56:48.720 align:middle line:84% is a subset of the negative integers. 00:56:48.720 --> 00:56:49.860 align:middle line:90% I take their minuses. 00:56:49.860 --> 00:56:54.120 align:middle line:84% This is now a subset of the natural numbers. 00:56:54.120 --> 00:57:07.460 align:middle line:84% And since by the well-ordering property 00:57:07.460 --> 00:57:14.390 align:middle line:84% of the natural numbers, there exists an element 00:57:14.390 --> 00:57:21.590 align:middle line:84% in minus E such that it sits below everything 00:57:21.590 --> 00:57:39.490 align:middle line:84% in minus E for all x in E, which means that for all x in E, 00:57:39.490 --> 00:57:41.560 align:middle line:84% x is less than or equal to minus m. 00:57:41.560 --> 00:57:46.690 align:middle line:90% 00:57:46.690 --> 00:57:51.070 align:middle line:84% And I'll let you think about this just for a minute. 00:57:51.070 --> 00:58:07.560 align:middle line:84% But so m is in minus E, therefore minus m is in E. 00:58:07.560 --> 00:58:09.870 align:middle line:84% And m less than or equal to minus x 00:58:09.870 --> 00:58:17.480 align:middle line:84% for all x in E implies that for all x in capital E is less than 00:58:17.480 --> 00:58:20.180 align:middle line:90% or equal to minus m. 00:58:20.180 --> 00:58:21.980 align:middle line:84% And therefore, I found an element actually 00:58:21.980 --> 00:58:23.660 align:middle line:84% in the set capital E that's bigger 00:58:23.660 --> 00:58:26.390 align:middle line:84% than or equal to everything in the set. 00:58:26.390 --> 00:58:29.420 align:middle line:84% You can convince yourself or write down 00:58:29.420 --> 00:58:36.500 align:middle line:84% a formal proof, if you like, that this implies that minus 00:58:36.500 --> 00:58:37.705 align:middle line:90% m is therefore the supremum. 00:58:37.705 --> 00:58:54.960 align:middle line:90% 00:58:54.960 --> 00:59:00.390 align:middle line:84% And so using this trickery of going from one 00:59:00.390 --> 00:59:10.650 align:middle line:84% set that's bounded above to a different set, a subset 00:59:10.650 --> 00:59:14.910 align:middle line:84% of the natural numbers, you can also show, for example, 00:59:14.910 --> 00:59:27.020 align:middle line:84% Z has the least upper bound property, 00:59:27.020 --> 00:59:30.710 align:middle line:84% which I'm going to now shorten because those 00:59:30.710 --> 00:59:32.120 align:middle line:90% are a lot of things to write. 00:59:32.120 --> 00:59:34.550 align:middle line:84% LUBP, least upper bound property-- 00:59:34.550 --> 00:59:38.252 align:middle line:84% the integers also have this property. 00:59:38.252 --> 00:59:40.460 align:middle line:84% But the integers are not all that interesting, again, 00:59:40.460 --> 00:59:43.700 align:middle line:84% because I cannot divide by an integer and stay in the set. 00:59:43.700 --> 00:59:48.264 align:middle line:90% 00:59:48.264 --> 00:59:54.240 align:middle line:84% For what we want to do, we want to be able to add, multiply, 00:59:54.240 --> 00:59:56.610 align:middle line:90% subtract, and divide. 00:59:56.610 --> 00:59:58.260 align:middle line:84% And we can't do that in the integers. 00:59:58.260 --> 01:00:01.875 align:middle line:84% We can do that in the rational numbers. 01:00:01.875 --> 01:00:04.920 align:middle line:90% 01:00:04.920 --> 01:00:15.180 align:middle line:84% However, Q does not have this property. 01:00:15.180 --> 01:00:18.090 align:middle line:90% 01:00:18.090 --> 01:00:19.380 align:middle line:90% And we're going to prove this. 01:00:19.380 --> 01:00:27.650 align:middle line:84% So this is what we're going to prove in a couple of theorems 01:00:27.650 --> 01:00:31.130 align:middle line:84% here, but let me just put this out in front. 