WEBVTT 00:00:00.000 --> 00:00:01.455 align:middle line:90% [SQUEAKING] 00:00:01.455 --> 00:00:03.395 align:middle line:90% [RUSTLING] 00:00:03.395 --> 00:00:04.365 align:middle line:90% [CLICKING] 00:00:04.365 --> 00:00:09.627 align:middle line:90% 00:00:09.627 --> 00:00:10.460 align:middle line:90% ZACHARY ABEL: Hello. 00:00:10.460 --> 00:00:11.220 align:middle line:90% Good morning. 00:00:11.220 --> 00:00:12.120 align:middle line:90% Good afternoon. 00:00:12.120 --> 00:00:13.560 align:middle line:90% Hope everyone's doing well. 00:00:13.560 --> 00:00:17.180 align:middle line:90% Welcome back to 6.1200. 00:00:17.180 --> 00:00:20.790 align:middle line:84% Today, we are going to be talking more about proofs. 00:00:20.790 --> 00:00:23.630 align:middle line:84% We spent all of last lecture going through 2/3 00:00:23.630 --> 00:00:27.080 align:middle line:84% of our definition of a mathematical proof. 00:00:27.080 --> 00:00:30.050 align:middle line:84% And the last term we haven't talked about yet 00:00:30.050 --> 00:00:31.566 align:middle line:90% is logical deductions. 00:00:31.566 --> 00:00:38.120 align:middle line:90% 00:00:38.120 --> 00:00:40.130 align:middle line:84% So we said that mathematical proofs 00:00:40.130 --> 00:00:43.665 align:middle line:84% are built from our verifications of propositions. 00:00:43.665 --> 00:00:45.290 align:middle line:84% We talked a lot about what propositions 00:00:45.290 --> 00:00:49.640 align:middle line:84% are using a sequence of logical deductions 00:00:49.640 --> 00:00:51.780 align:middle line:84% starting from a base set of axioms. 00:00:51.780 --> 00:00:53.400 align:middle line:84% And we talked about what axioms are. 00:00:53.400 --> 00:00:56.310 align:middle line:84% So finally, we get to talk about logical deductions. 00:00:56.310 --> 00:01:00.240 align:middle line:84% This is going to be the heart of how we build and construct 00:01:00.240 --> 00:01:02.190 align:middle line:90% and check our proofs. 00:01:02.190 --> 00:01:04.230 align:middle line:84% So this is where most of the proofing 00:01:04.230 --> 00:01:06.900 align:middle line:90% happens inside of our proofs. 00:01:06.900 --> 00:01:08.420 align:middle line:90% So what is a logical deduction? 00:01:08.420 --> 00:01:08.920 align:middle line:90% Let's see. 00:01:08.920 --> 00:01:10.205 align:middle line:90% So an inference rule-- 00:01:10.205 --> 00:01:16.530 align:middle line:90% 00:01:16.530 --> 00:01:26.610 align:middle line:84% let's see-- is a rule for combining 00:01:26.610 --> 00:01:33.916 align:middle line:84% true propositions to form other true propositions. 00:01:33.916 --> 00:01:43.650 align:middle line:90% 00:01:43.650 --> 00:01:46.290 align:middle line:84% For example, one of the classic ones, 00:01:46.290 --> 00:01:50.533 align:middle line:84% and one we've talked about briefly, is called modus ponens. 00:01:50.533 --> 00:01:54.210 align:middle line:90% 00:01:54.210 --> 00:01:56.370 align:middle line:84% And I'm writing this just to say let's not 00:01:56.370 --> 00:01:58.750 align:middle line:90% ever use those words again. 00:01:58.750 --> 00:02:02.450 align:middle line:90% The concept is intuitive enough. 00:02:02.450 --> 00:02:10.870 align:middle line:84% If we know P and we know P implies Q, if we know these, 00:02:10.870 --> 00:02:14.000 align:middle line:90% then we know Q. 00:02:14.000 --> 00:02:14.500 align:middle line:90% Yeah. 00:02:14.500 --> 00:02:17.120 align:middle line:84% If we know P, and we know P implies Q, 00:02:17.120 --> 00:02:21.590 align:middle line:84% then we know Q. If I think, and thinking implies amming, 00:02:21.590 --> 00:02:23.710 align:middle line:90% then I am. 00:02:23.710 --> 00:02:24.650 align:middle line:90% Very intuitive. 00:02:24.650 --> 00:02:26.900 align:middle line:84% Let's again never mention this by name. 00:02:26.900 --> 00:02:31.540 align:middle line:84% That's way too precise for our purposes. 00:02:31.540 --> 00:02:33.550 align:middle line:84% There are some other famous ones. 00:02:33.550 --> 00:02:40.090 align:middle line:84% Like if we know P implies Q, and we know not Q-- 00:02:40.090 --> 00:02:45.910 align:middle line:84% oh, sorry-- not Q. Then all of this implies, all of that 00:02:45.910 --> 00:02:52.610 align:middle line:84% implies not P. This is the opposite direction. 00:02:52.610 --> 00:02:54.800 align:middle line:84% If we know P implies Q, but Q is false, 00:02:54.800 --> 00:02:57.410 align:middle line:90% then P better be false as well. 00:02:57.410 --> 00:02:59.600 align:middle line:90% Here's another fun one. 00:02:59.600 --> 00:03:06.770 align:middle line:84% If we know that not P implies false, 00:03:06.770 --> 00:03:12.630 align:middle line:84% then that implies P. Basically, saying if P isn't false, 00:03:12.630 --> 00:03:15.350 align:middle line:90% then it's true. 00:03:15.350 --> 00:03:17.220 align:middle line:90% Hopefully, intuitive. 00:03:17.220 --> 00:03:20.540 align:middle line:84% But we have the tools to not just rely on intuition here. 00:03:20.540 --> 00:03:23.360 align:middle line:84% We can verify these statements are 00:03:23.360 --> 00:03:26.360 align:middle line:84% actual inference rules, for example, 00:03:26.360 --> 00:03:28.490 align:middle line:90% by drawing a truth table. 00:03:28.490 --> 00:03:31.460 align:middle line:84% Let's do that last one just because it's smaller. 00:03:31.460 --> 00:03:32.100 align:middle line:90% Let's see. 00:03:32.100 --> 00:03:37.580 align:middle line:84% So we have P, and it has two possible values. 00:03:37.580 --> 00:03:39.390 align:middle line:84% And then we have our big old formula. 00:03:39.390 --> 00:03:43.280 align:middle line:90% So not P implies false. 00:03:43.280 --> 00:03:47.690 align:middle line:84% Parens imply P. I want to make sure 00:03:47.690 --> 00:03:49.940 align:middle line:84% that this formula on the right is always true 00:03:49.940 --> 00:03:53.700 align:middle line:90% no matter what P is. 00:03:53.700 --> 00:03:58.500 align:middle line:84% Well, a useful way to fill out these truth tables 00:03:58.500 --> 00:04:01.170 align:middle line:90% is to go one step at a time. 00:04:01.170 --> 00:04:05.100 align:middle line:84% And I like to do that by drawing directly under the symbol 00:04:05.100 --> 00:04:06.100 align:middle line:90% I'm working with. 00:04:06.100 --> 00:04:07.560 align:middle line:90% So like not P-- 00:04:07.560 --> 00:04:09.880 align:middle line:90% when P is true, not P is false. 00:04:09.880 --> 00:04:12.220 align:middle line:84% And when P is false, not P is true. 00:04:12.220 --> 00:04:14.320 align:middle line:84% And I put that right under the P bar, 00:04:14.320 --> 00:04:16.529 align:middle line:84% so I know I'm talking about the P bar. 00:04:16.529 --> 00:04:19.170 align:middle line:84% Now let's talk about this implies. 00:04:19.170 --> 00:04:22.380 align:middle line:90% Does false imply false? 00:04:22.380 --> 00:04:22.890 align:middle line:90% Yeah. 00:04:22.890 --> 00:04:24.100 align:middle line:90% Yeah, that's true. 00:04:24.100 --> 00:04:26.970 align:middle line:90% Does true imply false? 00:04:26.970 --> 00:04:30.160 align:middle line:84% No, that's the case we need to avoid for implication 00:04:30.160 --> 00:04:34.740 align:middle line:84% So finally, this implies is really 00:04:34.740 --> 00:04:38.190 align:middle line:84% this whole parenthetical clause there. 00:04:38.190 --> 00:04:41.680 align:middle line:90% Does true imply true? 00:04:41.680 --> 00:04:44.210 align:middle line:84% Because remember, this is true and false here. 00:04:44.210 --> 00:04:46.470 align:middle line:90% Does true imply true? 00:04:46.470 --> 00:04:47.010 align:middle line:90% Yeah. 00:04:47.010 --> 00:04:49.390 align:middle line:84% So this implies in this case is true. 00:04:49.390 --> 00:04:52.490 align:middle line:90% Does false imply false? 00:04:52.490 --> 00:04:53.420 align:middle line:90% Also yes. 00:04:53.420 --> 00:04:54.270 align:middle line:90% OK. 00:04:54.270 --> 00:04:57.650 align:middle line:84% So the column that corresponds to the whole formula 00:04:57.650 --> 00:04:59.960 align:middle line:90% is always true no matter what. 00:04:59.960 --> 00:05:03.020 align:middle line:84% And that is how we verify that this statement right here 00:05:03.020 --> 00:05:04.260 align:middle line:90% is always true. 00:05:04.260 --> 00:05:07.220 align:middle line:84% This is one possible way to verify that. 00:05:07.220 --> 00:05:10.657 align:middle line:84% These other inference rules, and lots of other ones 00:05:10.657 --> 00:05:12.240 align:middle line:84% that we're going to use in this class, 00:05:12.240 --> 00:05:15.470 align:middle line:84% you can also verify this way if you wish. 00:05:15.470 --> 00:05:18.930 align:middle line:84% Please don't do this in the middle of a proof. 00:05:18.930 --> 00:05:23.300 align:middle line:84% Let's just use these rules as rules that we're familiar with. 00:05:23.300 --> 00:05:25.050 align:middle line:84% You don't have to reprove them every time. 00:05:25.050 --> 00:05:26.930 align:middle line:84% I'm just showing you that we have the tools to do it 00:05:26.930 --> 00:05:27.835 align:middle line:90% if we wanted to. 00:05:27.835 --> 00:05:31.400 align:middle line:90% 00:05:31.400 --> 00:05:34.700 align:middle line:84% In general, when writing proofs in this class, 00:05:34.700 --> 00:05:38.280 align:middle line:84% we're going to want each step to be clear and logical. 00:05:38.280 --> 00:05:44.450 align:middle line:84% We're going to want to use a valid inference rule. 00:05:44.450 --> 00:05:46.580 align:middle line:90% But we just want to use it. 00:05:46.580 --> 00:05:49.320 align:middle line:84% It should be clear from your writing what rule is being used, 00:05:49.320 --> 00:05:51.230 align:middle line:84% but you don't have to call it out every time. 00:05:51.230 --> 00:05:52.970 align:middle line:84% Like, modus ponens, modus tollens. 00:05:52.970 --> 00:05:56.110 align:middle line:84% I didn't even tell you what that second one means. 00:05:56.110 --> 00:05:57.440 align:middle line:90% Just go ahead and use it. 00:05:57.440 --> 00:06:01.090 align:middle line:84% We're going to have a familiar set of tools and deductions 00:06:01.090 --> 00:06:02.240 align:middle line:90% when writing these proofs. 00:06:02.240 --> 00:06:05.740 align:middle line:84% So just-- if you use those or you're doing something else that 00:06:05.740 --> 00:06:08.120 align:middle line:90% is clear enough, use that. 00:06:08.120 --> 00:06:12.160 align:middle line:84% And if it's less clear, explain it more. 00:06:12.160 --> 00:06:14.690 align:middle line:84% Of course, you shall be explaining your steps. 00:06:14.690 --> 00:06:18.170 align:middle line:84% Usually, you're relying on facts that we proved before, 00:06:18.170 --> 00:06:21.350 align:middle line:84% or axioms that maybe we haven't explicitly written down yet. 00:06:21.350 --> 00:06:22.900 align:middle line:84% And you should be identifying those. 00:06:22.900 --> 00:06:24.440 align:middle line:84% When you're relying on something, 00:06:24.440 --> 00:06:25.785 align:middle line:90% tell us what you're relying on. 00:06:25.785 --> 00:06:28.900 align:middle line:90% 00:06:28.900 --> 00:06:30.020 align:middle line:90% Let's see. 00:06:30.020 --> 00:06:32.990 align:middle line:84% Please don't use what's called proof by intimidation. 00:06:32.990 --> 00:06:34.520 align:middle line:90% Clearly, this is true. 00:06:34.520 --> 00:06:35.540 align:middle line:90% This is obvious. 00:06:35.540 --> 00:06:37.240 align:middle line:90% This is intuitive. 00:06:37.240 --> 00:06:39.580 align:middle line:84% In fact, I would strongly recommend 00:06:39.580 --> 00:06:42.250 align:middle line:84% striking those words from your mathematical vocabulary 00:06:42.250 --> 00:06:43.870 align:middle line:90% entirely. 00:06:43.870 --> 00:06:46.762 align:middle line:84% Anything that is clear and obvious to you, 00:06:46.762 --> 00:06:48.220 align:middle line:84% well, first of all, it might not be 00:06:48.220 --> 00:06:51.380 align:middle line:84% clear and obvious to your readers who 00:06:51.380 --> 00:06:54.980 align:middle line:84% might be your graders, who might be your peers. 00:06:54.980 --> 00:06:57.260 align:middle line:84% And that can lose you points because you 00:06:57.260 --> 00:06:58.680 align:middle line:90% didn't explain yourself fully. 00:06:58.680 --> 00:07:01.410 align:middle line:84% That can be demotivating for whoever's reading it. 00:07:01.410 --> 00:07:03.480 align:middle line:84% Like, wait, this isn't obvious to me. 00:07:03.480 --> 00:07:06.530 align:middle line:84% Am I not as smart as I thought I was? 00:07:06.530 --> 00:07:08.960 align:middle line:84% And, of course, it runs the risk that the thing 00:07:08.960 --> 00:07:12.620 align:middle line:90% you said is obvious is wrong. 00:07:12.620 --> 00:07:16.340 align:middle line:84% In fact, this is a common source of mistakes because instead 00:07:16.340 --> 00:07:19.310 align:middle line:84% of proving it, you just glossed over it. 00:07:19.310 --> 00:07:21.920 align:middle line:84% The proof is there to help you check that things are correct 00:07:21.920 --> 00:07:23.315 align:middle line:90% and help you catch mistakes. 00:07:23.315 --> 00:07:24.690 align:middle line:84% And if you're skipping the proof, 00:07:24.690 --> 00:07:26.780 align:middle line:84% then you're skipping that redundancy. 00:07:26.780 --> 00:07:28.800 align:middle line:90% So things are never obvious. 00:07:28.800 --> 00:07:29.940 align:middle line:90% Things are never clear. 00:07:29.940 --> 00:07:32.270 align:middle line:90% Please explain it. 00:07:32.270 --> 00:07:34.910 align:middle line:90% Doo, doo, doo, doo, doo. 00:07:34.910 --> 00:07:37.430 align:middle line:84% I mentioned last time that we're going 00:07:37.430 --> 00:07:39.320 align:middle line:84% to take basically all of high school math 00:07:39.320 --> 00:07:42.050 align:middle line:90% as our common set of axioms. 00:07:42.050 --> 00:07:47.100 align:middle line:84% The one exception to that is if we ask you to prove a theorem, 00:07:47.100 --> 00:07:49.530 align:middle line:84% and you say, I know this from high school math. 00:07:49.530 --> 00:07:50.220 align:middle line:90% It's an axiom. 00:07:50.220 --> 00:07:51.170 align:middle line:90% I'm done. 00:07:51.170 --> 00:07:56.670 align:middle line:84% That's maybe not adhering to the spirit of the assignment. 00:07:56.670 --> 00:07:59.300 align:middle line:84% If we ask you to prove a statement that you already know 00:07:59.300 --> 00:08:02.720 align:middle line:90% is true, please go and prove it. 00:08:02.720 --> 00:08:06.650 align:middle line:84% We're asking you to think more deeply about why it's true. 00:08:06.650 --> 00:08:08.120 align:middle line:90% Does that make sense? 00:08:08.120 --> 00:08:09.410 align:middle line:90% Cool. 00:08:09.410 --> 00:08:10.440 align:middle line:90% All right. 00:08:10.440 --> 00:08:14.910 align:middle line:84% So using these logical deductions and things like them, 00:08:14.910 --> 00:08:18.756 align:middle line:84% I want to talk about a couple very common proof outlines-- 00:08:18.756 --> 00:08:21.470 align:middle line:90% 00:08:21.470 --> 00:08:24.290 align:middle line:84% proof outlines-- oop, that's not a Q, that's 00:08:24.290 --> 00:08:31.010 align:middle line:84% a U-- outlines for proving some of the most common shapes 00:08:31.010 --> 00:08:33.380 align:middle line:90% that our theorems might take. 00:08:33.380 --> 00:08:38.120 align:middle line:84% For example, if we have a theorem, 00:08:38.120 --> 00:08:45.240 align:middle line:84% there exists a natural number N, such that n is at least 10, 00:08:45.240 --> 00:08:47.525 align:middle line:90% and n is prime. 00:08:47.525 --> 00:08:51.900 align:middle line:90% 00:08:51.900 --> 00:08:53.080 align:middle line:90% Here's our theorem. 00:08:53.080 --> 00:08:56.560 align:middle line:84% There exists a number that's at least 10 and is prime. 00:08:56.560 --> 00:09:00.870 align:middle line:84% There exists a prime bigger than 9. 00:09:00.870 --> 00:09:02.270 align:middle line:90% Can someone prove this for me? 00:09:02.270 --> 00:09:08.280 align:middle line:90% 00:09:08.280 --> 00:09:09.160 align:middle line:90% Yes? 00:09:09.160 --> 00:09:10.080 align:middle line:90% AUDIENCE: n is 17. 00:09:10.080 --> 00:09:10.975 align:middle line:90% ZACHARY ABEL: 17. 00:09:10.975 --> 00:09:12.850 align:middle line:84% That's actually the same example in my notes. 00:09:12.850 --> 00:09:14.100 align:middle line:90% Well done. 00:09:14.100 --> 00:09:18.040 align:middle line:90% Proof-- n equals 17. 00:09:18.040 --> 00:09:20.490 align:middle line:90% Is that a full proof? 