01:00:31.130 --> 01:00:42.230 align:middle line:90% 01:00:42.230 --> 01:00:44.390 align:middle line:84% Q does not have the least upper bound property. 01:00:44.390 --> 01:00:48.640 align:middle line:90% 01:00:48.640 --> 01:00:52.690 align:middle line:90% And where does this come from? 01:00:52.690 --> 01:00:57.880 align:middle line:84% This comes from the simple fact which if you believe your-- 01:00:57.880 --> 01:01:01.206 align:middle line:90% 01:01:01.206 --> 01:01:02.940 align:middle line:84% I don't want to call it Greek mythology, 01:01:02.940 --> 01:01:06.900 align:middle line:84% but maybe it's Greek mythology strictly speaking-- 01:01:06.900 --> 01:01:10.860 align:middle line:84% that some young guy discovered that the square root of 2 01:01:10.860 --> 01:01:15.840 align:middle line:84% is not a rational number and then he got thrown off a cliff. 01:01:15.840 --> 01:01:20.560 align:middle line:84% That's who we have to thank for showing us 01:01:20.560 --> 01:01:23.350 align:middle line:84% that Q is not perfect, that it does 01:01:23.350 --> 01:01:26.513 align:middle line:84% have this algebraic property which I just referenced-- 01:01:26.513 --> 01:01:28.430 align:middle line:84% the fact that you can add, subtract, multiply, 01:01:28.430 --> 01:01:30.940 align:middle line:84% and divide and stay within the set. 01:01:30.940 --> 01:01:36.860 align:middle line:84% But it does not have square roots of prime numbers, 01:01:36.860 --> 01:01:38.450 align:middle line:90% but square root of 2 in it. 01:01:38.450 --> 01:01:42.020 align:middle line:84% And this then manifests itself in this least 01:01:42.020 --> 01:01:49.170 align:middle line:84% upper bound property by giving an example of a set which 01:01:49.170 --> 01:01:56.231 align:middle line:84% is bounded above which does not have a supremum in the set Q. 01:01:56.231 --> 01:02:10.105 align:middle line:84% So basically, if E is the set of rational numbers-- 01:02:10.105 --> 01:02:12.480 align:middle line:84% I have not stated a theorem yet, but I'm just telling you 01:02:12.480 --> 01:02:14.130 align:middle line:90% what we're about to do-- 01:02:14.130 --> 01:02:17.580 align:middle line:84% if E is the subset of rational numbers 01:02:17.580 --> 01:02:28.480 align:middle line:84% where positive and q squared is less than 2, 01:02:28.480 --> 01:02:38.140 align:middle line:84% then sup E does not exist in the rational numbers. 01:02:38.140 --> 01:02:39.610 align:middle line:84% So therefore, we would have found 01:02:39.610 --> 01:02:44.230 align:middle line:84% a set which is bounded above which 01:02:44.230 --> 01:02:46.900 align:middle line:84% does not have a supremum in the rational numbers. 01:02:46.900 --> 01:02:48.400 align:middle line:84% And therefore, the rational numbers 01:02:48.400 --> 01:02:51.380 align:middle line:84% do not have the least upper bound property. 01:02:51.380 --> 01:02:55.150 align:middle line:84% So this is what we're going to do. 01:02:55.150 --> 01:02:59.530 align:middle line:84% And I'll prove-- state a couple of theorems 01:02:59.530 --> 01:03:01.550 align:middle line:90% that spell all this out. 01:03:01.550 --> 01:03:10.900 align:middle line:84% So first, I'm going to prove a theorem about a what 01:03:10.900 --> 01:03:14.440 align:middle line:84% the supremum of such a set would have to satisfy. 01:03:14.440 --> 01:03:20.810 align:middle line:90% So if x is in Q-- 01:03:20.810 --> 01:03:22.520 align:middle line:84% this is a statement of a theorem-- 01:03:22.520 --> 01:03:40.880 align:middle line:84% if x is in Q, and if x equals the supremum of the set, 01:03:40.880 --> 01:03:50.060 align:middle line:84% then x is bigger than or equal to 1, and x squared equals 2. 01:03:50.060 --> 01:04:01.450 align:middle line:90% 01:04:01.450 --> 01:04:03.100 align:middle line:90% So we're just saying-- 01:04:03.100 --> 01:04:09.040 align:middle line:84% don't try to take this as necessarily contradicting what 01:04:09.040 --> 01:04:12.920 align:middle line:84% I wrote there because that was not a statement of a theorem 01:04:12.920 --> 01:04:13.420 align:middle line:90% yet. 01:04:13.420 --> 01:04:17.890 align:middle line:84% I'm stating this theorem that if I have such a supremum in Q 01:04:17.890 --> 01:04:21.460 align:middle line:84% of the set, then it would have to square and give me 2. 01:04:21.460 --> 01:04:24.440 align:middle line:90% 01:04:24.440 --> 01:04:26.960 align:middle line:84% I'm not saying such an element exists. 01:04:26.960 --> 01:04:30.230 align:middle line:84% I'm just saying if there is any element of Q that 01:04:30.230 --> 01:04:32.360 align:middle line:84% is the supremum of this set, then 01:04:32.360 --> 01:04:34.760 align:middle line:84% it has to be bigger than or equal to 1, and it squares 2. 01:04:34.760 --> 01:04:37.310 align:middle line:90% So let's give a proof of this. 01:04:37.310 --> 01:04:42.790 align:middle line:84% And because I don't want to keep writing the set again, 01:04:42.790 --> 01:04:49.470 align:middle line:84% so I'm going to use the notation that I used in the comments. 01:04:49.470 --> 01:04:51.930 align:middle line:84% Let E be the set that I'm interested in. 01:04:51.930 --> 01:04:57.490 align:middle line:90% 01:04:57.490 --> 01:04:58.350 align:middle line:90% So this should be-- 01:04:58.350 --> 01:05:10.300 align:middle line:90% 01:05:10.300 --> 01:05:26.721 align:middle line:84% and suppose we have an element of Q 01:05:26.721 --> 01:05:32.130 align:middle line:90% which is the supremum of E. 01:05:32.130 --> 01:05:34.710 align:middle line:84% So first off, what would be one element of E? 01:05:34.710 --> 01:05:38.250 align:middle line:84% What's one rational number whose square is less than 2? 01:05:38.250 --> 01:05:39.690 align:middle line:90% 1. 01:05:39.690 --> 01:05:47.460 align:middle line:84% So since 1 is in E, square is less than 2, 01:05:47.460 --> 01:05:56.370 align:middle line:84% and x is the supremum of E, meaning 01:05:56.370 --> 01:05:58.350 align:middle line:84% it's an upper bound for E, it must 01:05:58.350 --> 01:06:02.940 align:middle line:84% be bigger than or equal to everything an E, 01:06:02.940 --> 01:06:04.110 align:middle line:90% in particular for 1. 01:06:04.110 --> 01:06:07.140 align:middle line:90% 01:06:07.140 --> 01:06:09.090 align:middle line:84% That implies x is bigger than or equal to 1. 01:06:09.090 --> 01:06:11.460 align:middle line:84% So that's the first part of the theorem I want to prove. 01:06:11.460 --> 01:06:14.760 align:middle line:90% 01:06:14.760 --> 01:06:20.110 align:middle line:84% So now we're going to prove two inequalities 01:06:20.110 --> 01:06:23.230 align:middle line:84% to show that x squared is actually equal to 2. 01:06:23.230 --> 01:06:24.160 align:middle line:90% We're going to prove-- 01:06:24.160 --> 01:06:28.970 align:middle line:84% so this is a common trick in analysis, 01:06:28.970 --> 01:06:30.650 align:middle line:84% that if I have two things that I would 01:06:30.650 --> 01:06:34.170 align:middle line:84% like to show equal to each other, sometimes a way 01:06:34.170 --> 01:06:38.110 align:middle line:84% to show that is by showing one is less than or equal to that. 01:06:38.110 --> 01:06:40.890 align:middle line:84% And this is less than or equal to that. 01:06:40.890 --> 01:06:44.700 align:middle line:84% So one side is less than or equal to the other side 01:06:44.700 --> 01:06:49.150 align:middle line:84% and vice versa, which immediately implies 01:06:49.150 --> 01:06:50.320 align:middle line:90% they must equal each other. 