00:09:20.490 --> 00:09:21.700 align:middle line:90% I mean, kind of. 00:09:21.700 --> 00:09:23.120 align:middle line:90% Yes? 00:09:23.120 --> 00:09:24.300 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:09:24.300 --> 00:09:29.000 align:middle line:90% 00:09:29.000 --> 00:09:30.500 align:middle line:84% ZACHARY ABEL: If we-- one more time. 00:09:30.500 --> 00:09:31.680 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:09:31.680 --> 00:09:34.152 align:middle line:90% 00:09:34.152 --> 00:09:34.860 align:middle line:90% ZACHARY ABEL: OK. 00:09:34.860 --> 00:09:36.810 align:middle line:84% So the suggestion was we should be 00:09:36.810 --> 00:09:39.450 align:middle line:84% going through all of the possible factors of 17 00:09:39.450 --> 00:09:40.830 align:middle line:84% and show that they're not factors 00:09:40.830 --> 00:09:43.300 align:middle line:90% to verify that 17 is prime. 00:09:43.300 --> 00:09:44.440 align:middle line:90% Absolutely. 00:09:44.440 --> 00:09:46.960 align:middle line:84% That is something we haven't done. 00:09:46.960 --> 00:09:50.000 align:middle line:84% Also, we haven't checked that explicitly. 00:09:50.000 --> 00:09:52.120 align:middle line:90% We probably want to do that. 00:09:52.120 --> 00:09:55.780 align:middle line:84% But let me back up, and let me write down 00:09:55.780 --> 00:09:59.020 align:middle line:84% what the general proof usually looks like. 00:09:59.020 --> 00:10:00.550 align:middle line:90% Let's see-- proof. 00:10:00.550 --> 00:10:01.050 align:middle line:90% Oops. 00:10:01.050 --> 00:10:03.880 align:middle line:90% 00:10:03.880 --> 00:10:08.850 align:middle line:84% We'll show that n equals fill in the blank works. 00:10:08.850 --> 00:10:12.790 align:middle line:90% 00:10:12.790 --> 00:10:17.570 align:middle line:84% This n is in the set we asked for. 00:10:17.570 --> 00:10:18.860 align:middle line:90% It's in the natural numbers. 00:10:18.860 --> 00:10:23.080 align:middle line:84% In this case, we're going to secretly be using 17. 00:10:23.080 --> 00:10:24.410 align:middle line:90% So it's pretty clear. 00:10:24.410 --> 00:10:26.750 align:middle line:84% But in general, it might be less clear. 00:10:26.750 --> 00:10:29.173 align:middle line:90% So for reasons. 00:10:29.173 --> 00:10:34.240 align:middle line:90% 00:10:34.240 --> 00:10:41.583 align:middle line:90% And n is prime because reasons. 00:10:41.583 --> 00:10:44.930 align:middle line:90% 00:10:44.930 --> 00:10:47.180 align:middle line:84% So this is in general, how will we 00:10:47.180 --> 00:10:50.100 align:middle line:84% recommend writing a proof of an exists? 00:10:50.100 --> 00:10:54.060 align:middle line:84% If you want to show something exists in a set with a property, 00:10:54.060 --> 00:10:55.457 align:middle line:90% you tell us what it is. 00:10:55.457 --> 00:10:56.790 align:middle line:90% You show us why it's in the set. 00:10:56.790 --> 00:10:59.990 align:middle line:84% You show us why it has the property. 00:10:59.990 --> 00:11:01.680 align:middle line:90% Pretty intuitive, right? 00:11:01.680 --> 00:11:02.720 align:middle line:90% Hopefully. 00:11:02.720 --> 00:11:04.340 align:middle line:84% I know I just used that word that I 00:11:04.340 --> 00:11:05.935 align:middle line:90% said we aren't supposed to use. 00:11:05.935 --> 00:11:08.630 align:middle line:90% 00:11:08.630 --> 00:11:12.500 align:middle line:84% But this is inextricably tied with what 00:11:12.500 --> 00:11:13.920 align:middle line:90% this symbol means in general. 00:11:13.920 --> 00:11:16.070 align:middle line:90% There exists a natural number. 00:11:16.070 --> 00:11:18.940 align:middle line:84% To prove that there exists a number with this property, 00:11:18.940 --> 00:11:20.940 align:middle line:84% you have to show me a number with this property. 00:11:20.940 --> 00:11:23.180 align:middle line:84% That's the most straightforward way to do it. 00:11:23.180 --> 00:11:26.600 align:middle line:84% And that's what this proof outline is letting us do. 00:11:26.600 --> 00:11:28.790 align:middle line:84% In this particular problem, we're going to say, 00:11:28.790 --> 00:11:31.410 align:middle line:90% we'll show n equals 17 works. 00:11:31.410 --> 00:11:35.900 align:middle line:84% This number, 17, is a natural number because well, yeah-- 00:11:35.900 --> 00:11:38.180 align:middle line:84% I don't know how to convince you of that 00:11:38.180 --> 00:11:40.560 align:middle line:90% if you're not sure of that. 00:11:40.560 --> 00:11:43.650 align:middle line:84% And n equals 17 is prime because-- 00:11:43.650 --> 00:11:46.830 align:middle line:84% and this is where we should, if we want to be super careful, 00:11:46.830 --> 00:11:49.290 align:middle line:90% check all the possible factors. 00:11:49.290 --> 00:11:52.500 align:middle line:84% It's not divisible by 2, by 3, by 4, et cetera. 00:11:52.500 --> 00:11:53.460 align:middle line:90% Eric? 00:11:53.460 --> 00:11:55.585 align:middle line:84% AUDIENCE: Are you missing n greater or equal to 10? 00:11:55.585 --> 00:11:58.320 align:middle line:84% ZACHARY ABEL: I am missing n greater or equal to 10. 00:11:58.320 --> 00:12:02.160 align:middle line:84% Yeah, we need to show that n satisfies this whole property, 00:12:02.160 --> 00:12:14.063 align:middle line:84% and n is prime and greater or equal to 10 because reasons. 00:12:14.063 --> 00:12:22.110 align:middle line:90% 00:12:22.110 --> 00:12:22.990 align:middle line:90% All right. 00:12:22.990 --> 00:12:25.530 align:middle line:84% So more abstractly, if our theorem 00:12:25.530 --> 00:12:32.920 align:middle line:84% is there exists x in some set such that P of x holds, 00:12:32.920 --> 00:12:39.080 align:middle line:84% then our proof can often look like choose X 00:12:39.080 --> 00:12:44.780 align:middle line:84% equals some specific value that you have to put in. 00:12:44.780 --> 00:12:52.130 align:middle line:84% Then x is in S because big to-do there. 00:12:52.130 --> 00:12:55.820 align:middle line:90% These are to-dos. 00:12:55.820 --> 00:13:02.690 align:middle line:84% And P of x is true because to-do. 00:13:02.690 --> 00:13:05.480 align:middle line:90% 00:13:05.480 --> 00:13:07.250 align:middle line:90% All right. 00:13:07.250 --> 00:13:13.673 align:middle line:84% Similarly, what if we're asked to prove a for all theorem? 00:13:13.673 --> 00:13:17.450 align:middle line:90% 00:13:17.450 --> 00:13:21.510 align:middle line:84% For all numbers, x in the reals-- 00:13:21.510 --> 00:13:23.780 align:middle line:90% so for all real numbers x. 00:13:23.780 --> 00:13:31.550 align:middle line:84% x squared minus 6x is greater than negative 10. 00:13:31.550 --> 00:13:35.390 align:middle line:84% So this theorem is about a for all, not in exists. 00:13:35.390 --> 00:13:38.620 align:middle line:84% When we're asking about exist, we need just a single example. 00:13:38.620 --> 00:13:42.490 align:middle line:84% But when we're asking about for all, examples aren't enough. 00:13:42.490 --> 00:13:44.280 align:middle line:84% Remember, last time when we talked about n 00:13:44.280 --> 00:13:47.620 align:middle line:84% squared plus n plus 41 being prime lots of times. 00:13:47.620 --> 00:13:49.950 align:middle line:90% Well, it's not prime every time. 00:13:49.950 --> 00:13:51.520 align:middle line:84% If we're trying to prove it for all, 00:13:51.520 --> 00:13:53.430 align:middle line:84% we need to show that the property is 00:13:53.430 --> 00:13:57.330 align:middle line:84% true for every possible choice of x. 00:13:57.330 --> 00:14:00.790 align:middle line:84% And, in general, let's see what this outline might look like. 00:14:00.790 --> 00:14:08.160 align:middle line:84% So if our theorem is for all x in some set, P of x is true, 00:14:08.160 --> 00:14:10.270 align:middle line:84% our proof usually looks like this. 00:14:10.270 --> 00:14:24.750 align:middle line:84% Proof-- suppose x is a generic element of S. Then 00:14:24.750 --> 00:14:29.506 align:middle line:90% P of x is true because reasons. 00:14:29.506 --> 00:14:34.618 align:middle line:90% 00:14:34.618 --> 00:14:37.160 align:middle line:84% I think it makes sense to put little question marks in there. 00:14:37.160 --> 00:14:40.070 align:middle line:84% So we know that we have to figure out what's going on. 00:14:40.070 --> 00:14:41.350 align:middle line:90% There we go. 00:14:41.350 --> 00:14:45.100 align:middle line:84% This first line-- suppose x is some element of S. Which 00:14:45.100 --> 00:14:45.640 align:middle line:90% element? 00:14:45.640 --> 00:14:46.880 align:middle line:90% We're not allowed to know. 00:14:46.880 --> 00:14:48.860 align:middle line:84% We're not allowed to choose a particular one. 00:14:48.860 --> 00:14:51.910 align:middle line:84% We need to prove that this is true for every possible choice 00:14:51.910 --> 00:14:55.510 align:middle line:84% of x, which means in our proof, the only thing we assume 00:14:55.510 --> 00:14:58.150 align:middle line:90% about x is that it's in S. 00:14:58.150 --> 00:15:01.000 align:middle line:84% So you start by taking some thing in S 00:15:01.000 --> 00:15:03.260 align:middle line:84% and assuming nothing else about it, 00:15:03.260 --> 00:15:05.740 align:middle line:84% and then going and concluding the property 00:15:05.740 --> 00:15:07.750 align:middle line:90% that we need to show. 00:15:07.750 --> 00:15:14.980 align:middle line:84% Again, these might look very similar to you, and they are. 00:15:14.980 --> 00:15:17.510 align:middle line:84% This says everything in S has this thing. 00:15:17.510 --> 00:15:21.670 align:middle line:84% This says a generic thing in S has this property. 00:15:21.670 --> 00:15:25.480 align:middle line:84% The difference is that suppose x is a generic element of S, 00:15:25.480 --> 00:15:29.150 align:middle line:84% or however you want to phrase that, makes it concrete. 00:15:29.150 --> 00:15:32.840 align:middle line:84% It gives us a particular x to talk about. 00:15:32.840 --> 00:15:35.930 align:middle line:84% So now we have something to use and talk about and reason 00:15:35.930 --> 00:15:37.800 align:middle line:90% about inside of our proof. 00:15:37.800 --> 00:15:42.030 align:middle line:84% Whereas, up here, how do we approach every-- 00:15:42.030 --> 00:15:43.290 align:middle line:90% that's a little daunting. 00:15:43.290 --> 00:15:45.300 align:middle line:84% I only know how to deal with one. 00:15:45.300 --> 00:15:46.800 align:middle line:90% So this gives us one. 00:15:46.800 --> 00:15:47.790 align:middle line:90% It's called x. 00:15:47.790 --> 00:15:50.070 align:middle line:84% We know nothing about it except that it's in S, 00:15:50.070 --> 00:15:53.150 align:middle line:84% but at least we have the one, and we can work with it. 00:15:53.150 --> 00:15:56.730 align:middle line:84% So whenever you see a theorem that starts with a for all, 00:15:56.730 --> 00:15:59.400 align:middle line:84% your first line is almost always going to be this. 00:15:59.400 --> 00:16:02.420 align:middle line:84% In order to bring some thing into scope 00:16:02.420 --> 00:16:03.920 align:middle line:90% that we can talk about. 00:16:03.920 --> 00:16:05.600 align:middle line:90% Sound good? 00:16:05.600 --> 00:16:06.320 align:middle line:90% Awesome. 00:16:06.320 --> 00:16:07.780 align:middle line:90% So how do we prove this theorem? 00:16:07.780 --> 00:16:11.090 align:middle line:90% 00:16:11.090 --> 00:16:13.310 align:middle line:90% Proof-- let's see. 00:16:13.310 --> 00:16:16.250 align:middle line:90% Let's see. 00:16:16.250 --> 00:16:19.435 align:middle line:84% Suppose x is an element of the real numbers. 00:16:19.435 --> 00:16:22.370 align:middle line:90% 00:16:22.370 --> 00:16:29.390 align:middle line:84% Then, let's see. x minus 3 squared is greater or equal to 0 00:16:29.390 --> 00:16:36.183 align:middle line:84% because all reals have non-negative squares. 00:16:36.183 --> 00:16:40.290 align:middle line:90% 00:16:40.290 --> 00:16:42.670 align:middle line:84% This can be a fact we're citing from high school. 00:16:42.670 --> 00:16:46.330 align:middle line:84% We know how the real numbers multiply to each other. 00:16:46.330 --> 00:16:48.390 align:middle line:90% And once we know this-- 00:16:48.390 --> 00:16:56.490 align:middle line:84% let's see-- equivalently, this thing says that x squared minus 00:16:56.490 --> 00:17:02.100 align:middle line:84% 6x is greater or equal to minus 9, which, in turn, 00:17:02.100 --> 00:17:05.330 align:middle line:84% is greater or equal to minus 10 as needed. 00:17:05.330 --> 00:17:08.520 align:middle line:90% 00:17:08.520 --> 00:17:12.390 align:middle line:84% So we started with pick some element of R 00:17:12.390 --> 00:17:14.160 align:middle line:90% without specifying which one. 00:17:14.160 --> 00:17:17.589 align:middle line:84% We ended with the property we need is true-- 00:17:17.589 --> 00:17:19.530 align:middle line:90% QED. 00:17:19.530 --> 00:17:20.760 align:middle line:90% Sound good? 00:17:20.760 --> 00:17:26.260 align:middle line:84% QED, by the way, is a common way to mark the end of a proof. 00:17:26.260 --> 00:17:29.250 align:middle line:84% It's Latin for quod erat demonstrandum, 00:17:29.250 --> 00:17:31.150 align:middle line:90% which was to be shown. 00:17:31.150 --> 00:17:35.870 align:middle line:84% So it's a synonym for as needed or as desired. 00:17:35.870 --> 00:17:38.140 align:middle line:90% QED-- all the same thing. 00:17:38.140 --> 00:17:39.670 align:middle line:90% All right. 00:17:39.670 --> 00:17:40.900 align:middle line:90% Question? 00:17:40.900 --> 00:17:43.570 align:middle line:84% AUDIENCE: Could you like show that, like, 00:17:43.570 --> 00:17:46.540 align:middle line:84% for all the negative numbers, [INAUDIBLE] 00:17:46.540 --> 00:17:49.733 align:middle line:84% all the negative numbers will not [INAUDIBLE]? 00:17:49.733 --> 00:17:55.048 align:middle line:90% 00:17:55.048 --> 00:17:55.840 align:middle line:90% ZACHARY ABEL: Sure. 00:17:55.840 --> 00:17:58.090 align:middle line:84% So the question was, could we do this a different way? 00:17:58.090 --> 00:18:00.630 align:middle line:84% Could we look at all the negative numbers, 00:18:00.630 --> 00:18:01.630 align:middle line:90% and they behave one way. 00:18:01.630 --> 00:18:04.205 align:middle line:84% And then look at all the numbers between 0 and 3 00:18:04.205 --> 00:18:06.580 align:middle line:84% or 0 and 6 or something, and they behave a different way. 00:18:06.580 --> 00:18:07.830 align:middle line:90% And then look at all the rest. 00:18:07.830 --> 00:18:10.130 align:middle line:84% Yes, absolutely. we can do something like that. 00:18:10.130 --> 00:18:11.510 align:middle line:90% That's called proof by cases. 00:18:11.510 --> 00:18:15.100 align:middle line:84% We're going to talk about it in next lecture. 00:18:15.100 --> 00:18:19.490 align:middle line:84% But for now, let's move on to the next common outline, 00:18:19.490 --> 00:18:23.620 align:middle line:90% which is proving implications. 00:18:23.620 --> 00:18:30.270 align:middle line:84% If we have a theorem that says P implies Q. I mentioned, 00:18:30.270 --> 00:18:33.110 align:middle line:84% we're going to be using implies all the time. 00:18:33.110 --> 00:18:35.960 align:middle line:84% And often, theorems say, if this, then that. 00:18:35.960 --> 00:18:38.970 align:middle line:84% If P, then Q. How do we prove this? 00:18:38.970 --> 00:18:47.090 align:middle line:84% Our proof often looks like assume P, 00:18:47.090 --> 00:18:52.553 align:middle line:90% then Q is true because reasons. 00:18:52.553 --> 00:18:56.120 align:middle line:90% 00:18:56.120 --> 00:18:58.790 align:middle line:84% Again, looks pretty similar to the kind 00:18:58.790 --> 00:19:00.030 align:middle line:90% of thing we're talking about. 00:19:00.030 --> 00:19:03.740 align:middle line:84% But this first sentence, assume P-- assume P is true-- 00:19:03.740 --> 00:19:05.160 align:middle line:90% makes things concrete. 00:19:05.160 --> 00:19:09.650 align:middle line:84% It instantiates the thing we're supposed to be assuming. 00:19:09.650 --> 00:19:12.000 align:middle line:84% It gives us something concrete to work with. 00:19:12.000 --> 00:19:14.730 align:middle line:84% We can now live in this hypothetical 00:19:14.730 --> 00:19:17.930 align:middle line:84% where we know for certain that P is true, 00:19:17.930 --> 00:19:20.630 align:middle line:84% and then work with it as if it's true because inside 00:19:20.630 --> 00:19:24.500 align:middle line:84% of this proof, for the rest of this proof, it is. 00:19:24.500 --> 00:19:27.008 align:middle line:84% Assume P. Use that to show Q. Whenever 00:19:27.008 --> 00:19:28.550 align:middle line:84% we're trying to prove an implication, 00:19:28.550 --> 00:19:30.370 align:middle line:90% this is a common way to do it. 00:19:30.370 --> 00:19:31.813 align:middle line:90% This is called a direct proof. 00:19:31.813 --> 00:19:39.700 align:middle line:90% 00:19:39.700 --> 00:19:43.600 align:middle line:84% We're going to see indirect proofs a bit later. 00:19:43.600 --> 00:19:46.510 align:middle line:84% But there is another common way-- 00:19:46.510 --> 00:19:47.200 align:middle line:90% actually, sorry. 00:19:47.200 --> 00:19:51.310 align:middle line:84% Before that, let's do an example of proving something 00:19:51.310 --> 00:19:52.640 align:middle line:90% in this fashion. 00:19:52.640 --> 00:20:06.040 align:middle line:84% Theorem-- if n is a multiple of 10, then it is even. 00:20:06.040 --> 00:20:08.950 align:middle line:84% If n is a multiple of 10, then it is even. 00:20:08.950 --> 00:20:14.320 align:middle line:84% We're proving an implication. n multiple of 10 implies even. 00:20:14.320 --> 00:20:15.320 align:middle line:90% P implies Q. 00:20:15.320 --> 00:20:19.383 align:middle line:84% So even without having to think very hard, proof-- 00:20:19.383 --> 00:20:22.600 align:middle line:90% 00:20:22.600 --> 00:20:29.570 align:middle line:90% assume n is a multiple of 10. 00:20:29.570 --> 00:20:31.470 align:middle line:84% We know we're going to start with that. 00:20:31.470 --> 00:20:33.050 align:middle line:90% Assume the first thing. 00:20:33.050 --> 00:20:36.240 align:middle line:84% We know we're going to end with the last thing. 00:20:36.240 --> 00:20:38.540 align:middle line:90% Therefore, n is even. 00:20:38.540 --> 00:20:43.190 align:middle line:84% So question mark, question mark, question mark. 00:20:43.190 --> 00:20:46.130 align:middle line:90% Therefore, n is even. 00:20:46.130 --> 00:20:48.470 align:middle line:84% And if we can connect the dots from the first line 00:20:48.470 --> 00:20:51.350 align:middle line:84% to the last line, we'll have our proof. 00:20:51.350 --> 00:20:54.810 align:middle line:84% Notice, by the way, that was kind of an automatic process. 00:20:54.810 --> 00:20:56.550 align:middle line:84% If the theorem is an implication, 00:20:56.550 --> 00:20:59.280 align:middle line:84% then this is what our proof is commonly going to look like. 00:20:59.280 --> 00:21:00.600 align:middle line:90% So we can just write it down. 00:21:00.600 --> 00:21:02.660 align:middle line:84% And now we have a clearer, more concrete sense 00:21:02.660 --> 00:21:04.710 align:middle line:84% of the task we're supposed to do. 00:21:04.710 --> 00:21:06.350 align:middle line:84% Starting from the fact that n is even, 00:21:06.350 --> 00:21:08.640 align:middle line:84% we're supposed to derive the fact that it's-- sorry. 00:21:08.640 --> 00:21:10.800 align:middle line:84% Starting from the fact that it's a multiple of 10, 00:21:10.800 --> 00:21:12.770 align:middle line:90% derive the fact that it's even. 00:21:12.770 --> 00:21:14.490 align:middle line:90% So let's go ahead and do that. 00:21:14.490 --> 00:21:17.330 align:middle line:90% Assume n is a multiple of 10. 00:21:17.330 --> 00:21:19.940 align:middle line:84% What does it mean to be a multiple of 10? 00:21:19.940 --> 00:21:25.790 align:middle line:84% So n is 10 times k for some integer k. 00:21:25.790 --> 00:21:30.384 align:middle line:90% 00:21:30.384 --> 00:21:37.530 align:middle line:84% But then n equals 2 times 5 times k. 00:21:37.530 --> 00:21:39.400 align:middle line:90% And so it's 2 times an integer. 00:21:39.400 --> 00:21:40.720 align:middle line:90% Therefore, it's even. 00:21:40.720 --> 00:21:41.470 align:middle line:90% There's our proof. 00:21:41.470 --> 00:21:42.720 align:middle line:90% We're done. 00:21:42.720 --> 00:21:44.880 align:middle line:84% Just by identifying where we're supposed to start 00:21:44.880 --> 00:21:48.600 align:middle line:84% and where we're supposed to end, we were able to bridge the gap. 00:21:48.600 --> 00:21:50.250 align:middle line:90% Sound good? 00:21:50.250 --> 00:21:52.020 align:middle line:90% Nice. 00:21:52.020 --> 00:21:56.190 align:middle line:84% So there's another common way of proving implications. 00:21:56.190 --> 00:21:59.481 align:middle line:84% This is called proof by contrapositive. 00:21:59.481 --> 00:22:04.050 align:middle line:90% 00:22:04.050 --> 00:22:07.230 align:middle line:84% Recall last lecture, we said that P-- 00:22:07.230 --> 00:22:14.880 align:middle line:84% oops-- P implies Q is equivalent to its contrapositive. 00:22:14.880 --> 00:22:15.610 align:middle line:90% Let's see. 00:22:15.610 --> 00:22:20.620 align:middle line:84% Not Q implies not P. This right here 00:22:20.620 --> 00:22:23.272 align:middle line:84% is the contrapositive-- contrapositive. 00:22:23.272 --> 00:22:26.710 align:middle line:90% 00:22:26.710 --> 00:22:30.160 align:middle line:84% And so if we're supposed to be proving P implies Q, 00:22:30.160 --> 00:22:33.010 align:middle line:84% sometimes it's more convenient to instead prove 00:22:33.010 --> 00:22:39.610 align:middle line:84% not Q implies not P. And so our proof outline 00:22:39.610 --> 00:22:42.745 align:middle line:84% would say proof by contrapositive. 00:22:42.745 --> 00:22:47.200 align:middle line:90% 00:22:47.200 --> 00:22:52.010 align:middle line:90% So assume Q is false. 00:22:52.010 --> 00:22:54.650 align:middle line:90% Assume not Q. Let's see. 00:22:54.650 --> 00:22:56.400 align:middle line:90% Assume Q is false. 00:22:56.400 --> 00:23:00.940 align:middle line:90% 00:23:00.940 --> 00:23:06.240 align:middle line:84% Then P is false because question mark. 00:23:06.240 --> 00:23:09.790 align:middle line:90% 00:23:09.790 --> 00:23:13.100 align:middle line:84% Same as we did for direct implications, 00:23:13.100 --> 00:23:15.470 align:middle line:84% but we're proving this implication instead. 00:23:15.470 --> 00:23:19.070 align:middle line:84% So assume not Q. Use it to prove not P. 00:23:19.070 --> 00:23:22.040 align:middle line:84% And let's see an example of this. 00:23:22.040 --> 00:23:26.490 align:middle line:90% Theorem-- let's see. 00:23:26.490 --> 00:23:30.260 align:middle line:90% So for all integers n-- 00:23:30.260 --> 00:23:33.530 align:middle line:90% let's see. 00:23:33.530 --> 00:23:41.900 align:middle line:84% If n squared is even, then n is even. 00:23:41.900 --> 00:23:47.360 align:middle line:84% For all natural numbers n, if n squared is even, then n is even. 00:23:47.360 --> 00:23:49.610 align:middle line:84% Now automatically, oh, we're proving it for all. 00:23:49.610 --> 00:23:50.940 align:middle line:90% I know how to write that proof. 00:23:50.940 --> 00:23:57.090 align:middle line:90% Proof-- suppose n is an integer. 00:23:57.090 --> 00:24:00.140 align:middle line:90% Now we have an n to work with. 00:24:00.140 --> 00:24:04.370 align:middle line:84% We need to prove this implication. 00:24:04.370 --> 00:24:05.130 align:middle line:90% Let's see. 00:24:05.130 --> 00:24:09.510 align:middle line:90% So proof by contrapositive. 00:24:09.510 --> 00:24:13.520 align:middle line:90% 00:24:13.520 --> 00:24:16.470 align:middle line:84% It's always nice to say what proof technique you're using, 00:24:16.470 --> 00:24:19.810 align:middle line:84% if it's maybe not as self-evident as it might be. 00:24:19.810 --> 00:24:22.230 align:middle line:84% So now we're indicating we're proving this implication 00:24:22.230 --> 00:24:23.710 align:middle line:90% by contrapositive. 00:24:23.710 --> 00:24:26.490 align:middle line:84% And once we've said that, both you, the writer, 00:24:26.490 --> 00:24:29.340 align:middle line:84% and your audience, the readers, know 00:24:29.340 --> 00:24:31.290 align:middle line:90% what's supposed to come next. 00:24:31.290 --> 00:24:36.780 align:middle line:84% Proof by contrapositive-- so assume this thing on the right 00:24:36.780 --> 00:24:38.200 align:middle line:90% is false. 00:24:38.200 --> 00:24:43.610 align:middle line:84% Assume n is odd because that's the opposite of n being even. 00:24:43.610 --> 00:24:46.200 align:middle line:90% 00:24:46.200 --> 00:24:53.850 align:middle line:84% We want to show n squared is not even. n squared is odd. 00:24:53.850 --> 00:24:57.330 align:middle line:90% This, by the way, WTS-- 00:24:57.330 --> 00:24:59.830 align:middle line:90% Want to Show. 00:24:59.830 --> 00:25:02.770 align:middle line:84% This, I find, is really useful for me when planning out proofs, 00:25:02.770 --> 00:25:06.480 align:middle line:84% thinking about proofs, sometimes even when writing proofs. 00:25:06.480 --> 00:25:10.000 align:middle line:84% This is an indicator that this is where I'm trying to go. 00:25:10.000 --> 00:25:11.620 align:middle line:90% This is the end of our proof. 00:25:11.620 --> 00:25:13.560 align:middle line:90% This is my goal. 00:25:13.560 --> 00:25:17.920 align:middle line:84% And it signals both to you that this is work that's left to do. 00:25:17.920 --> 00:25:21.140 align:middle line:84% It signals to your reader that, OK, great. 00:25:21.140 --> 00:25:23.690 align:middle line:84% We agree that we're working towards the right thing. 00:25:23.690 --> 00:25:25.900 align:middle line:90% So now let's go and do it. 00:25:25.900 --> 00:25:28.540 align:middle line:84% And also, as you're writing scratch work 00:25:28.540 --> 00:25:31.505 align:middle line:84% and solving these proofs, and maybe you get 00:25:31.505 --> 00:25:33.130 align:middle line:84% distracted and have to come back later, 00:25:33.130 --> 00:25:36.350 align:middle line:84% and you see a bunch of statements on your page. 00:25:36.350 --> 00:25:38.230 align:middle line:84% Wait, which of these statements did we prove, 00:25:38.230 --> 00:25:40.460 align:middle line:84% and which are we trying to work towards? 00:25:40.460 --> 00:25:43.490 align:middle line:84% So as I'm working on these proofs, 00:25:43.490 --> 00:25:46.250 align:middle line:84% I really like to be organized and say, we know this. 00:25:46.250 --> 00:25:46.900 align:middle line:90% We know that. 00:25:46.900 --> 00:25:49.330 align:middle line:84% We want to show this other thing. 00:25:49.330 --> 00:25:54.140 align:middle line:84% So I know that we haven't proven it yet, and we want to. 00:25:54.140 --> 00:25:55.670 align:middle line:90% So proof by contrapositive. 00:25:55.670 --> 00:25:56.720 align:middle line:90% So assume n is odd. 00:25:56.720 --> 00:26:01.640 align:middle line:84% We want to show that n squared is odd 00:26:01.640 --> 00:26:04.700 align:middle line:84% because that's what the proof method tells us to do. 00:26:04.700 --> 00:26:06.260 align:middle line:90% And now we can go do it. 00:26:06.260 --> 00:26:08.470 align:middle line:90% So assume n is odd. 00:26:08.470 --> 00:26:11.320 align:middle line:90% So n is 2k plus 1. 00:26:11.320 --> 00:26:18.140 align:middle line:84% That's what it means to be odd for some integer k. 00:26:18.140 --> 00:26:23.850 align:middle line:84% Then n squared is 4k squared plus 4k plus 1, 00:26:23.850 --> 00:26:30.055 align:middle line:84% which is 2 times an integer plus 1, which is odd. 00:26:30.055 --> 00:26:34.550 align:middle line:90% 00:26:34.550 --> 00:26:36.650 align:middle line:90% QED. 00:26:36.650 --> 00:26:39.690 align:middle line:84% And just to be precise about it, 2k-- 00:26:39.690 --> 00:26:42.860 align:middle line:84% oops, sorry-- 2k squared plus 2k. 00:26:42.860 --> 00:26:46.047 align:middle line:84% But 2 times an integer plus 1 is what it means to be odd. 00:26:46.047 --> 00:26:46.630 align:middle line:90% So we're done. 00:26:46.630 --> 00:26:50.120 align:middle line:90% 00:26:50.120 --> 00:26:53.490 align:middle line:90% So we have proven implication. 00:26:53.490 --> 00:26:55.520 align:middle line:84% We've shown two different proof outlines 00:26:55.520 --> 00:26:58.160 align:middle line:84% for how to show P implies Q. One direct-- 00:26:58.160 --> 00:27:03.680 align:middle line:84% you assume P. Use it to conclude Q. One flipped around. 00:27:03.680 --> 00:27:07.160 align:middle line:84% Assume not Q. Use it to derive not P. 00:27:07.160 --> 00:27:10.760 align:middle line:84% There's a related proof technique called proof 00:27:10.760 --> 00:27:11.950 align:middle line:90% by contradiction. 00:27:11.950 --> 00:27:18.182 align:middle line:90% 00:27:18.182 --> 00:27:20.140 align:middle line:84% Now, when growing up, you might have been told, 00:27:20.140 --> 00:27:21.700 align:middle line:90% don't use double negatives. 00:27:21.700 --> 00:27:23.290 align:middle line:90% They're not as clear. 00:27:23.290 --> 00:27:27.040 align:middle line:84% It's always much stronger to more directly say what you mean. 00:27:27.040 --> 00:27:29.880 align:middle line:84% But sometimes in math, a double negative 00:27:29.880 --> 00:27:33.870 align:middle line:84% isn't what we don't want to avoid. 00:27:33.870 --> 00:27:36.100 align:middle line:84% I think there were four negatives in there. 00:27:36.100 --> 00:27:36.700 align:middle line:90% Got it right. 00:27:36.700 --> 00:27:37.650 align:middle line:90% Yeah. 00:27:37.650 --> 00:27:40.440 align:middle line:84% Double negatives can sometimes work to our favor. 00:27:40.440 --> 00:27:44.610 align:middle line:84% And proof by contradiction is exactly that technique 00:27:44.610 --> 00:27:46.300 align:middle line:90% as a proof outline. 00:27:46.300 --> 00:27:48.930 align:middle line:84% So let's talk about what this does-- 00:27:48.930 --> 00:27:50.470 align:middle line:90% proof by contradiction. 00:27:50.470 --> 00:27:56.370 align:middle line:84% So theorem P. Whatever P is, the proof by contradiction 00:27:56.370 --> 00:27:59.490 align:middle line:84% can be used to prove anything you want sometimes. 00:27:59.490 --> 00:28:00.515 align:middle line:90% Sometimes it works. 00:28:00.515 --> 00:28:01.390 align:middle line:90% Sometimes it doesn't. 00:28:01.390 --> 00:28:02.590 align:middle line:90% Sometimes it's nicer. 00:28:02.590 --> 00:28:07.560 align:middle line:84% But it's not assuming a specific form of theorem. 00:28:07.560 --> 00:28:09.860 align:middle line:90% Our proof looks like this. 00:28:09.860 --> 00:28:10.800 align:middle line:90% Let's see. 00:28:10.800 --> 00:28:23.533 align:middle line:84% For the sake of contradiction, assume P is false. 00:28:23.533 --> 00:28:26.540 align:middle line:90% 00:28:26.540 --> 00:28:27.210 align:middle line:90% All right. 00:28:27.210 --> 00:28:29.000 align:middle line:90% Assume P is false. 00:28:29.000 --> 00:28:31.216 align:middle line:90% Then the proof continues. 00:28:31.216 --> 00:28:33.800 align:middle line:90% 00:28:33.800 --> 00:28:36.890 align:middle line:84% Then some other statement R. Any statement R 00:28:36.890 --> 00:28:38.610 align:middle line:90% we want we get to choose. 00:28:38.610 --> 00:28:49.920 align:middle line:84% Then R is both true and false, which is bad. 00:28:49.920 --> 00:28:51.270 align:middle line:90% That breaks math. 00:28:51.270 --> 00:28:56.120 align:middle line:84% Propositions are supposed to be true or false, not both. 00:28:56.120 --> 00:28:59.310 align:middle line:84% That's a sign that our set of axioms is inconsistent. 00:28:59.310 --> 00:29:02.840 align:middle line:84% And remember, that's the one thing we don't want them to be. 00:29:02.840 --> 00:29:05.090 align:middle line:84% If something's true and false, then everything is true 00:29:05.090 --> 00:29:07.620 align:middle line:84% and everything is false, and that's bad. 00:29:07.620 --> 00:29:11.676 align:middle line:84% Let's not break math, which is a contradiction. 00:29:11.676 --> 00:29:17.550 align:middle line:90% 00:29:17.550 --> 00:29:19.720 align:middle line:90% And then to explain it further. 00:29:19.720 --> 00:29:31.415 align:middle line:84% So our assumption is wrong, so P is true. 00:29:31.415 --> 00:29:35.040 align:middle line:90% 00:29:35.040 --> 00:29:37.660 align:middle line:84% So we started by assuming P is false. 00:29:37.660 --> 00:29:39.130 align:middle line:90% We broke math. 00:29:39.130 --> 00:29:42.360 align:middle line:84% So the assumption that P is false must be wrong. 00:29:42.360 --> 00:29:45.450 align:middle line:90% So if P is false, it isn't true. 00:29:45.450 --> 00:29:49.180 align:middle line:84% Now, I put this last sentence in parentheses because it's clear. 00:29:49.180 --> 00:29:51.960 align:middle line:84% It's clearer than if you leave it out. 00:29:51.960 --> 00:29:53.730 align:middle line:84% But if you're writing between people 00:29:53.730 --> 00:29:55.620 align:middle line:84% that all agree on how proof by contradiction 00:29:55.620 --> 00:29:58.230 align:middle line:84% works and understand what's going on, then just 00:29:58.230 --> 00:30:00.480 align:middle line:84% including this word contradiction at the beginning 00:30:00.480 --> 00:30:04.620 align:middle line:84% and saying what the contradiction that we've reached 00:30:04.620 --> 00:30:05.230 align:middle line:90% is. 00:30:05.230 --> 00:30:06.370 align:middle line:90% That's usually enough. 00:30:06.370 --> 00:30:08.920 align:middle line:84% You don't have to have that summary statement at the end 00:30:08.920 --> 00:30:10.420 align:middle line:90% if you don't want it. 00:30:10.420 --> 00:30:12.230 align:middle line:84% That's a matter of personal style. 