01:06:50.320 --> 01:06:55.310 align:middle line:90% 01:06:55.310 --> 01:07:00.850 align:middle line:84% So that's what we're going to do. 01:07:00.850 --> 01:07:04.450 align:middle line:84% And we're going to now prove that x squared 01:07:04.450 --> 01:07:07.205 align:middle line:90% is bigger than or equal to 2. 01:07:07.205 --> 01:07:10.270 align:middle line:90% 01:07:10.270 --> 01:07:14.020 align:middle line:84% We'll then prove that x squared is less than or equal to 2, 01:07:14.020 --> 01:07:16.780 align:middle line:84% and therefore, x squared equals 2. 01:07:16.780 --> 01:07:19.960 align:middle line:84% So to do this, we'll do this by contradiction. 01:07:19.960 --> 01:07:24.710 align:middle line:90% 01:07:24.710 --> 01:07:32.200 align:middle line:84% So let's assume-- everything else we've done so far 01:07:32.200 --> 01:07:34.750 align:middle line:84% is still true, but now we want to prove this statement. 01:07:34.750 --> 01:07:37.000 align:middle line:84% So we're going to assume that this statement is false. 01:07:37.000 --> 01:07:40.720 align:middle line:90% 01:07:40.720 --> 01:07:45.120 align:middle line:84% I assume that x squared is less than 2. 01:07:45.120 --> 01:07:55.630 align:middle line:90% 01:07:55.630 --> 01:08:04.580 align:middle line:84% So we'll define a certain rational number, h, 01:08:04.580 --> 01:08:11.090 align:middle line:84% which is a smaller of two rational numbers. 01:08:11.090 --> 01:08:15.590 align:middle line:84% It's going to be the smaller of 1/2 and 2 01:08:15.590 --> 01:08:21.620 align:middle line:84% minus x squared over 2 over 2x plus 1. 01:08:21.620 --> 01:08:24.410 align:middle line:84% And let me just reiterate this is less than 1 01:08:24.410 --> 01:08:27.229 align:middle line:84% because it's the smaller of 1/2 and this number. 01:08:27.229 --> 01:08:30.609 align:middle line:90% 01:08:30.609 --> 01:08:33.130 align:middle line:84% Now, when you write a proof, as you'll see, 01:08:33.130 --> 01:08:35.410 align:middle line:84% it's going to be magic that somehow 01:08:35.410 --> 01:08:39.580 align:middle line:90% this h does something magical. 01:08:39.580 --> 01:08:42.040 align:middle line:84% That's not exactly how you come up with proofs. 01:08:42.040 --> 01:08:46.120 align:middle line:84% How it comes up is you take an inequality 01:08:46.120 --> 01:08:48.640 align:middle line:84% that you want to mess with, you fiddle around with it, 01:08:48.640 --> 01:08:52.060 align:middle line:84% and you see that if h is given by something, 01:08:52.060 --> 01:08:56.569 align:middle line:84% then it breaks the inequality or it 01:08:56.569 --> 01:08:58.790 align:middle line:84% satisfies the inequality, which whichever one you're 01:08:58.790 --> 01:08:59.600 align:middle line:90% trying to do. 01:08:59.600 --> 01:09:02.370 align:middle line:90% 01:09:02.370 --> 01:09:13.970 align:middle line:84% So since x squared is less than 2, h is positive. 01:09:13.970 --> 01:09:16.649 align:middle line:84% It's the minimum of a 1/2 and this number. 01:09:16.649 --> 01:09:18.439 align:middle line:90% So if it's a 1/2, it's positive. 01:09:18.439 --> 01:09:23.290 align:middle line:84% If it's this one, then it's still positive. 