00:30:12.230 --> 00:30:14.680 align:middle line:84% But you should be saying right at the beginning 00:30:14.680 --> 00:30:16.990 align:middle line:84% that you're doing a proof by contradiction, 00:30:16.990 --> 00:30:21.040 align:middle line:84% and you should say at the end what your contradiction is. 00:30:21.040 --> 00:30:23.440 align:middle line:90% Does that make sense? 00:30:23.440 --> 00:30:25.030 align:middle line:90% Cool. 00:30:25.030 --> 00:30:27.280 align:middle line:90% Yes? 00:30:27.280 --> 00:30:28.417 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:30:28.417 --> 00:30:31.930 align:middle line:90% 00:30:31.930 --> 00:30:35.320 align:middle line:84% Are you really just saying that [INAUDIBLE]? 00:30:35.320 --> 00:30:36.580 align:middle line:90% ZACHARY ABEL: Yeah. 00:30:36.580 --> 00:30:37.760 align:middle line:90% That's exactly right. 00:30:37.760 --> 00:30:41.140 align:middle line:84% This inference rule that we analyzed at the beginning 00:30:41.140 --> 00:30:45.010 align:middle line:84% is exactly what's justifying proof by contradiction 00:30:45.010 --> 00:30:47.080 align:middle line:90% as a proof outline. 00:30:47.080 --> 00:30:53.470 align:middle line:84% If not, P implies that false is true, then P has to be true. 00:30:53.470 --> 00:30:54.928 align:middle line:84% That's exactly what this is saying. 00:30:54.928 --> 00:30:56.803 align:middle line:84% And that's what the proof technique is doing. 00:30:56.803 --> 00:30:57.590 align:middle line:90% Very well-spotted. 00:30:57.590 --> 00:30:59.260 align:middle line:90% Thank you. 00:30:59.260 --> 00:31:01.160 align:middle line:84% I meant to say that and might have forgotten. 00:31:01.160 --> 00:31:03.550 align:middle line:90% So thank you for the reminder. 00:31:03.550 --> 00:31:06.050 align:middle line:84% A couple other things about this. 00:31:06.050 --> 00:31:09.830 align:middle line:84% For the sake of contradiction, there are lots of other ways 00:31:09.830 --> 00:31:10.430 align:middle line:90% to say that. 00:31:10.430 --> 00:31:12.322 align:middle line:84% You can just say proof by contradiction, 00:31:12.322 --> 00:31:14.030 align:middle line:84% and that's enough to signal to the reader 00:31:14.030 --> 00:31:16.820 align:middle line:84% that you're doing a proof by contradiction. 00:31:16.820 --> 00:31:24.540 align:middle line:84% I've also seen BWOC, By Way of Contradiction, 00:31:24.540 --> 00:31:27.470 align:middle line:90% assume P is false. 00:31:27.470 --> 00:31:28.310 align:middle line:90% I have seen that. 00:31:28.310 --> 00:31:28.980 align:middle line:90% I like it. 00:31:28.980 --> 00:31:33.440 align:middle line:84% I don't use it as much because I don't think it's as standard. 00:31:33.440 --> 00:31:35.690 align:middle line:90% But it's silly, and I like it. 00:31:35.690 --> 00:31:38.330 align:middle line:84% At the end, when you find your contradiction, 00:31:38.330 --> 00:31:40.350 align:middle line:84% and you want to say, this is a contradiction. 00:31:40.350 --> 00:31:41.430 align:middle line:90% We've broken math. 00:31:41.430 --> 00:31:43.760 align:middle line:84% We're done-- however you want to say that part. 00:31:43.760 --> 00:31:45.770 align:middle line:90% I've also seen this symbol-- 00:31:45.770 --> 00:31:49.550 align:middle line:84% equals x equals because it looks like two arrows pointing 00:31:49.550 --> 00:31:51.770 align:middle line:84% at each other contradicting each other. 00:31:51.770 --> 00:31:55.550 align:middle line:84% That's another fun embellishment if you want. 00:31:55.550 --> 00:31:59.360 align:middle line:84% But now that we've seen the proof outline, 00:31:59.360 --> 00:32:01.847 align:middle line:84% hey, that looks like a smiley face. 00:32:01.847 --> 00:32:03.430 align:middle line:84% Now that we've seen the proof outline, 00:32:03.430 --> 00:32:07.000 align:middle line:90% let's use it on a real theorem. 00:32:07.000 --> 00:32:14.280 align:middle line:84% And that is that the square root of 2 is not a rational number. 00:32:14.280 --> 00:32:16.530 align:middle line:84% So remember, this blackboard bold Q 00:32:16.530 --> 00:32:19.560 align:middle line:84% is the set of rational numbers of integer 00:32:19.560 --> 00:32:21.220 align:middle line:90% divided by nonzero integer-- 00:32:21.220 --> 00:32:23.430 align:middle line:90% the set of all such things. 00:32:23.430 --> 00:32:27.540 align:middle line:84% And the theorem is that square root of 2 is not rational. 00:32:27.540 --> 00:32:29.050 align:middle line:90% So let's prove it. 00:32:29.050 --> 00:32:32.650 align:middle line:84% Often, when your theorem has a not on the outside, 00:32:32.650 --> 00:32:34.950 align:middle line:90% it is not true that property. 00:32:34.950 --> 00:32:38.820 align:middle line:84% Contradiction is a common way to handle that. 00:32:38.820 --> 00:32:41.230 align:middle line:84% Because our proof by contradiction is going to say, 00:32:41.230 --> 00:32:51.480 align:middle line:84% well, proof by contradiction, assume-- 00:32:51.480 --> 00:32:53.550 align:middle line:90% And, again, I want to emphasize. 00:32:53.550 --> 00:32:57.210 align:middle line:84% The moment we say this is a proof by contradiction, 00:32:57.210 --> 00:32:58.670 align:middle line:90% what we assume is fixed. 00:32:58.670 --> 00:33:00.670 align:middle line:84% There's only one thing we're supposed to assume. 00:33:00.670 --> 00:33:02.900 align:middle line:84% The outline tells us what we have to assume, 00:33:02.900 --> 00:33:06.040 align:middle line:84% which is that the entire theorem is false. 00:33:06.040 --> 00:33:08.000 align:middle line:84% That's the only thing we're supposed to do. 00:33:08.000 --> 00:33:11.410 align:middle line:84% We should still say it because we want you and the reader 00:33:11.410 --> 00:33:14.450 align:middle line:84% to agree that the correct procedure is being followed. 00:33:14.450 --> 00:33:16.460 align:middle line:84% You're assuming the correct thing. 00:33:16.460 --> 00:33:18.860 align:middle line:90% So assume that this is false. 00:33:18.860 --> 00:33:21.580 align:middle line:84% In other words, assume that square root of 2 00:33:21.580 --> 00:33:24.465 align:middle line:90% is a rational number. 00:33:24.465 --> 00:33:27.400 align:middle line:90% 00:33:27.400 --> 00:33:29.400 align:middle line:90% Ah-- is, not isn't. 00:33:29.400 --> 00:33:32.050 align:middle line:90% 00:33:32.050 --> 00:33:35.650 align:middle line:90% Do what I say, not what I write. 00:33:35.650 --> 00:33:37.247 align:middle line:90% And now let's keep going. 00:33:37.247 --> 00:33:38.830 align:middle line:84% What does it mean for square root of 2 00:33:38.830 --> 00:33:41.210 align:middle line:90% to be a rational number? 00:33:41.210 --> 00:33:48.820 align:middle line:84% So square root of 2 equals a over b for some integers a 00:33:48.820 --> 00:33:52.870 align:middle line:90% and b, where b does not equal 0. 00:33:52.870 --> 00:33:55.790 align:middle line:84% And as we might recall from high school math, 00:33:55.790 --> 00:34:01.070 align:middle line:84% we can always assume that a fraction is in lowest terms-- 00:34:01.070 --> 00:34:03.853 align:middle line:90% in lowest terms. 00:34:03.853 --> 00:34:06.410 align:middle line:90% 00:34:06.410 --> 00:34:11.352 align:middle line:84% So a and b have no common divisors. 00:34:11.352 --> 00:34:19.190 align:middle line:90% 00:34:19.190 --> 00:34:21.870 align:middle line:84% We can always reduce a fraction to lowest terms. 00:34:21.870 --> 00:34:25.670 align:middle line:84% So let's take that fraction already written in lowest terms, 00:34:25.670 --> 00:34:27.120 align:middle line:90% and let's keep going from there. 00:34:27.120 --> 00:34:34.070 align:middle line:84% So a over b equals square root of 2 means a equals square root 00:34:34.070 --> 00:34:40.530 align:middle line:84% of 2 times b, which means a squared equals 2b squared, 00:34:40.530 --> 00:34:43.735 align:middle line:90% which means a squared is even. 00:34:43.735 --> 00:34:47.449 align:middle line:90% 00:34:47.449 --> 00:34:51.980 align:middle line:84% And we already proved something about if the square of a number 00:34:51.980 --> 00:34:54.920 align:middle line:84% is even, then the number itself is even. 00:34:54.920 --> 00:34:57.450 align:middle line:84% It's very convenient that we already have that there. 00:34:57.450 --> 00:35:01.200 align:middle line:84% So since a squared is even, this theorem 00:35:01.200 --> 00:35:06.720 align:middle line:84% shows us that a is even by the theorem over there 00:35:06.720 --> 00:35:09.420 align:middle line:90% on the previous board. 00:35:09.420 --> 00:35:11.040 align:middle line:90% Yeah? 00:35:11.040 --> 00:35:11.830 align:middle line:90% OK. 00:35:11.830 --> 00:35:14.070 align:middle line:84% What does it mean for a to be even? 00:35:14.070 --> 00:35:17.495 align:middle line:84% So a equals 2 times some integer c. 00:35:17.495 --> 00:35:20.760 align:middle line:90% 00:35:20.760 --> 00:35:22.630 align:middle line:90% For some c-- oops, sorry. 00:35:22.630 --> 00:35:23.760 align:middle line:90% For some integer-- 00:35:23.760 --> 00:35:26.850 align:middle line:84% I did it again-- c in the integers. 00:35:26.850 --> 00:35:29.700 align:middle line:90% 00:35:29.700 --> 00:35:31.420 align:middle line:84% That's what it means for a to be even. 00:35:31.420 --> 00:35:33.280 align:middle line:90% It's 2 times an integer. 00:35:33.280 --> 00:35:34.450 align:middle line:90% So let's keep going. 00:35:34.450 --> 00:35:38.190 align:middle line:84% Let's put this back into our formula. 00:35:38.190 --> 00:35:47.230 align:middle line:84% So now, this says that 2c squared equals 2b squared. 00:35:47.230 --> 00:35:51.100 align:middle line:84% So this says that 4c squared equals 2b squared. 00:35:51.100 --> 00:35:52.270 align:middle line:90% We can divide by 2. 00:35:52.270 --> 00:35:56.930 align:middle line:84% This says 2c squared equals b squared. 00:35:56.930 --> 00:35:58.420 align:middle line:90% But now b squared is even. 00:35:58.420 --> 00:36:02.330 align:middle line:90% 00:36:02.330 --> 00:36:06.320 align:middle line:84% And as we know by our Lemma over there again, 00:36:06.320 --> 00:36:13.040 align:middle line:84% B is now even by the same theorem. 00:36:13.040 --> 00:36:14.930 align:middle line:90% Yeah? 00:36:14.930 --> 00:36:15.660 align:middle line:90% OK. 00:36:15.660 --> 00:36:20.455 align:middle line:90% So a and b are now both even. 00:36:20.455 --> 00:36:26.390 align:middle line:90% 00:36:26.390 --> 00:36:33.685 align:middle line:84% So they have a common factor of 2. 00:36:33.685 --> 00:36:37.250 align:middle line:90% 00:36:37.250 --> 00:36:42.350 align:middle line:84% To be more precise, no common divisor is greater than 1. 00:36:42.350 --> 00:36:44.490 align:middle line:84% And if a and b are in lowest terms, 00:36:44.490 --> 00:36:46.370 align:middle line:84% then there's no integer bigger than 1 00:36:46.370 --> 00:36:47.940 align:middle line:90% that divides both of them. 00:36:47.940 --> 00:36:51.110 align:middle line:84% But we just found that 2 divides both of them. 00:36:51.110 --> 00:36:53.310 align:middle line:90% So this is our contradiction. 00:36:53.310 --> 00:36:58.190 align:middle line:84% This statement here contradicts this statement here. 00:36:58.190 --> 00:37:01.590 align:middle line:90% 00:37:01.590 --> 00:37:03.420 align:middle line:84% We have a single statement that we 00:37:03.420 --> 00:37:06.360 align:middle line:90% showed is both true and false. 00:37:06.360 --> 00:37:07.890 align:middle line:84% So we've reached a contradiction, 00:37:07.890 --> 00:37:10.650 align:middle line:90% and our proof is done. 00:37:10.650 --> 00:37:13.260 align:middle line:90% Does that make sense? 00:37:13.260 --> 00:37:15.030 align:middle line:90% Awesome. 00:37:15.030 --> 00:37:18.450 align:middle line:90% Very nice, very nice. 00:37:18.450 --> 00:37:22.980 align:middle line:84% Let me take one more minute to talk about these proof outline 00:37:22.980 --> 00:37:25.380 align:middle line:90% techniques in general. 00:37:25.380 --> 00:37:28.230 align:middle line:84% Now, these aren't meant to restrict 00:37:28.230 --> 00:37:30.960 align:middle line:84% the way you're allowed to prove things or restrict 00:37:30.960 --> 00:37:33.280 align:middle line:84% the way you're allowed to express things. 00:37:33.280 --> 00:37:35.310 align:middle line:84% They're there as scaffolds to make 00:37:35.310 --> 00:37:37.950 align:middle line:84% sure you're proving the right things 00:37:37.950 --> 00:37:39.990 align:middle line:90% and to direct your thinking. 00:37:39.990 --> 00:37:42.120 align:middle line:84% The more you can break down a proof 00:37:42.120 --> 00:37:46.270 align:middle line:84% into smaller and smaller to-dos, first of all, 00:37:46.270 --> 00:37:49.020 align:middle line:84% the easier it is to make sure that your proof is correct. 00:37:49.020 --> 00:37:51.390 align:middle line:84% And secondly, the easier it's going 00:37:51.390 --> 00:37:53.810 align:middle line:84% to be to think about those individual tasks. 00:37:53.810 --> 00:37:58.570 align:middle line:84% Smaller tasks are easier to handle than larger tasks. 00:37:58.570 --> 00:38:02.320 align:middle line:84% Also, a common mistake when writing proofs in general 00:38:02.320 --> 00:38:05.630 align:middle line:84% is when you're presented with a theorem to dive right in. 00:38:05.630 --> 00:38:06.470 align:middle line:90% Why is this true? 00:38:06.470 --> 00:38:09.040 align:middle line:84% How am I going to figure this out 00:38:09.040 --> 00:38:12.760 align:middle line:84% without taking the time to realize that actually, what's 00:38:12.760 --> 00:38:15.220 align:middle line:84% going on in your head, what you're trying to prove 00:38:15.220 --> 00:38:18.140 align:middle line:84% is not actually what the theorem is asking you to prove. 00:38:18.140 --> 00:38:20.860 align:middle line:84% And so that's just a bunch of wasted effort and lost 00:38:20.860 --> 00:38:23.500 align:middle line:90% points on your homework. 00:38:23.500 --> 00:38:27.130 align:middle line:84% If you had taken a few minutes, or with practice, 00:38:27.130 --> 00:38:33.730 align:middle line:84% a few seconds to do these standard outlining procedures 00:38:33.730 --> 00:38:37.628 align:middle line:84% to figure out what the structure of the proof is supposed to be, 00:38:37.628 --> 00:38:39.670 align:middle line:84% then now you know what the structure of the proof 00:38:39.670 --> 00:38:40.640 align:middle line:90% is supposed to be. 00:38:40.640 --> 00:38:44.953 align:middle line:84% And you can spend your precious thinking cycles 00:38:44.953 --> 00:38:46.870 align:middle line:84% thinking about the things that you're actually 00:38:46.870 --> 00:38:49.060 align:middle line:90% supposed to prove. 00:38:49.060 --> 00:38:51.920 align:middle line:84% And let me demonstrate this, that with proof you 00:38:51.920 --> 00:38:53.130 align:middle line:90% can make this automatic. 00:38:53.130 --> 00:38:54.420 align:middle line:90% You can make this fast. 00:38:54.420 --> 00:38:58.430 align:middle line:84% You can really use it to help inform your proof writing 00:38:58.430 --> 00:38:59.250 align:middle line:90% endeavors. 00:38:59.250 --> 00:39:02.450 align:middle line:90% So let's say we have a theorem. 00:39:02.450 --> 00:39:03.060 align:middle line:90% Let's see. 00:39:03.060 --> 00:39:16.190 align:middle line:84% So an integer n is fooish precisely when n plus 1 00:39:16.190 --> 00:39:17.200 align:middle line:90% is barsome. 00:39:17.200 --> 00:39:20.630 align:middle line:90% 00:39:20.630 --> 00:39:22.650 align:middle line:84% What do these underlined words mean? 00:39:22.650 --> 00:39:24.052 align:middle line:90% Absolutely nothing. 00:39:24.052 --> 00:39:26.010 align:middle line:84% But let's imagine they actually mean something. 00:39:26.010 --> 00:39:27.770 align:middle line:84% Let's imagine this theorem is meaningful, 00:39:27.770 --> 00:39:30.260 align:middle line:84% and we've been asked to prove it. 00:39:30.260 --> 00:39:32.190 align:middle line:84% I claim there's a lot of progress 00:39:32.190 --> 00:39:35.480 align:middle line:84% we can make in writing and structuring this proof, 00:39:35.480 --> 00:39:40.040 align:middle line:84% even before we go and look up what these two weird terms mean. 00:39:40.040 --> 00:39:41.762 align:middle line:90% So let's do that. 00:39:41.762 --> 00:39:49.510 align:middle line:84% So this theorem, we can phrase it as for all n in z, 00:39:49.510 --> 00:39:57.052 align:middle line:84% n is fooish if and only if n plus 1 is barsome. 00:39:57.052 --> 00:39:57.900 align:middle line:90% Mm. 00:39:57.900 --> 00:39:59.790 align:middle line:90% Excuse me. 00:39:59.790 --> 00:40:03.460 align:middle line:84% Now, this-- sorry, I prefer to write it like this. 00:40:03.460 --> 00:40:06.550 align:middle line:90% Fn, if and only if Bn plus 1. 00:40:06.550 --> 00:40:09.370 align:middle line:84% I don't remember if we talked about if and only if on Tuesday. 00:40:09.370 --> 00:40:09.870 align:middle line:90% Great. 00:40:09.870 --> 00:40:12.450 align:middle line:90% Let's talk about it now. 00:40:12.450 --> 00:40:18.300 align:middle line:84% So P if and only if Q. also written as P if and only 00:40:18.300 --> 00:40:24.650 align:middle line:84% if Q means that P and Q are both true or both false. 00:40:24.650 --> 00:40:31.445 align:middle line:90% P and Q both true or both false. 00:40:31.445 --> 00:40:34.020 align:middle line:90% 00:40:34.020 --> 00:40:36.730 align:middle line:84% It means they're expressing the same condition. 00:40:36.730 --> 00:40:39.550 align:middle line:84% They're true in exactly the same times. 00:40:39.550 --> 00:40:41.770 align:middle line:84% They're false in exactly the same times. 00:40:41.770 --> 00:40:45.300 align:middle line:90% So they're logically equivalent. 00:40:45.300 --> 00:40:47.290 align:middle line:90% One way you can define this-- 00:40:47.290 --> 00:40:50.390 align:middle line:84% we could just draw the truth table like we did last time. 00:40:50.390 --> 00:40:55.180 align:middle line:90% But also commonly-- oops-- 00:40:55.180 --> 00:41:04.040 align:middle line:84% this can be defined as P implies Q and Q implies P. If P is true, 00:41:04.040 --> 00:41:04.850 align:middle line:90% then Q is true. 00:41:04.850 --> 00:41:06.345 align:middle line:84% And if Q is true, then P is true. 00:41:06.345 --> 00:41:09.520 align:middle line:90% 00:41:09.520 --> 00:41:12.760 align:middle line:84% And this, if and only if, implies 00:41:12.760 --> 00:41:15.820 align:middle line:84% in both directions this concept is what we mean 00:41:15.820 --> 00:41:18.340 align:middle line:90% when we say precisely when. 00:41:18.340 --> 00:41:21.880 align:middle line:84% These are true or false together. 00:41:21.880 --> 00:41:23.590 align:middle line:90% Sound good? 00:41:23.590 --> 00:41:25.246 align:middle line:90% OK. 00:41:25.246 --> 00:41:27.050 align:middle line:90% So we have our theorem. 00:41:27.050 --> 00:41:28.280 align:middle line:90% Let's prove it. 00:41:28.280 --> 00:41:30.950 align:middle line:90% Proof-- OK. 00:41:30.950 --> 00:41:32.470 align:middle line:84% The theorem starts with a for all. 00:41:32.470 --> 00:41:33.945 align:middle line:90% I know how to handle a for all. 00:41:33.945 --> 00:41:37.390 align:middle line:90% 00:41:37.390 --> 00:41:40.215 align:middle line:90% Suppose n is an integer. 00:41:40.215 --> 00:41:42.910 align:middle line:90% 00:41:42.910 --> 00:41:53.790 align:middle line:84% We want to show Fn, if and only if, Bn plus 1. 00:41:53.790 --> 00:41:56.310 align:middle line:84% That was one of the first outlines we talked about. 00:41:56.310 --> 00:41:59.180 align:middle line:84% If you have a for all, you introduce a generic one, 00:41:59.180 --> 00:42:01.445 align:middle line:84% and you want to show the property for that n. 00:42:01.445 --> 00:42:04.310 align:middle line:90% 00:42:04.310 --> 00:42:07.880 align:middle line:84% Well, now we want to show an if and only if, which we know 00:42:07.880 --> 00:42:09.870 align:middle line:90% is defined like this. 00:42:09.870 --> 00:42:13.100 align:middle line:84% So really, instead of all of that, 00:42:13.100 --> 00:42:15.150 align:middle line:84% we can be a little more precise with it. 00:42:15.150 --> 00:42:26.270 align:middle line:84% We want to show that Fn implies Bn plus 1, and that Bn plus 1 00:42:26.270 --> 00:42:28.820 align:middle line:90% implies Fn. 00:42:28.820 --> 00:42:31.020 align:middle line:84% And once we've done both of those tasks, 00:42:31.020 --> 00:42:34.580 align:middle line:84% we will have proved the if and only if. 00:42:34.580 --> 00:42:37.080 align:middle line:84% Well, now we're trying to prove implications. 00:42:37.080 --> 00:42:38.910 align:middle line:84% We know how to prove implications. 00:42:38.910 --> 00:42:42.050 align:middle line:84% Let's use a direct method for each of these. 00:42:42.050 --> 00:42:48.090 align:middle line:90% So assume Fn is true. 00:42:48.090 --> 00:42:50.885 align:middle line:90% Now we want to show Bn plus 1. 00:42:50.885 --> 00:42:54.000 align:middle line:90% 00:42:54.000 --> 00:42:59.220 align:middle line:84% So these are separate sections in our proof. 00:42:59.220 --> 00:43:03.630 align:middle line:84% In this part, we are temporarily assuming that Fn is true 00:43:03.630 --> 00:43:07.200 align:middle line:84% and using that to prove that Bn plus 1 is true. 00:43:07.200 --> 00:43:17.050 align:middle line:84% And now down here, assume instead that Bn plus 1 is true. 00:43:17.050 --> 00:43:21.930 align:middle line:84% And now we want to show Fn is true. 00:43:21.930 --> 00:43:24.160 align:middle line:84% And now our tasks are very clear to us. 00:43:24.160 --> 00:43:27.450 align:middle line:84% We need to show that if Fn, then Bn plus 1. 00:43:27.450 --> 00:43:29.040 align:middle line:90% We need to show-- 00:43:29.040 --> 00:43:34.470 align:middle line:84% in addition to that, we need to show that if Bn plus 1, then Fn. 00:43:34.470 --> 00:43:35.940 align:middle line:84% And now at this point, there's not 00:43:35.940 --> 00:43:38.400 align:middle line:84% really much else we can do without going to look up 00:43:38.400 --> 00:43:40.870 align:middle line:84% what these terms mean, use their definitions, 00:43:40.870 --> 00:43:42.780 align:middle line:84% figure out why this thing is true. 00:43:42.780 --> 00:43:44.968 align:middle line:84% But this is where the interesting part of the proof 00:43:44.968 --> 00:43:46.760 align:middle line:84% happens, where the unique part of the proof 00:43:46.760 --> 00:43:49.623 align:middle line:84% happens, the part that's specific to this problem. 00:43:49.623 --> 00:43:52.880 align:middle line:90% 00:43:52.880 --> 00:43:54.950 align:middle line:90% Yeah. 00:43:54.950 --> 00:43:56.450 align:middle line:84% And that's a lot of progress that we 00:43:56.450 --> 00:43:59.510 align:middle line:84% made even before the theorem even makes sense. 00:43:59.510 --> 00:44:02.030 align:middle line:84% So once you get better at this task, 00:44:02.030 --> 00:44:05.060 align:middle line:84% the easier it will be to set things up correctly and set 00:44:05.060 --> 00:44:07.250 align:middle line:90% yourself up for success. 00:44:07.250 --> 00:44:10.280 align:middle line:90% Questions on that? 00:44:10.280 --> 00:44:11.790 align:middle line:90% Awesome. 00:44:11.790 --> 00:44:12.660 align:middle line:90% All right. 00:44:12.660 --> 00:44:14.400 align:middle line:84% So the other main topic for today, 00:44:14.400 --> 00:44:16.430 align:middle line:84% we talked a lot about proof outlines, proof 00:44:16.430 --> 00:44:17.930 align:middle line:90% by contradiction. 00:44:17.930 --> 00:44:20.556 align:middle line:84% The other main topic is proof by induction. 00:44:20.556 --> 00:44:24.260 align:middle line:90% 00:44:24.260 --> 00:44:27.050 align:middle line:84% And interesting, I didn't I didn't hear the groan 00:44:27.050 --> 00:44:28.670 align:middle line:90% that I usually get. 00:44:28.670 --> 00:44:31.340 align:middle line:84% Proof by induction is one of those topics 00:44:31.340 --> 00:44:34.220 align:middle line:84% that some students come in with the impression 00:44:34.220 --> 00:44:38.600 align:middle line:84% that it's difficult, that it's hard to grasp. 00:44:38.600 --> 00:44:41.260 align:middle line:90% And there are reasons for that. 00:44:41.260 --> 00:44:43.340 align:middle line:84% Even before telling you what induction is, 00:44:43.340 --> 00:44:47.080 align:middle line:84% I can tell you that it's kind of a shame 00:44:47.080 --> 00:44:50.050 align:middle line:84% that we have to teach you induction so early. 00:44:50.050 --> 00:44:53.290 align:middle line:84% Because we're now in the process of telling you 00:44:53.290 --> 00:44:58.960 align:middle line:84% that proofs are supposed to be clear and self-evident and easy 00:44:58.960 --> 00:44:59.840 align:middle line:90% to follow. 00:44:59.840 --> 00:45:01.538 align:middle line:84% And then we come to proof by induction, 00:45:01.538 --> 00:45:03.580 align:middle line:84% which has a weird outline that has lots of things 00:45:03.580 --> 00:45:06.750 align:middle line:84% you have to say in weird orders for some arcane region, 00:45:06.750 --> 00:45:08.960 align:middle line:84% like we're summoning an ancient demon. 00:45:08.960 --> 00:45:11.720 align:middle line:84% Why do we have to say this in that particular way, 00:45:11.720 --> 00:45:12.770 align:middle line:90% in this place? 00:45:12.770 --> 00:45:13.640 align:middle line:90% I don't know. 00:45:13.640 --> 00:45:14.470 align:middle line:90% It's confusing. 00:45:14.470 --> 00:45:15.740 align:middle line:90% And it is confusing. 00:45:15.740 --> 00:45:17.440 align:middle line:84% And there's a reason it's confusing 00:45:17.440 --> 00:45:21.430 align:middle line:84% that the reason induction works, and the reason 00:45:21.430 --> 00:45:26.260 align:middle line:84% that the proof is structured the way it's structured 00:45:26.260 --> 00:45:29.890 align:middle line:84% isn't explained in every proof by induction 00:45:29.890 --> 00:45:32.230 align:middle line:84% because everyone who's writing a proof by induction 00:45:32.230 --> 00:45:35.590 align:middle line:84% assumes that everyone who's reading their proof by induction 00:45:35.590 --> 00:45:39.200 align:middle line:84% knows what proof by induction is and why it works. 00:45:39.200 --> 00:45:43.070 align:middle line:84% So every proof by induction you see is completely opaque 00:45:43.070 --> 00:45:45.530 align:middle line:84% and makes absolutely no sense until you 00:45:45.530 --> 00:45:47.160 align:middle line:90% know what induction is. 00:45:47.160 --> 00:45:51.140 align:middle line:84% And I think that's why it can be difficult to approach. 00:45:51.140 --> 00:45:54.660 align:middle line:84% What we're going to talk about right now is-- 00:45:54.660 --> 00:45:57.960 align:middle line:84% first of all, before I tell you what proof by induction is, 00:45:57.960 --> 00:46:00.020 align:middle line:84% let's solve a problem together that's going 00:46:00.020 --> 00:46:05.400 align:middle line:84% to lead us to the why and how and what of induction. 00:46:05.400 --> 00:46:07.100 align:middle line:84% It's going to tell us what we need to do 00:46:07.100 --> 00:46:09.030 align:middle line:90% and what tool we need to do it. 00:46:09.030 --> 00:46:10.730 align:middle line:84% And then induction will be there right 00:46:10.730 --> 00:46:14.720 align:middle line:84% at the end to say I'm the tool for the job. 00:46:14.720 --> 00:46:19.910 align:middle line:90% So here is a theorem. 00:46:19.910 --> 00:46:26.160 align:middle line:90% Theorem-- for all n and z-- 00:46:26.160 --> 00:46:26.660 align:middle line:90% sorry. 00:46:26.660 --> 00:46:33.980 align:middle line:84% For all natural numbers n 1 plus 2 plus 3 plus up to n 00:46:33.980 --> 00:46:39.200 align:middle line:84% equals n times n plus 1 divided by 2. 00:46:39.200 --> 00:46:40.980 align:middle line:84% This might be familiar to some of you. 00:46:40.980 --> 00:46:42.730 align:middle line:84% There are lots of ways to prove it. 00:46:42.730 --> 00:46:43.980 align:middle line:90% You can prove it by induction. 00:46:43.980 --> 00:46:45.147 align:middle line:90% You can prove it by picture. 00:46:45.147 --> 00:46:46.680 align:middle line:90% You can prove it by counting. 00:46:46.680 --> 00:46:48.777 align:middle line:90% There are lots and lots of ways. 00:46:48.777 --> 00:46:51.110 align:middle line:84% We're going to talk through one particular way of making 00:46:51.110 --> 00:46:52.700 align:middle line:90% sense of this theorem. 00:46:52.700 --> 00:46:55.010 align:middle line:90% And let's just go by intuition. 00:46:55.010 --> 00:46:56.850 align:middle line:84% We're not doing induction right now. 00:46:56.850 --> 00:47:00.732 align:middle line:84% We're just we're just figuring out why this thing is true. 00:47:00.732 --> 00:47:02.690 align:middle line:84% So, first of all, let's make sure we understand 00:47:02.690 --> 00:47:04.880 align:middle line:90% what this thing is even saying. 00:47:04.880 --> 00:47:09.830 align:middle line:84% For example, so e.g., when n equals 4, 00:47:09.830 --> 00:47:13.580 align:middle line:84% this is claiming that 1 plus 2 plus 3 plus 4 00:47:13.580 --> 00:47:18.630 align:middle line:84% equals 4 times 5 over 2, which looks pretty good. 00:47:18.630 --> 00:47:19.680 align:middle line:90% This is 10. 00:47:19.680 --> 00:47:20.850 align:middle line:90% This is 20 over 2. 00:47:20.850 --> 00:47:22.430 align:middle line:90% Yeah, they're equal. 00:47:22.430 --> 00:47:25.310 align:middle line:84% And we're supposed to prove this kind of thing 00:47:25.310 --> 00:47:28.010 align:middle line:90% for every natural number n. 00:47:28.010 --> 00:47:29.540 align:middle line:90% Yeah? 00:47:29.540 --> 00:47:33.035 align:middle line:84% Even all the way down to just n equals 1-- 00:47:33.035 --> 00:47:34.680 align:middle line:90% e.g., n equals 1. 00:47:34.680 --> 00:47:38.250 align:middle line:90% What's the sum from 1 up to 1? 00:47:38.250 --> 00:47:40.590 align:middle line:90% Just the 1 itself. 00:47:40.590 --> 00:47:44.550 align:middle line:90% 1 is the sum of one term. 00:47:44.550 --> 00:47:46.290 align:middle line:84% And that's supposed to equal 1 times 2 00:47:46.290 --> 00:47:53.820 align:middle line:84% over 2, which looks like it does, even down to n equals 0. 00:47:53.820 --> 00:47:56.650 align:middle line:84% What is the sum from 1 up to zero? 00:47:56.650 --> 00:47:59.280 align:middle line:90% Oh, that's weird. 00:47:59.280 --> 00:48:00.700 align:middle line:90% That is a little weird. 00:48:00.700 --> 00:48:04.510 align:middle line:84% Turns out, we can make sense of it in this context. 00:48:04.510 --> 00:48:08.130 align:middle line:84% So another way to write this is with what's 00:48:08.130 --> 00:48:11.450 align:middle line:90% called sigma notation. 00:48:11.450 --> 00:48:15.330 align:middle line:90% k goes from 1 up to n of k. 00:48:15.330 --> 00:48:18.720 align:middle line:84% And you can think of this like a for loop. 00:48:18.720 --> 00:48:22.330 align:middle line:84% You start at k equals 1, and you write down the value of k. 00:48:22.330 --> 00:48:23.260 align:middle line:90% In that case, 1. 00:48:23.260 --> 00:48:25.620 align:middle line:84% Then you go up to k equals 2, and write down the value, 00:48:25.620 --> 00:48:27.030 align:middle line:84% k equals 3, write down the value, 00:48:27.030 --> 00:48:28.870 align:middle line:90% all the way up to this part. 00:48:28.870 --> 00:48:29.890 align:middle line:90% You go up to k equals n. 00:48:29.890 --> 00:48:30.780 align:middle line:90% Write down the value. 00:48:30.780 --> 00:48:33.860 align:middle line:84% And the sigma says you add all these together. 00:48:33.860 --> 00:48:38.080 align:middle line:90% Sigma is the Greek S for sum. 00:48:38.080 --> 00:48:42.280 align:middle line:84% So this is notation for exactly this sum from 1 to n. 00:48:42.280 --> 00:48:45.520 align:middle line:84% And by convention, and there are good reasons 00:48:45.520 --> 00:48:46.430 align:middle line:90% for this convention. 00:48:46.430 --> 00:48:52.870 align:middle line:84% But by convention, the sum from k 00:48:52.870 --> 00:48:57.730 align:middle line:84% equals 1 to 0, the sum of 0 terms of whatever you 00:48:57.730 --> 00:49:01.780 align:middle line:84% want in here, but now, we're talking about the sum k, 00:49:01.780 --> 00:49:02.810 align:middle line:90% is just zero. 00:49:02.810 --> 00:49:03.980 align:middle line:90% The empty sum is zero. 00:49:03.980 --> 00:49:08.230 align:middle line:84% If you're adding nothing, you have a total of zero. 00:49:08.230 --> 00:49:11.440 align:middle line:84% And, in that case, the empty sum is 00:49:11.440 --> 00:49:14.960 align:middle line:84% zero, which is supposed to equal 0 times 1 divided by 2. 00:49:14.960 --> 00:49:16.100 align:middle line:90% And it does. 00:49:16.100 --> 00:49:16.600 align:middle line:90% Hooray. 00:49:16.600 --> 00:49:18.820 align:middle line:84% This theorem is true when n equals 0. 00:49:18.820 --> 00:49:22.780 align:middle line:84% And 1 and 4-- we verified three cases, which 00:49:22.780 --> 00:49:26.920 align:middle line:84% is a pretty good 0% of all the cases we're trying to show. 00:49:26.920 --> 00:49:28.250 align:middle line:90% But it's a good start. 00:49:28.250 --> 00:49:31.870 align:middle line:90% So let's see how to continue. 00:49:31.870 --> 00:49:45.950 align:middle line:90% 00:49:45.950 --> 00:49:52.270 align:middle line:84% So we are trying to show that 0 equals 0 times 1 divided by 2. 00:49:52.270 --> 00:49:54.990 align:middle line:90% 1 equals 1 times 2 divided by 2. 