01:09:23.290 --> 01:09:32.140 align:middle line:84% And what I'm going to prove now is that x plus h 01:09:32.140 --> 01:09:36.410 align:middle line:84% is in E. Its square is less than 2. 01:09:36.410 --> 01:09:38.270 align:middle line:84% This will give us our contradiction, 01:09:38.270 --> 01:09:41.689 align:middle line:84% because x is supposed to be an upper bound for E, 01:09:41.689 --> 01:09:43.790 align:middle line:84% and therefore, x is supposed to be bigger 01:09:43.790 --> 01:09:45.319 align:middle line:90% than or equal to x plus h. 01:09:45.319 --> 01:09:48.050 align:middle line:84% But this is x plus a positive number. 01:09:48.050 --> 01:09:51.469 align:middle line:84% x plus a positive number is bigger than x. 01:09:51.469 --> 01:09:54.560 align:middle line:90% 01:09:54.560 --> 01:09:56.870 align:middle line:90% So how do we do this? 01:09:56.870 --> 01:09:59.900 align:middle line:84% We have to compute the square of this and show it's less than 2. 01:09:59.900 --> 01:10:04.880 align:middle line:84% And that's where our choice of h comes from. 01:10:04.880 --> 01:10:09.220 align:middle line:90% 01:10:09.220 --> 01:10:15.530 align:middle line:84% We compute that x plus h squared-- 01:10:15.530 --> 01:10:18.715 align:middle line:84% this is x squared plus 2xh plus h squared-- 01:10:18.715 --> 01:10:21.320 align:middle line:90% 01:10:21.320 --> 01:10:32.355 align:middle line:84% now this is less than x squared plus 2xh plus h. 01:10:32.355 --> 01:10:33.180 align:middle line:90% Why? 01:10:33.180 --> 01:10:37.965 align:middle line:84% Because h times h is less than h since h is less than 1. 01:10:37.965 --> 01:10:40.830 align:middle line:90% 01:10:40.830 --> 01:10:45.690 align:middle line:84% So this is why we chose h to be less than 1, 01:10:45.690 --> 01:10:47.710 align:middle line:84% so that I can get rid of this square, 01:10:47.710 --> 01:10:50.610 align:middle line:84% and somehow just have a single h floating around, 01:10:50.610 --> 01:10:56.970 align:middle line:84% which I can then use to show x plus h is in E. 01:10:56.970 --> 01:11:02.120 align:middle line:84% So when I write the string of inequalities, 01:11:02.120 --> 01:11:04.370 align:middle line:84% this thing is supposed to be equal to the next thing. 01:11:04.370 --> 01:11:06.890 align:middle line:84% This is not saying x plus h squared is equal to what 01:11:06.890 --> 01:11:08.840 align:middle line:90% I'm about to write now. 01:11:08.840 --> 01:11:15.800 align:middle line:84% So this is equal to x squared plus 2x plus 1 times h. 01:11:15.800 --> 01:11:19.910 align:middle line:84% Now, h is the minimum of these two numbers. 01:11:19.910 --> 01:11:26.090 align:middle line:84% So h is going to be smaller than or equal to this thing. 01:11:26.090 --> 01:11:30.690 align:middle line:90% 01:11:30.690 --> 01:11:35.455 align:middle line:84% And so this is 2 minus x squared over 2. 01:11:35.455 --> 01:11:38.850 align:middle line:90% 01:11:38.850 --> 01:11:46.820 align:middle line:84% Write it this way, so I had what I had before. 01:11:46.820 --> 01:11:53.760 align:middle line:84% And then times 2 minus x squared over 2, 2x plus 1. 01:11:53.760 --> 01:11:55.500 align:middle line:90% So now some magic is happening. 01:11:55.500 --> 01:11:56.980 align:middle line:90% This cancels with that. 01:11:56.980 --> 01:12:01.080 align:middle line:84% And I have 2 minus x squared over 2, which is less than 2 01:12:01.080 --> 01:12:05.190 align:middle line:84% minus x squared, so less than x squared 01:12:05.190 --> 01:12:12.670 align:middle line:84% plus 2 minus x squared, because I took a 1/2 of it, equals 2. 01:12:12.670 --> 01:12:15.570 align:middle line:90% 01:12:15.570 --> 01:12:19.060 align:middle line:84% So summarizing, we started off with x plus h squared, 01:12:19.060 --> 01:12:20.460 align:middle line:84% and we showed it was less than 2. 01:12:20.460 --> 01:12:26.940 align:middle line:90% 01:12:26.940 --> 01:12:42.170 align:middle line:84% And therefore, x plus h is in E. But if I have an element of E-- 01:12:42.170 --> 01:12:45.720 align:middle line:90% 01:12:45.720 --> 01:12:47.550 align:middle line:90% let's see where are we on space. 01:12:47.550 --> 01:13:10.140 align:middle line:90% 01:13:10.140 --> 01:13:18.110 align:middle line:84% But I have an element of E which is bigger than x. 01:13:18.110 --> 01:13:25.040 align:middle line:84% So that implies that x is not equal to the supremum 01:13:25.040 --> 01:13:26.330 align:middle line:90% of the set. 01:13:26.330 --> 01:13:28.940 align:middle line:84% Remember by definition, something is the supremum 01:13:28.940 --> 01:13:32.550 align:middle line:84% if it's an upper bound for the set. 01:13:32.550 --> 01:13:36.590 align:middle line:84% Something's not an upper bound if you can find something 01:13:36.590 --> 01:13:40.620 align:middle line:90% in the set bigger than b0. 01:13:40.620 --> 01:13:42.470 align:middle line:84% And this is also a good exercise to do 01:13:42.470 --> 01:13:45.110 align:middle line:84% when you come across a new definition, 01:13:45.110 --> 01:13:47.150 align:middle line:84% is to try and negate it to understand 01:13:47.150 --> 01:13:49.550 align:middle line:90% it a little bit better. 01:13:49.550 --> 01:13:52.850 align:middle line:84% So let me write here next to b is an upper bound for E-- 01:13:52.850 --> 01:13:57.365 align:middle line:84% let me write what this means, actually, here. 01:13:57.365 --> 01:14:01.540 align:middle line:90% 01:14:01.540 --> 01:14:02.755 align:middle line:90% Let me write this here. 01:14:02.755 --> 01:14:05.540 align:middle line:90% 01:14:05.540 --> 01:14:10.095 align:middle line:84% b0 is not-- so what's the negation of being an upper 01:14:10.095 --> 01:14:10.595 align:middle line:90% bound-- 01:14:10.595 --> 01:14:14.570 align:middle line:90% 01:14:14.570 --> 01:14:21.770 align:middle line:84% so not an upper bound for E if there exists an x in E 01:14:21.770 --> 01:14:25.520 align:middle line:90% such that x is bigger than b0. 01:14:25.520 --> 01:14:28.300 align:middle line:90% 01:14:28.300 --> 01:14:30.550 align:middle line:90% So we have found an element-- 01:14:30.550 --> 01:14:32.290 align:middle line:84% so going back to our proof, we have 01:14:32.290 --> 01:14:35.170 align:middle line:84% found an element of our set E which 01:14:35.170 --> 01:14:38.110 align:middle line:84% is bigger than x, which means that x is not 01:14:38.110 --> 01:14:41.470 align:middle line:84% the supremum of E, which is a contradiction 01:14:41.470 --> 01:14:43.390 align:middle line:90% to our assumption. 01:14:43.390 --> 01:14:47.660 align:middle line:84% These are assumptions for the theorem. 01:14:47.660 --> 01:14:49.435 align:middle line:90% So we're always assuming this. 01:14:49.435 --> 01:14:53.740 align:middle line:90% 01:14:53.740 --> 01:14:55.390 align:middle line:90% So this is a contradiction. 01:14:55.390 --> 01:15:00.370 align:middle line:90% 01:15:00.370 --> 01:15:03.655 align:middle line:84% Thus our assumption that x squared 01:15:03.655 --> 01:15:05.530 align:middle line:90% is bigger than or equal to 2-- 01:15:05.530 --> 01:15:09.940 align:middle line:84% I mean less than 2, which is different from our assumptions 01:15:09.940 --> 01:15:11.860 align:middle line:90% of the theorem, is false. 