00:49:54.990 --> 00:50:00.000 align:middle line:84% 1 plus 2 is 2 times 3 divided by 2. 00:50:00.000 --> 00:50:05.010 align:middle line:84% 1 plus 2 plus 3 is 3 times 4 divided by 2. 00:50:05.010 --> 00:50:07.020 align:middle line:84% I'll stop in a minute, I promise. 00:50:07.020 --> 00:50:12.810 align:middle line:84% 1 plus 2 plus 3 plus 4 equals 4 times 5 divided by 2. 00:50:12.810 --> 00:50:14.070 align:middle line:90% Let's stop there for now. 00:50:14.070 --> 00:50:16.800 align:middle line:84% But we're supposed to prove all of these statements. 00:50:16.800 --> 00:50:20.300 align:middle line:84% That's what the theorem is asking from us. 00:50:20.300 --> 00:50:24.110 align:middle line:84% And we can think about this a little bit. 00:50:24.110 --> 00:50:28.400 align:middle line:84% If we stare at this, we can notice that on the left side, 00:50:28.400 --> 00:50:30.990 align:middle line:84% to get from this to this, we're adding 1. 00:50:30.990 --> 00:50:33.760 align:middle line:84% To get from this to this, we're adding 2. 00:50:33.760 --> 00:50:36.150 align:middle line:84% It's right there in how this sum is created. 00:50:36.150 --> 00:50:39.190 align:middle line:84% To get from here to here, we're adding 3. 00:50:39.190 --> 00:50:41.800 align:middle line:84% Again, we're just adding one more term to the sum. 00:50:41.800 --> 00:50:43.620 align:middle line:90% Then we're adding 4. 00:50:43.620 --> 00:50:48.430 align:middle line:84% Then we're adding 5, then 6, then 7, and then 8, and so on. 00:50:48.430 --> 00:50:50.580 align:middle line:84% There's a very simple pattern that 00:50:50.580 --> 00:50:52.980 align:middle line:84% explains how these numbers are changing from one row 00:50:52.980 --> 00:50:54.810 align:middle line:90% to the next. 00:50:54.810 --> 00:50:58.270 align:middle line:84% So wouldn't it be convenient if the numbers on the right side 00:50:58.270 --> 00:51:00.300 align:middle line:90% followed that same pattern? 00:51:00.300 --> 00:51:04.350 align:middle line:84% And if they did, it would make a lot of sense, at least to me, 00:51:04.350 --> 00:51:06.990 align:middle line:84% that since they start at the same number, 00:51:06.990 --> 00:51:10.140 align:middle line:84% and we're making the same changes to both sides every time 00:51:10.140 --> 00:51:14.100 align:middle line:84% we move down a row, that all the rows are going to be equal, 00:51:14.100 --> 00:51:17.350 align:middle line:84% that the left side is going to be the right side in every row. 00:51:17.350 --> 00:51:17.850 align:middle line:90% Yeah? 00:51:17.850 --> 00:51:22.650 align:middle line:84% So we're hoping that that's a plus 1, and that's 00:51:22.650 --> 00:51:27.920 align:middle line:84% a plus 2, and that's a plus 3, and that's a plus 4, and so on. 00:51:27.920 --> 00:51:29.920 align:middle line:84% These are question marks because we don't really 00:51:29.920 --> 00:51:31.620 align:middle line:90% know whether that's true yet. 00:51:31.620 --> 00:51:34.450 align:middle line:90% 00:51:34.450 --> 00:51:37.340 align:middle line:84% So let's write down what we're hoping is true. 00:51:37.340 --> 00:51:42.220 align:middle line:84% 1 plus 2 plus up to n is supposed to equal n 00:51:42.220 --> 00:51:44.090 align:middle line:90% times n plus 1 over 2. 00:51:44.090 --> 00:51:45.250 align:middle line:90% And then the next row-- 00:51:45.250 --> 00:51:49.300 align:middle line:84% 1 plus 2 plus n plus the next term, n plus 1, 00:51:49.300 --> 00:51:55.610 align:middle line:84% is supposed to equal n plus 1 times n plus 2 divided by 2. 00:51:55.610 --> 00:51:59.020 align:middle line:84% These are two consecutive rows in our big list. 00:51:59.020 --> 00:52:01.300 align:middle line:84% And what we are saying is to get from here to there, 00:52:01.300 --> 00:52:04.390 align:middle line:84% we're definitely adding n plus 1. 00:52:04.390 --> 00:52:06.500 align:middle line:84% We're just adding this one more term over here. 00:52:06.500 --> 00:52:09.940 align:middle line:84% So is it true that to get from here to here, we're adding n 00:52:09.940 --> 00:52:10.560 align:middle line:90% plus 1? 00:52:10.560 --> 00:52:13.240 align:middle line:90% 00:52:13.240 --> 00:52:14.480 align:middle line:90% And we can check that. 00:52:14.480 --> 00:52:15.310 align:middle line:90% Let's see. 00:52:15.310 --> 00:52:20.860 align:middle line:84% n plus 1 times n plus 2 divided by 2 minus n times 00:52:20.860 --> 00:52:23.690 align:middle line:90% n plus 1 divided by 2. 00:52:23.690 --> 00:52:26.650 align:middle line:84% Well, we can factor in n plus 1 from both sides. 00:52:26.650 --> 00:52:33.840 align:middle line:84% That's n plus 1 times n plus 2 over 2 minus n over 2. 00:52:33.840 --> 00:52:35.500 align:middle line:90% That's looking pretty good. 00:52:35.500 --> 00:52:39.160 align:middle line:84% That's going to be n plus 1 times 2 over 2 times 1. 00:52:39.160 --> 00:52:40.160 align:middle line:90% That's just n plus 1. 00:52:40.160 --> 00:52:42.750 align:middle line:90% 00:52:42.750 --> 00:52:46.110 align:middle line:84% So our hunch about that pattern is true. 00:52:46.110 --> 00:52:49.620 align:middle line:84% To go from the n-th row to the n plus first row, 00:52:49.620 --> 00:52:51.720 align:middle line:90% we're adding n plus 1. 00:52:51.720 --> 00:52:53.280 align:middle line:90% Yeah? 00:52:53.280 --> 00:52:55.270 align:middle line:90% So what we said earlier is true. 00:52:55.270 --> 00:52:57.180 align:middle line:84% This pattern add 1, then 2, then 3, then 00:52:57.180 --> 00:53:01.330 align:middle line:84% 4, then 5 that we saw on the left is recreated on the right. 00:53:01.330 --> 00:53:04.470 align:middle line:84% Add 1, add 2, add 3, add 4, add 5, and so on. 00:53:04.470 --> 00:53:08.400 align:middle line:84% So at least to me, this feels like an explanation that 00:53:08.400 --> 00:53:09.960 align:middle line:84% should make sense, and an explanation 00:53:09.960 --> 00:53:12.120 align:middle line:84% that we should be able to write down somehow 00:53:12.120 --> 00:53:14.340 align:middle line:90% as a mathematical proof. 00:53:14.340 --> 00:53:17.050 align:middle line:84% Does anyone have questions about this idea? 00:53:17.050 --> 00:53:19.720 align:middle line:84% I haven't said anything about how we're going to write this. 00:53:19.720 --> 00:53:20.825 align:middle line:90% But is the idea clear? 00:53:20.825 --> 00:53:24.700 align:middle line:90% 00:53:24.700 --> 00:53:26.740 align:middle line:90% Awesome. 00:53:26.740 --> 00:53:28.210 align:middle line:90% Let's see. 00:53:28.210 --> 00:53:32.560 align:middle line:84% So let's dig in a little more and see what exactly 00:53:32.560 --> 00:53:33.765 align:middle line:90% we accomplished over there. 00:53:33.765 --> 00:53:40.660 align:middle line:90% 00:53:40.660 --> 00:53:45.190 align:middle line:84% So this argument here on the right that we did over here, 00:53:45.190 --> 00:53:50.020 align:middle line:84% let me write down a little more carefully what we accomplished. 00:53:50.020 --> 00:53:52.720 align:middle line:84% And it's going to look like this. 00:53:52.720 --> 00:54:01.190 align:middle line:84% If 1 plus 2 plus n equals n times n plus 1 over 2, 00:54:01.190 --> 00:54:13.240 align:middle line:84% if that's true, then 1 plus 2 plus n plus 1 equals 00:54:13.240 --> 00:54:25.700 align:middle line:84% n plus 1, n plus 2 over 2 because we just added n plus 1 00:54:25.700 --> 00:54:26.665 align:middle line:90% to both sides. 00:54:26.665 --> 00:54:34.220 align:middle line:90% 00:54:34.220 --> 00:54:35.960 align:middle line:84% You would also want the algebra work 00:54:35.960 --> 00:54:37.850 align:middle line:84% to show that going from this to this 00:54:37.850 --> 00:54:39.860 align:middle line:84% really is just adding n plus 1, and that's 00:54:39.860 --> 00:54:41.430 align:middle line:90% exactly what we did over there. 00:54:41.430 --> 00:54:43.138 align:middle line:84% I don't want to write it again, so I just 00:54:43.138 --> 00:54:45.050 align:middle line:90% wrote the word algebra. 00:54:45.050 --> 00:54:48.320 align:middle line:84% Does it make sense that this is what we accomplished? 00:54:48.320 --> 00:54:52.800 align:middle line:84% If this row of our list is true, if left side equals right side, 00:54:52.800 --> 00:54:55.490 align:middle line:90% then the next row is also true. 00:54:55.490 --> 00:54:57.230 align:middle line:90% Yeah? 00:54:57.230 --> 00:54:58.050 align:middle line:90% Cool. 00:54:58.050 --> 00:55:01.850 align:middle line:84% So let's put some notation to this. 00:55:01.850 --> 00:55:07.110 align:middle line:84% I'm going to say that P of n is the statement. 00:55:07.110 --> 00:55:09.680 align:middle line:84% By the way, this colon equals is a way 00:55:09.680 --> 00:55:13.170 align:middle line:84% that we'll commonly say P of n is defined to be this thing. 00:55:13.170 --> 00:55:16.100 align:middle line:84% So not only is it equal to this, but this line right here 00:55:16.100 --> 00:55:18.410 align:middle line:90% is the definition of P of n. 00:55:18.410 --> 00:55:21.690 align:middle line:84% So let's define P of n to be the statement 00:55:21.690 --> 00:55:29.460 align:middle line:84% that 1 plus 2 plus up to n equals n times n plus 1 over 2. 00:55:29.460 --> 00:55:31.420 align:middle line:90% So Pn is a predicate. 00:55:31.420 --> 00:55:33.660 align:middle line:90% It's not a theorem on its own. 00:55:33.660 --> 00:55:40.860 align:middle line:84% What is a theorem is that for all n greater equals 0, P of n 00:55:40.860 --> 00:55:42.420 align:middle line:90% is true. 00:55:42.420 --> 00:55:44.410 align:middle line:84% That's what we're being asked to prove. 00:55:44.410 --> 00:55:46.980 align:middle line:84% That's just a restatement of our theorem 00:55:46.980 --> 00:55:50.550 align:middle line:90% from the top board over there. 00:55:50.550 --> 00:55:55.380 align:middle line:84% And now what we accomplished over on that board-- actually, 00:55:55.380 --> 00:55:57.750 align:middle line:90% I want a fresh board for this. 00:55:57.750 --> 00:55:59.345 align:middle line:84% That's going to be this board up here. 00:55:59.345 --> 00:56:12.750 align:middle line:90% 00:56:12.750 --> 00:56:23.650 align:middle line:84% What we accomplished is, first of all, we proved P of 0. 00:56:23.650 --> 00:56:27.400 align:middle line:90% That's the first line up here. 00:56:27.400 --> 00:56:33.550 align:middle line:84% And then these statements up here, what that's saying 00:56:33.550 --> 00:56:37.750 align:middle line:90% is that Pn implies Pn plus 1. 00:56:37.750 --> 00:56:42.590 align:middle line:90% So we proved P0 implies P1. 00:56:42.590 --> 00:56:50.990 align:middle line:84% We proved P1 implies P2, P2 implies P3, P3-- 00:56:50.990 --> 00:56:55.990 align:middle line:84% I'll stop, I promise-- implies P4, and all the way down. 00:56:55.990 --> 00:56:58.510 align:middle line:84% This is what we really accomplished 00:56:58.510 --> 00:57:00.700 align:middle line:90% on these last couple boards. 00:57:00.700 --> 00:57:03.760 align:middle line:84% But this isn't the theorem we're aiming for, right? 00:57:03.760 --> 00:57:10.420 align:middle line:84% The theorem we're aiming for asks for P of 0-- 00:57:10.420 --> 00:57:18.147 align:middle line:84% oops-- and P of 1, and P of 2, and P of 3, and P of 4, 00:57:18.147 --> 00:57:20.480 align:middle line:84% and all the way down-- we want to know that all of these 00:57:20.480 --> 00:57:21.510 align:middle line:90% are true. 00:57:21.510 --> 00:57:25.610 align:middle line:84% These implications are not what we're asking for. 00:57:25.610 --> 00:57:29.220 align:middle line:84% However, it kind of feels like this should be enough, right? 00:57:29.220 --> 00:57:33.150 align:middle line:84% Because once we know P of 0, and we know P of 0 implies P of 1, 00:57:33.150 --> 00:57:35.430 align:middle line:84% then modus ponens, we know P of 1. 00:57:35.430 --> 00:57:39.900 align:middle line:84% And once we know P of 1, this implication tells us P of 2. 00:57:39.900 --> 00:57:43.020 align:middle line:84% Once we know P of 2, pair with this, we get P of 3. 00:57:43.020 --> 00:57:43.950 align:middle line:90% Then we get P of 4. 00:57:43.950 --> 00:57:44.867 align:middle line:90% Then all the way down. 00:57:44.867 --> 00:57:46.430 align:middle line:90% We get all of them. 00:57:46.430 --> 00:57:49.650 align:middle line:84% I don't know how to write that yet, 00:57:49.650 --> 00:57:54.050 align:middle line:84% but that's the intuition that I have guiding this. 00:57:54.050 --> 00:57:58.350 align:middle line:84% And so it really feels like if we accomplish the left side, 00:57:58.350 --> 00:58:00.330 align:middle line:84% we should be able to conclude the right side. 00:58:00.330 --> 00:58:02.840 align:middle line:90% Do we agree? 00:58:02.840 --> 00:58:04.130 align:middle line:90% All right. 00:58:04.130 --> 00:58:14.420 align:middle line:84% And now we have the principle of induction, k which just 00:58:14.420 --> 00:58:18.780 align:middle line:90% says what we wanted to say. 00:58:18.780 --> 00:58:21.300 align:middle line:84% The left-hand side of this table implies the right-hand side 00:58:21.300 --> 00:58:22.000 align:middle line:90% of this table. 00:58:22.000 --> 00:58:28.440 align:middle line:84% If you know all of these, then you conclude all of these. 00:58:28.440 --> 00:58:34.080 align:middle line:84% So when we're structuring a proof by induction, 00:58:34.080 --> 00:58:39.610 align:middle line:84% we must prove everything here on the left. 00:58:39.610 --> 00:58:41.290 align:middle line:90% That is our responsibility. 00:58:41.290 --> 00:58:50.020 align:middle line:84% And then everything on the right, 00:58:50.020 --> 00:58:53.550 align:middle line:84% we get for free by the principle of induction. 00:58:53.550 --> 00:58:58.050 align:middle line:84% To be more precise the principle of induction says, 00:58:58.050 --> 00:59:07.440 align:middle line:84% if you know P of 0 and for all n greater or equal to 0, P of n 00:59:07.440 --> 00:59:14.890 align:middle line:84% implies P of n plus 1, that implies for all n great 00:59:14.890 --> 00:59:18.220 align:middle line:90% equals 0, P of n. 00:59:18.220 --> 00:59:21.640 align:middle line:84% So that's exactly what's written in this table just in formula 00:59:21.640 --> 00:59:22.450 align:middle line:90% form. 00:59:22.450 --> 00:59:25.370 align:middle line:84% P of 0 is this first one over here. 00:59:25.370 --> 00:59:29.050 align:middle line:84% And then all of the Pn's imply Pn plus 1s are all 00:59:29.050 --> 00:59:30.230 align:middle line:90% of the remaining stuff. 00:59:30.230 --> 00:59:33.070 align:middle line:84% And it says, if we accomplish all of these, 00:59:33.070 --> 00:59:36.280 align:middle line:90% then we can conclude these. 00:59:36.280 --> 00:59:37.235 align:middle line:90% Yes? 00:59:37.235 --> 00:59:38.860 align:middle line:84% AUDIENCE: Is the principle of induction 00:59:38.860 --> 00:59:40.780 align:middle line:84% just not like the axiom [INAUDIBLE]? 00:59:40.780 --> 00:59:42.653 align:middle line:90% 00:59:42.653 --> 00:59:43.820 align:middle line:90% ZACHARY ABEL: Good question. 00:59:43.820 --> 00:59:46.160 align:middle line:84% So is the principle of induction a new axiom, 00:59:46.160 --> 00:59:51.550 align:middle line:84% or is it just a name for this intuitive fact that we have? 00:59:51.550 --> 00:59:53.980 align:middle line:84% In truth, what it's really an axiom about 00:59:53.980 --> 00:59:57.550 align:middle line:84% is how the natural numbers are constructed. 00:59:57.550 --> 01:00:00.460 align:middle line:84% Really, what it's saying is that if you start from 0, 01:00:00.460 --> 01:00:05.720 align:middle line:84% and then add 1, add 1, add 1, add 1, add 1 to infinity, 01:00:05.720 --> 01:00:08.980 align:middle line:84% you will hit all the natural numbers. 01:00:08.980 --> 01:00:10.940 align:middle line:84% That's one way of thinking about it. 01:00:10.940 --> 01:00:13.690 align:middle line:84% But in practice, when we use the principle of induction, 01:00:13.690 --> 01:00:16.650 align:middle line:90% we're using it like this. 01:00:16.650 --> 01:00:18.760 align:middle line:84% So yeah, you can take it as an axiom. 01:00:18.760 --> 01:00:21.120 align:middle line:84% You can take it as following from the definition 01:00:21.120 --> 01:00:22.270 align:middle line:90% of the natural numbers. 01:00:22.270 --> 01:00:24.360 align:middle line:84% We didn't really go that far down 01:00:24.360 --> 01:00:27.220 align:middle line:90% in defining our fundamentals. 01:00:27.220 --> 01:00:29.410 align:middle line:84% So we'll just leave it like this. 01:00:29.410 --> 01:00:31.020 align:middle line:90% Good question. 01:00:31.020 --> 01:00:31.780 align:middle line:90% All right. 01:00:31.780 --> 01:00:34.440 align:middle line:84% So we have the principle of induction. 