01:15:11.860 --> 01:15:58.020 align:middle line:90% 01:15:58.020 --> 01:16:02.130 align:middle line:84% So now, we want to prove that x squared equals 2. 01:16:02.130 --> 01:16:12.630 align:middle line:84% So we now show x squared equals 2. 01:16:12.630 --> 01:16:16.240 align:middle line:90% 01:16:16.240 --> 01:16:19.878 align:middle line:84% Since x squared-- so this is not exactly the proof. 01:16:19.878 --> 01:16:21.920 align:middle line:84% Just I'm going to rewrite it a little differently 01:16:21.920 --> 01:16:30.360 align:middle line:84% so it's maybe clear that instead of showing also x squared is 01:16:30.360 --> 01:16:33.930 align:middle line:84% less than or equal to 2, which we know the less than cannot 01:16:33.930 --> 01:16:37.360 align:middle line:84% happen, let's just show that x squared cannot be bigger than 01:16:37.360 --> 01:16:37.860 align:middle line:90% 2. 01:16:37.860 --> 01:16:45.935 align:middle line:90% 01:16:45.935 --> 01:16:47.810 align:middle line:84% Since x squared is bigger than or equal to 2, 01:16:47.810 --> 01:16:50.240 align:middle line:84% this means either x squared equals 2, 01:16:50.240 --> 01:16:54.140 align:middle line:84% which we want to show, or x squared is bigger than 2. 01:16:54.140 --> 01:16:59.330 align:middle line:90% 01:16:59.330 --> 01:17:01.625 align:middle line:90% So let's rule out this case. 01:17:01.625 --> 01:17:11.786 align:middle line:90% 01:17:11.786 --> 01:17:15.670 align:middle line:84% We now show that the case x squared bigger than 2 cannot 01:17:15.670 --> 01:17:16.870 align:middle line:90% hold. 01:17:16.870 --> 01:17:18.040 align:middle line:90% So suppose otherwise. 01:17:18.040 --> 01:17:19.660 align:middle line:84% We're going to do this-- this is going 01:17:19.660 --> 01:17:23.350 align:middle line:84% to be a proof by contradiction, as well. 01:17:23.350 --> 01:17:26.170 align:middle line:84% I guess when I say x squared bigger than 2 cannot hold, 01:17:26.170 --> 01:17:29.530 align:middle line:84% I'm also saying x squared must be less than or equal to 2. 01:17:29.530 --> 01:17:33.640 align:middle line:84% But anyway, so let's show this cannot hold. 01:17:33.640 --> 01:17:36.280 align:middle line:84% So we're going to do this by contradiction, as well. 01:17:36.280 --> 01:17:40.300 align:middle line:90% 01:17:40.300 --> 01:17:42.940 align:middle line:84% So we're trying to show this cannot hold. 01:17:42.940 --> 01:17:45.870 align:middle line:84% So let's assume that it does hold. 01:17:45.870 --> 01:17:50.550 align:middle line:90% 01:17:50.550 --> 01:17:53.130 align:middle line:84% So that's the negation of what we 01:17:53.130 --> 01:17:55.080 align:middle line:84% want to show, which is this statement, 01:17:55.080 --> 01:18:00.000 align:middle line:84% that x squared greater than 2 does not hold. 01:18:00.000 --> 01:18:01.700 align:middle line:84% So assume x squared is bigger than 2. 01:18:01.700 --> 01:18:04.043 align:middle line:90% 01:18:04.043 --> 01:18:05.460 align:middle line:84% So what we're going to do is we're 01:18:05.460 --> 01:18:09.390 align:middle line:84% going to find an upper bound for the set 01:18:09.390 --> 01:18:11.235 align:middle line:84% E which is strictly smaller than x. 01:18:11.235 --> 01:18:14.670 align:middle line:90% 01:18:14.670 --> 01:18:18.760 align:middle line:84% And we have to do that next time because I think 01:18:18.760 --> 01:18:22.000 align:middle line:90% I'm about to run out of time. 01:18:22.000 --> 01:18:27.160 align:middle line:84% So I hate to stop the proof here, 01:18:27.160 --> 01:18:30.980 align:middle line:84% but we'll finish this in the next lecture. 01:18:30.980 --> 01:18:39.480 align:middle line:90%