01:00:34.440 --> 01:00:37.360 align:middle line:84% We know what it's accomplishing and why. 01:00:37.360 --> 01:00:39.270 align:middle line:84% So how do we actually write using 01:00:39.270 --> 01:00:40.810 align:middle line:90% the principle of induction? 01:00:40.810 --> 01:00:42.785 align:middle line:84% Let's actually write this proof now. 01:00:42.785 --> 01:00:46.080 align:middle line:90% 01:00:46.080 --> 01:00:49.650 align:middle line:84% And the thing to always keep in mind 01:00:49.650 --> 01:00:53.730 align:middle line:84% when writing a proof by induction is this table. 01:00:53.730 --> 01:00:57.250 align:middle line:84% We are responsible for proving everything on the left side, 01:00:57.250 --> 01:01:00.510 align:middle line:84% and then we're able to conclude everything on the right side. 01:01:00.510 --> 01:01:03.150 align:middle line:84% So the theorem that we're trying to prove 01:01:03.150 --> 01:01:05.520 align:middle line:90% should be the right side. 01:01:05.520 --> 01:01:08.160 align:middle line:84% And then we're going to do the left side to get it. 01:01:08.160 --> 01:01:13.046 align:middle line:90% So proof by induction-- 01:01:13.046 --> 01:01:17.080 align:middle line:90% 01:01:17.080 --> 01:01:21.470 align:middle line:90% we need to say what P of n-- 01:01:21.470 --> 01:01:21.970 align:middle line:90% let's see. 01:01:21.970 --> 01:01:30.860 align:middle line:84% P of n is defined as the sum and n plus 1 over 2. 01:01:30.860 --> 01:01:34.330 align:middle line:90% Same way we defined Pn before. 01:01:34.330 --> 01:01:44.710 align:middle line:84% And we'll show that Pn is true for every n greater 01:01:44.710 --> 01:01:46.640 align:middle line:90% or equal to 0 by induction. 01:01:46.640 --> 01:01:51.670 align:middle line:90% 01:01:51.670 --> 01:01:56.020 align:middle line:84% So the first thing we're responsible for is P0. 01:01:56.020 --> 01:02:00.010 align:middle line:84% That's usually called the base case. 01:02:00.010 --> 01:02:05.890 align:middle line:90% Base case-- want to show P of 0. 01:02:05.890 --> 01:02:10.620 align:middle line:84% Well, P of 0 says that 0 equals 0 times 1 over 2, which is true. 01:02:10.620 --> 01:02:12.440 align:middle line:90% So we verified P of 0. 01:02:12.440 --> 01:02:16.430 align:middle line:84% Usually, the base cases are pretty straightforward. 01:02:16.430 --> 01:02:19.250 align:middle line:90% It's the inductive step-- 01:02:19.250 --> 01:02:25.820 align:middle line:84% inductive step-- that we need to watch out for. 01:02:25.820 --> 01:02:30.860 align:middle line:84% We now need to prove this whole thing. 01:02:30.860 --> 01:02:33.810 align:middle line:84% And we usually write it like this. 01:02:33.810 --> 01:02:39.260 align:middle line:84% Assume n is greater or equal to 0. 01:02:39.260 --> 01:02:46.430 align:middle line:90% And assume Pn is true. 01:02:46.430 --> 01:02:49.960 align:middle line:84% We want to show Pn plus 1 is true. 01:02:49.960 --> 01:02:52.520 align:middle line:90% 01:02:52.520 --> 01:02:56.720 align:middle line:84% And notice, what we just wrote is exactly the unpacked outline 01:02:56.720 --> 01:02:57.900 align:middle line:90% for this theorem. 01:02:57.900 --> 01:02:59.040 align:middle line:90% It starts with for all. 01:02:59.040 --> 01:03:00.890 align:middle line:90% So assume n. 01:03:00.890 --> 01:03:02.370 align:middle line:90% It then has an implication. 01:03:02.370 --> 01:03:03.600 align:middle line:90% So assume the left side. 01:03:03.600 --> 01:03:05.755 align:middle line:90% Let's prove the right side. 01:03:05.755 --> 01:03:08.130 align:middle line:84% The things we write when we're doing a proof by induction 01:03:08.130 --> 01:03:09.160 align:middle line:90% are not magic. 01:03:09.160 --> 01:03:11.790 align:middle line:90% They're just unpacking this. 01:03:11.790 --> 01:03:12.910 align:middle line:90% And now we can do it. 01:03:12.910 --> 01:03:16.140 align:middle line:84% The next thing I always recommend when proving something 01:03:16.140 --> 01:03:18.420 align:middle line:90% by induction-- 01:03:18.420 --> 01:03:19.540 align:middle line:90% what is P of n? 01:03:19.540 --> 01:03:20.710 align:middle line:90% What is P of n plus 1? 01:03:20.710 --> 01:03:23.500 align:middle line:84% I know we defined it, but I can't remember what it is. 01:03:23.500 --> 01:03:24.990 align:middle line:84% I always find it much more useful 01:03:24.990 --> 01:03:28.150 align:middle line:84% to write these things down in context. 01:03:28.150 --> 01:03:36.630 align:middle line:90% So, in other words, assume-- 01:03:36.630 --> 01:03:39.160 align:middle line:90% we're supposed to assume P of n. 01:03:39.160 --> 01:03:46.210 align:middle line:84% So assume 1 plus 2 plus up to n equals n times n plus 1 over 2. 01:03:46.210 --> 01:03:56.430 align:middle line:84% And we want to show Pn plus 1, which says that the next row, n 01:03:56.430 --> 01:04:01.650 align:middle line:90% plus 1 times n plus 2 over 2. 01:04:01.650 --> 01:04:02.320 align:middle line:90% All right. 01:04:02.320 --> 01:04:04.170 align:middle line:84% And just by following the proof outline 01:04:04.170 --> 01:04:07.010 align:middle line:84% and by identifying the things we're supposed to identify, 01:04:07.010 --> 01:04:09.410 align:middle line:84% now we have a very clear task in front of us. 01:04:09.410 --> 01:04:10.820 align:middle line:90% Assume this row. 01:04:10.820 --> 01:04:13.300 align:middle line:90% Use it to prove this next row. 01:04:13.300 --> 01:04:14.480 align:middle line:90% And we've already done that. 01:04:14.480 --> 01:04:17.650 align:middle line:90% We know how to do that. 01:04:17.650 --> 01:04:20.800 align:middle line:84% We know that this thing equals that thing. 01:04:20.800 --> 01:04:23.840 align:middle line:84% And then this plus that equals that because of algebra. 01:04:23.840 --> 01:04:25.820 align:middle line:84% This is all the work we already did. 01:04:25.820 --> 01:04:29.500 align:middle line:84% And now we're seeing how the induction outline leads us 01:04:29.500 --> 01:04:33.460 align:middle line:84% to figuring out that that is the right work to do. 01:04:33.460 --> 01:04:34.670 align:middle line:90% Did that all make sense? 01:04:34.670 --> 01:04:36.510 align:middle line:90% Yes? 01:04:36.510 --> 01:04:39.130 align:middle line:84% AUDIENCE: Does this necessarily has to start at 0? 01:04:39.130 --> 01:04:40.962 align:middle line:90% Like, if it started [INAUDIBLE]. 01:04:40.962 --> 01:04:42.170 align:middle line:90% ZACHARY ABEL: Great question. 01:04:42.170 --> 01:04:45.940 align:middle line:84% Does this have to start at 0, or could it start somewhere else? 01:04:45.940 --> 01:04:46.480 align:middle line:90% Yeah. 01:04:46.480 --> 01:04:48.670 align:middle line:90% It can start wherever we want. 01:04:48.670 --> 01:04:52.720 align:middle line:84% If we didn't know that P of 0 really made sense, 01:04:52.720 --> 01:04:55.100 align:middle line:84% and we wanted to start at P of 1, 01:04:55.100 --> 01:04:59.450 align:middle line:84% we absolutely could have started at P of 1 as our base case. 01:04:59.450 --> 01:05:02.380 align:middle line:84% And then all the implications above that. 01:05:02.380 --> 01:05:05.970 align:middle line:84% Just keep in mind that by not mentioning P of 0, 01:05:05.970 --> 01:05:08.910 align:middle line:84% we didn't prove P of 0, and so we can't conclude it either. 01:05:08.910 --> 01:05:13.310 align:middle line:84% But wherever we started and all the implications after it, 01:05:13.310 --> 01:05:17.510 align:middle line:84% those fit together to give us the thing on the right side. 01:05:17.510 --> 01:05:19.490 align:middle line:84% Yes, that's a slightly more general form 01:05:19.490 --> 01:05:22.130 align:middle line:90% of the induction principle. 01:05:22.130 --> 01:05:26.060 align:middle line:84% Please go ahead and use that version as well. 01:05:26.060 --> 01:05:28.520 align:middle line:90% All right. 01:05:28.520 --> 01:05:32.150 align:middle line:84% So let's do one more proof by induction. 01:05:32.150 --> 01:05:33.630 align:middle line:90% This is a really pretty problem. 01:05:33.630 --> 01:05:37.070 align:middle line:84% This seems to be a mandatory problem for any textbook 01:05:37.070 --> 01:05:39.470 align:middle line:90% on discrete math. 01:05:39.470 --> 01:05:41.460 align:middle line:90% It's one of my favorites. 01:05:41.460 --> 01:05:43.320 align:middle line:90% And here is the problem. 01:05:43.320 --> 01:05:45.350 align:middle line:90% Which board did I leave on? 01:05:45.350 --> 01:05:46.195 align:middle line:90% This one. 01:05:46.195 --> 01:06:00.870 align:middle line:90% 01:06:00.870 --> 01:06:03.160 align:middle line:90% So the story is like this. 01:06:03.160 --> 01:06:13.650 align:middle line:84% We have a big old courtyard subdivided into lots and lots 01:06:13.650 --> 01:06:14.950 align:middle line:90% of tiny little cells. 01:06:14.950 --> 01:06:18.250 align:middle line:84% It's going to be 2 to the n tall and 2 to the n wide. 01:06:18.250 --> 01:06:20.920 align:middle line:90% So imagine 64 by 64. 01:06:20.920 --> 01:06:22.570 align:middle line:90% It's a big old checkerboard. 01:06:22.570 --> 01:06:25.350 align:middle line:90% 01:06:25.350 --> 01:06:27.042 align:middle line:84% I'm just going to draw 8 by 8 because I 01:06:27.042 --> 01:06:28.250 align:middle line:90% don't have that kind of time. 01:06:28.250 --> 01:06:30.780 align:middle line:90% 01:06:30.780 --> 01:06:37.560 align:middle line:84% And now we're going to take a single cell and get rid of it. 01:06:37.560 --> 01:06:38.498 align:middle line:90% Which cell? 01:06:38.498 --> 01:06:39.040 align:middle line:90% I don't know. 01:06:39.040 --> 01:06:40.830 align:middle line:90% Doesn't matter. 01:06:40.830 --> 01:06:44.610 align:middle line:84% The question is, can we then fill the remaining 01:06:44.610 --> 01:06:47.556 align:middle line:90% cells with these L trominoes. 01:06:47.556 --> 01:06:50.760 align:middle line:90% 01:06:50.760 --> 01:06:52.480 align:middle line:90% You might know the word domino. 01:06:52.480 --> 01:06:54.160 align:middle line:84% It's two squares next to each other. 01:06:54.160 --> 01:06:56.010 align:middle line:90% It's a fun game. 01:06:56.010 --> 01:06:59.230 align:middle line:84% This is like a domino, but it's 3 squares instead of 2, 01:06:59.230 --> 01:07:02.890 align:middle line:84% and it's in the shape of an L. So it's called the L tromino. 01:07:02.890 --> 01:07:06.100 align:middle line:84% And the question is, can we fill all of this board 01:07:06.100 --> 01:07:10.060 align:middle line:84% without that one missing cell using 01:07:10.060 --> 01:07:13.030 align:middle line:90% L trominoes and nothing else? 01:07:13.030 --> 01:07:14.690 align:middle line:90% So that is our theorem. 01:07:14.690 --> 01:07:25.130 align:middle line:84% Theorem-- a 2 to the n by 2 to the n board without one cell, 01:07:25.130 --> 01:07:39.400 align:middle line:84% without any one cell can be tiled with L trominoes. 01:07:39.400 --> 01:07:43.300 align:middle line:84% So that's what we're supposed to prove. 01:07:43.300 --> 01:07:45.423 align:middle line:84% And let's see if I can set this up quickly. 01:07:45.423 --> 01:07:48.520 align:middle line:90% 01:07:48.520 --> 01:07:50.530 align:middle line:90% Yeah, that's all right. 01:07:50.530 --> 01:07:53.120 align:middle line:84% Let's see if I can get the glare a little different. 01:07:53.120 --> 01:07:54.730 align:middle line:90% OK. 01:07:54.730 --> 01:07:56.060 align:middle line:90% I do want to plug-in. 01:07:56.060 --> 01:07:56.780 align:middle line:90% Thank you. 01:07:56.780 --> 01:07:58.953 align:middle line:84% I always forget this in Zoom meetings, too. 01:07:58.953 --> 01:07:59.995 align:middle line:90% Please share your screen. 01:07:59.995 --> 01:08:03.170 align:middle line:90% 01:08:03.170 --> 01:08:03.670 align:middle line:90% Yes. 01:08:03.670 --> 01:08:04.370 align:middle line:90% All right. 01:08:04.370 --> 01:08:08.500 align:middle line:84% So I laser cut some toys to help us play with this puzzle. 01:08:08.500 --> 01:08:12.190 align:middle line:84% So we're supposed to prove that a 2 to the n by 2 to the n board 01:08:12.190 --> 01:08:14.740 align:middle line:84% can always be tiled with L trominoes. 01:08:14.740 --> 01:08:19.270 align:middle line:84% Here, let me put something on there for you to enjoy. 01:08:19.270 --> 01:08:23.560 align:middle line:84% Well, let's phrase it like we were going to prove it 01:08:23.560 --> 01:08:25.220 align:middle line:90% with induction because we are. 01:08:25.220 --> 01:08:32.109 align:middle line:84% So let's say that P of n is going to be a 2 to the n by 2 01:08:32.109 --> 01:08:37.029 align:middle line:84% to the n board without one cell-- 01:08:37.029 --> 01:08:38.450 align:middle line:90% without any one cell. 01:08:38.450 --> 01:08:42.010 align:middle line:84% Doesn't matter which one you pick-- 01:08:42.010 --> 01:08:47.620 align:middle line:90% can be tiled with L's. 01:08:47.620 --> 01:08:49.880 align:middle line:84% I'm going to call them L's, not L trominoes. 01:08:49.880 --> 01:08:52.279 align:middle line:84% And notice I kind of just wrote the same thing twice. 01:08:52.279 --> 01:08:53.500 align:middle line:90% That often happens. 01:08:53.500 --> 01:08:56.029 align:middle line:90% P of n is not our theorem. 01:08:56.029 --> 01:09:00.460 align:middle line:84% It's the predicate, which is inside of the for all that 01:09:00.460 --> 01:09:01.819 align:middle line:90% is part of the theorem. 01:09:01.819 --> 01:09:07.029 align:middle line:84% So this is really every 2 to the n by 2 to the n board. 01:09:07.029 --> 01:09:10.090 align:middle line:84% So if p of n is this, then our theorem 01:09:10.090 --> 01:09:16.300 align:middle line:84% is that for all n greater or equal 1, p of n is true. 01:09:16.300 --> 01:09:17.979 align:middle line:90% All right? 01:09:17.979 --> 01:09:19.439 align:middle line:90% And now we need to prove this. 01:09:19.439 --> 01:09:19.939 align:middle line:90% OK. 01:09:19.939 --> 01:09:23.479 align:middle line:84% Well, let's start with n equals 1. 01:09:23.479 --> 01:09:25.580 align:middle line:84% You can think about what n equals 0 means. 01:09:25.580 --> 01:09:27.189 align:middle line:90% It turns out it's true. 01:09:27.189 --> 01:09:28.760 align:middle line:90% But let's start with n equals 1. 01:09:28.760 --> 01:09:32.740 align:middle line:84% So a 2 to the 1 by 2 to the 1 board-- that's a 2 by 2 board. 01:09:32.740 --> 01:09:36.439 align:middle line:84% And can you see the little crosshairs in there. 01:09:36.439 --> 01:09:37.930 align:middle line:90% It's a 2 by 2 grid. 01:09:37.930 --> 01:09:42.069 align:middle line:84% And we're supposed to say no matter which single piece-- 01:09:42.069 --> 01:09:44.450 align:middle line:84% no matter which single cell we remove, 01:09:44.450 --> 01:09:50.529 align:middle line:84% we can always tile what remains with these L shapes. 01:09:50.529 --> 01:09:53.990 align:middle line:90% Anyone see how to do it? 01:09:53.990 --> 01:09:54.890 align:middle line:90% Yeah. 01:09:54.890 --> 01:09:55.440 align:middle line:90% All right. 01:09:55.440 --> 01:09:57.410 align:middle line:90% I'm going to put that there. 01:09:57.410 --> 01:09:59.060 align:middle line:90% Solved. 01:09:59.060 --> 01:10:03.260 align:middle line:84% What if the missing cell was in a different place, like there? 01:10:03.260 --> 01:10:03.810 align:middle line:90% Yep. 01:10:03.810 --> 01:10:04.310 align:middle line:90% OK. 01:10:04.310 --> 01:10:05.340 align:middle line:90% Can still solve that. 01:10:05.340 --> 01:10:07.010 align:middle line:90% We're allowed to rotate. 01:10:07.010 --> 01:10:10.790 align:middle line:84% So the 2 by 2 case isn't very hard. 01:10:10.790 --> 01:10:13.400 align:middle line:84% So let's keep track of our progress-- 01:10:13.400 --> 01:10:14.670 align:middle line:90% P of 1. 01:10:14.670 --> 01:10:19.680 align:middle line:84% Actually, I'm going to write this as the 2 by 2 case. 01:10:19.680 --> 01:10:20.850 align:middle line:90% Not very hard. 01:10:20.850 --> 01:10:22.710 align:middle line:90% Let's step it up a little. 01:10:22.710 --> 01:10:26.930 align:middle line:90% What about the 4 by 4 case? 01:10:26.930 --> 01:10:31.393 align:middle line:84% And say we're trying to avoid that one cell. 01:10:31.393 --> 01:10:33.560 align:middle line:84% Well, we also need to solve it for avoiding that one 01:10:33.560 --> 01:10:34.700 align:middle line:90% cell and that one cell. 01:10:34.700 --> 01:10:37.520 align:middle line:90% We need to solve all 16 cases. 01:10:37.520 --> 01:10:39.140 align:middle line:84% Because the theorem says it doesn't 01:10:39.140 --> 01:10:40.500 align:middle line:90% matter which one you exclude. 01:10:40.500 --> 01:10:43.400 align:middle line:90% You can always tile the rest. 01:10:43.400 --> 01:10:45.120 align:middle line:84% Well, you can play around with this. 01:10:45.120 --> 01:10:49.100 align:middle line:84% I have lots and lots of pieces to play with. 01:10:49.100 --> 01:10:53.890 align:middle line:84% You can tell I had way too much fun designing this thing. 01:10:53.890 --> 01:10:57.210 align:middle line:84% And it turns out, you can make it work. 01:10:57.210 --> 01:11:00.730 align:middle line:84% Like this one-- this is one of the 16 examples. 01:11:00.730 --> 01:11:03.010 align:middle line:84% It turns out, if I put that one there, 01:11:03.010 --> 01:11:04.710 align:middle line:84% and then that one there and there 01:11:04.710 --> 01:11:08.110 align:middle line:84% and there and there, then I've solved it. 01:11:08.110 --> 01:11:12.360 align:middle line:84% I've solved this one out of 16 cases. 01:11:12.360 --> 01:11:15.930 align:middle line:84% But also, it turns out it's not too difficult to generalize 01:11:15.930 --> 01:11:16.595 align:middle line:90% from there. 01:11:16.595 --> 01:11:19.830 align:middle line:90% 01:11:19.830 --> 01:11:25.540 align:middle line:84% Like, if the missing cell in one of these four positions, 01:11:25.540 --> 01:11:27.720 align:middle line:90% then I know how to do it. 01:11:27.720 --> 01:11:30.490 align:middle line:84% If it's in one of these four positions, 01:11:30.490 --> 01:11:32.980 align:middle line:90% then I also know how to do it. 01:11:32.980 --> 01:11:35.080 align:middle line:84% If it's there, I can just put that there. 01:11:35.080 --> 01:11:37.380 align:middle line:84% If it's there, I can put that there. 01:11:37.380 --> 01:11:41.430 align:middle line:84% If it's in one of these four spots, can still do it. 01:11:41.430 --> 01:11:44.730 align:middle line:84% If it's one of these four spots, can still do it. 01:11:44.730 --> 01:11:48.430 align:middle line:84% So by looking at all of those 16 cases combined, 01:11:48.430 --> 01:11:51.520 align:middle line:84% we now have a way that can put that yellow cell anywhere 01:11:51.520 --> 01:11:54.340 align:middle line:84% we want and finish tiling the rest. 01:11:54.340 --> 01:11:58.310 align:middle line:84% Not a very nice or clean argument, 01:11:58.310 --> 01:12:00.260 align:middle line:84% but at least we did all 16 cases. 01:12:00.260 --> 01:12:02.420 align:middle line:90% So that's good enough for now. 01:12:02.420 --> 01:12:10.750 align:middle line:84% Let's call the 4 by 4 case solved for now and move on. 01:12:10.750 --> 01:12:12.850 align:middle line:84% And I want to emphasize this is actually 01:12:12.850 --> 01:12:16.390 align:middle line:90% a key moment in this proof. 01:12:16.390 --> 01:12:19.870 align:middle line:84% We just solved the 4 by 4 case by looking at all 16 cases 01:12:19.870 --> 01:12:22.120 align:middle line:84% and showing that they're all possible. 01:12:22.120 --> 01:12:24.790 align:middle line:84% I want you to hold that deep in your heart 01:12:24.790 --> 01:12:26.930 align:middle line:90% that we have solved 4 by 4. 01:12:26.930 --> 01:12:30.640 align:middle line:84% Any 4 by 4 you ever see with a single missing cell 01:12:30.640 --> 01:12:32.770 align:middle line:90% can be tiled. 01:12:32.770 --> 01:12:34.330 align:middle line:90% Yeah? 01:12:34.330 --> 01:12:37.840 align:middle line:84% I want you to hold that deeply in your heart and then not 01:12:37.840 --> 01:12:40.490 align:middle line:84% worry too much about how we did all the 16 cases. 01:12:40.490 --> 01:12:42.820 align:middle line:90% Just know that it's true. 01:12:42.820 --> 01:12:44.660 align:middle line:84% Are we comfortable with that for now? 01:12:44.660 --> 01:12:47.470 align:middle line:90% 4 by 4 case is fully solved. 01:12:47.470 --> 01:12:50.520 align:middle line:90% Let's move on to 8 by 8. 01:12:50.520 --> 01:12:51.670 align:middle line:90% We have an 8 by 8. 01:12:51.670 --> 01:12:53.650 align:middle line:84% There's a single cell we're trying to avoid. 01:12:53.650 --> 01:12:55.710 align:middle line:90% Maybe it's there. 01:12:55.710 --> 01:12:57.335 align:middle line:90% Maybe it's over here. 01:12:57.335 --> 01:12:58.210 align:middle line:90% Let's leave it there. 01:12:58.210 --> 01:12:59.580 align:middle line:90% I don't care. 01:12:59.580 --> 01:13:03.820 align:middle line:84% What we're really looking for here is a generalizable method. 01:13:03.820 --> 01:13:06.190 align:middle line:90% I don't want to do 64 cases. 01:13:06.190 --> 01:13:08.487 align:middle line:84% And in the 16 by 16 case, I don't want to do-- 01:13:08.487 --> 01:13:09.070 align:middle line:90% what is that-- 01:13:09.070 --> 01:13:10.810 align:middle line:90% 256 cases? 01:13:10.810 --> 01:13:12.930 align:middle line:90% Please don't make me. 01:13:12.930 --> 01:13:15.750 align:middle line:84% We want a generalizable process that we can 01:13:15.750 --> 01:13:18.870 align:middle line:90% use to solve every board size. 01:13:18.870 --> 01:13:21.943 align:middle line:84% And here is a cool idea that works nicely. 01:13:21.943 --> 01:13:24.840 align:middle line:90% 01:13:24.840 --> 01:13:28.110 align:middle line:84% So first thing we're going to do. 01:13:28.110 --> 01:13:30.070 align:middle line:84% We're going to shift our perspective. 01:13:30.070 --> 01:13:31.900 align:middle line:90% This isn't an 8 by 8 board. 01:13:31.900 --> 01:13:35.580 align:middle line:90% This is for 4 by 4 boards. 01:13:35.580 --> 01:13:37.230 align:middle line:90% All right? 01:13:37.230 --> 01:13:40.500 align:middle line:84% And now let me just make this one purple for now 01:13:40.500 --> 01:13:42.310 align:middle line:84% to remember that that's the special one. 01:13:42.310 --> 01:13:44.130 align:middle line:90% We cannot move that one. 01:13:44.130 --> 01:13:48.070 align:middle line:84% However, remember we have solved the 4 by 4 case. 01:13:48.070 --> 01:13:51.660 align:middle line:84% So if we're missing a cell somewhere in this quadrant, 01:13:51.660 --> 01:13:53.670 align:middle line:84% and missing a cell somewhere in this quadrant, 01:13:53.670 --> 01:13:57.180 align:middle line:84% and missing a cell somewhere in this quadrant, then now 01:13:57.180 --> 01:13:59.800 align:middle line:84% we just have four instances of the 4 by 4 problem. 01:13:59.800 --> 01:14:03.070 align:middle line:84% We have four 4 by 4s, each missing a single cell. 01:14:03.070 --> 01:14:05.550 align:middle line:84% And we know how to solve all of those. 01:14:05.550 --> 01:14:10.020 align:middle line:84% So we know how to solve all of the red that remains. 01:14:10.020 --> 01:14:12.150 align:middle line:84% Unfortunately, we're not actually 01:14:12.150 --> 01:14:13.960 align:middle line:84% allowed to use these individual cells. 01:14:13.960 --> 01:14:15.540 align:middle line:84% We're supposed to cover everything 01:14:15.540 --> 01:14:18.780 align:middle line:90% with our L-shaped tiles. 01:14:18.780 --> 01:14:22.380 align:middle line:84% But remember, doesn't matter where we put these yellow ones. 01:14:22.380 --> 01:14:25.200 align:middle line:84% Anywhere we put this in here, we know how to solve that quadrant. 01:14:25.200 --> 01:14:28.000 align:middle line:84% Anywhere we put this in here, we know how to solve that quadrant. 01:14:28.000 --> 01:14:29.920 align:middle line:84% The purple one was the one that's given to us. 01:14:29.920 --> 01:14:32.010 align:middle line:90% We're not allowed to move that. 01:14:32.010 --> 01:14:34.860 align:middle line:84% But the next clever idea is, what 01:14:34.860 --> 01:14:37.260 align:middle line:84% if we put the yellow ones right there, 01:14:37.260 --> 01:14:40.150 align:middle line:84% and then think of them as not three individual cells, 01:14:40.150 --> 01:14:42.480 align:middle line:90% but as an L in their own right? 01:14:42.480 --> 01:14:43.350 align:middle line:90% No. 01:14:43.350 --> 01:14:46.000 align:middle line:90% There we go. 01:14:46.000 --> 01:14:49.810 align:middle line:84% So now this is going to be our key idea. 01:14:49.810 --> 01:14:52.870 align:middle line:84% If we're trying to tile everything 01:14:52.870 --> 01:14:57.070 align:middle line:84% except the single special cell, step one-- 01:14:57.070 --> 01:14:59.630 align:middle line:84% think about it as four separate quadrants. 01:14:59.630 --> 01:15:05.080 align:middle line:84% Step two-- put an L right in the middle, directly in the middle, 01:15:05.080 --> 01:15:07.450 align:middle line:84% to make sure that each of the four quadrants 01:15:07.450 --> 01:15:09.610 align:middle line:90% is now missing one cell. 01:15:09.610 --> 01:15:11.320 align:middle line:90% Yeah? 01:15:11.320 --> 01:15:14.020 align:middle line:84% If this initial cell that we're supposed to avoid 01:15:14.020 --> 01:15:15.850 align:middle line:84% is somewhere else, well, then we just 01:15:15.850 --> 01:15:18.290 align:middle line:84% put the L to touch the other three quadrants. 01:15:18.290 --> 01:15:20.620 align:middle line:84% We just rotate it this way instead. 01:15:20.620 --> 01:15:23.950 align:middle line:84% And once again, every single quadrant, all four quadrants, 01:15:23.950 --> 01:15:25.750 align:middle line:90% are missing a single cell. 01:15:25.750 --> 01:15:30.640 align:middle line:84% To really drive that home, let's see. 01:15:30.640 --> 01:15:32.810 align:middle line:90% We have this shape here. 01:15:32.810 --> 01:15:34.610 align:middle line:84% That's a 4 by 4 missing a corner. 01:15:34.610 --> 01:15:37.190 align:middle line:84% This shape here is a 4 by 4 missing a corner. 01:15:37.190 --> 01:15:39.200 align:middle line:84% Here's a 4 by 4 missing a corner. 01:15:39.200 --> 01:15:42.190 align:middle line:84% Here's a 4 by 4 missing that initial cell. 01:15:42.190 --> 01:15:45.210 align:middle line:84% So we have four separate problems 01:15:45.210 --> 01:15:48.010 align:middle line:84% where if we can solve the four orange cases, 01:15:48.010 --> 01:15:51.250 align:middle line:84% we'll be able to piece them back together to form a solution. 01:15:51.250 --> 01:15:53.860 align:middle line:84% And this is why we stored that deep in our heart. 01:15:53.860 --> 01:15:56.830 align:middle line:84% We know we can solve 4 by 4s with a cell missing. 01:15:56.830 --> 01:16:01.800 align:middle line:84% So we know all four of these orange quadrants can be filled. 01:16:01.800 --> 01:16:07.200 align:middle line:84% So we already know that this bigger shape can also be filled. 01:16:07.200 --> 01:16:08.500 align:middle line:90% So we're done. 01:16:08.500 --> 01:16:12.640 align:middle line:84% No matter where this one started, we can do this thing. 01:16:12.640 --> 01:16:13.915 align:middle line:90% We put down the cross. 01:16:13.915 --> 01:16:15.040 align:middle line:90% We put the L in the middle. 01:16:15.040 --> 01:16:17.040 align:middle line:84% Now we have four quadrants, each missing a cell. 01:16:17.040 --> 01:16:18.640 align:middle line:90% We invoke the 4 by 4 case. 01:16:18.640 --> 01:16:22.770 align:middle line:84% And now we've solved any 8 by 8 case that's given to us. 01:16:22.770 --> 01:16:25.080 align:middle line:90% Does that make sense? 01:16:25.080 --> 01:16:28.500 align:middle line:84% So if we believe that we can solve every 4 by 4, 01:16:28.500 --> 01:16:31.930 align:middle line:84% then we also believe that we can solve every 8 by 8. 01:16:31.930 --> 01:16:33.570 align:middle line:90% Yeah? 01:16:33.570 --> 01:16:35.080 align:middle line:90% And this keeps going. 01:16:35.080 --> 01:16:37.120 align:middle line:84% If you instead had an 8 by 8-- sorry. 01:16:37.120 --> 01:16:40.505 align:middle line:84% If you instead had a 16 by 16, you can do the same thing. 01:16:40.505 --> 01:16:42.380 align:middle line:84% Draw the crosshairs, put the L in the middle. 01:16:42.380 --> 01:16:45.500 align:middle line:84% Now, all four 8 by 8 quadrants are missing a single cell. 01:16:45.500 --> 01:16:47.590 align:middle line:90% Invoke the 8 by 8 case. 01:16:47.590 --> 01:16:50.650 align:middle line:90% We can even see that here. 01:16:50.650 --> 01:16:51.735 align:middle line:90% Go away, webcam. 01:16:51.735 --> 01:16:55.060 align:middle line:90% 01:16:55.060 --> 01:17:00.410 align:middle line:84% So here is a 16 by 16 with one cell we're supposed to avoid. 01:17:00.410 --> 01:17:02.020 align:middle line:84% So let's cut it into four quadrants 01:17:02.020 --> 01:17:03.980 align:middle line:84% and put an L right there in the middle. 01:17:03.980 --> 01:17:07.310 align:middle line:84% Now we should think of this as four separate 8 by 8s, 01:17:07.310 --> 01:17:10.180 align:middle line:90% each missing a single cell. 01:17:10.180 --> 01:17:15.400 align:middle line:84% And we trust in our heart that we have solved the 8 by 8 case. 01:17:15.400 --> 01:17:17.690 align:middle line:84% Don't worry too much about how we solved it. 01:17:17.690 --> 01:17:21.160 align:middle line:84% Just be confident that we solved the 8 by 8 case, 01:17:21.160 --> 01:17:25.450 align:middle line:84% that there is a way to tile each of the 8 by 8 corners. 01:17:25.450 --> 01:17:26.920 align:middle line:84% So now we can piece them together 01:17:26.920 --> 01:17:29.990 align:middle line:90% to form our 16 by 16 solution. 01:17:29.990 --> 01:17:34.030 align:middle line:84% And now it's all L tiles, and we win. 01:17:34.030 --> 01:17:37.300 align:middle line:84% So this is the idea behind this proof 01:17:37.300 --> 01:17:50.180 align:middle line:84% that we're able to show that Pn always implies P of n plus 1. 01:17:50.180 --> 01:17:54.890 align:middle line:84% If you can tile a 2 to the n by 2 to the n missing one cell, 01:17:54.890 --> 01:18:01.760 align:middle line:84% then you can tile 2 to the n plus 1 by 2 to the n plus 1 01:18:01.760 --> 01:18:05.420 align:middle line:84% missing a tile using our strategy of taking 01:18:05.420 --> 01:18:09.860 align:middle line:84% the 2 to the n plus 1 by 2 to the n plus 1 01:18:09.860 --> 01:18:14.330 align:middle line:84% and cutting it into four quadrants of 2 to the n by 2 01:18:14.330 --> 01:18:15.000 align:middle line:90% to the n. 01:18:15.000 --> 01:18:16.460 align:middle line:90% Put an L in the middle. 01:18:16.460 --> 01:18:19.490 align:middle line:84% Make sure each quadrant is missing a cell. 01:18:19.490 --> 01:18:23.120 align:middle line:90% Invoke the inductive hypothesis. 01:18:23.120 --> 01:18:25.040 align:middle line:90% That was a little sketchy. 01:18:25.040 --> 01:18:27.008 align:middle line:84% I didn't write this down carefully. 01:18:27.008 --> 01:18:28.550 align:middle line:84% You can check the lecture notes later 01:18:28.550 --> 01:18:31.092 align:middle line:84% if you want to see this written down a little more carefully. 01:18:31.092 --> 01:18:34.890 align:middle line:84% But that is the core idea and the problem I want you to see. 01:18:34.890 --> 01:18:38.760 align:middle line:84% There's one more thing I want to mention before we go. 01:18:38.760 --> 01:18:42.300 align:middle line:84% So I kept saying, store it deep in your heart 01:18:42.300 --> 01:18:43.930 align:middle line:90% that we've solved the 4 by 4. 01:18:43.930 --> 01:18:46.860 align:middle line:84% And then use that to believe that we've solved 8 by 8. 01:18:46.860 --> 01:18:49.050 align:middle line:84% Believe that we solved 8 by 8 in order 01:18:49.050 --> 01:18:51.790 align:middle line:84% to believe that we've solved 16 by 16. 01:18:51.790 --> 01:18:54.460 align:middle line:84% But sometimes belief isn't quite enough. 01:18:54.460 --> 01:18:58.600 align:middle line:84% And you really want to actually run this algorithm. 01:18:58.600 --> 01:19:04.140 align:middle line:84% Well, for then, all you have to do is when we have the 16 by 16, 01:19:04.140 --> 01:19:07.210 align:middle line:84% and you break it down into 8 by 8s, well, 01:19:07.210 --> 01:19:09.730 align:middle line:84% let's actually remember how we solved the 8 by 8s. 01:19:09.730 --> 01:19:12.380 align:middle line:84% We did that by splitting them up into 4 by 4s. 01:19:12.380 --> 01:19:14.880 align:middle line:84% And then turns out you can do the same thing and split the 4 01:19:14.880 --> 01:19:16.660 align:middle line:90% by 4s into 2 by 2s. 01:19:16.660 --> 01:19:19.570 align:middle line:84% So now each 2 by 2 is missing a single cell. 01:19:19.570 --> 01:19:23.150 align:middle line:84% But a 2 by 2 missing a single cell is just an L So it's all 01:19:23.150 --> 01:19:23.650 align:middle line:90% L's. 01:19:23.650 --> 01:19:26.192 align:middle line:84% And then we just recombine, reverse what we did before, 01:19:26.192 --> 01:19:28.150 align:middle line:84% and now we've tiled everything, and we're done. 01:19:28.150 --> 01:19:29.140 align:middle line:90% And that's the algorithm. 01:19:29.140 --> 01:19:29.890 align:middle line:90% Thank you so much. 01:19:29.890 --> 01:19:33.140 align:middle line:90% [APPLAUSE] 01:19:33.140 --> 01:19:38.000 align:middle line:90%