WEBVTT 00:00:00.000 --> 00:00:02.470 align:middle line:90% [SQUEAKING] 00:00:02.470 --> 00:00:04.446 align:middle line:90% [RUSTLING] 00:00:04.446 --> 00:00:07.410 align:middle line:90% [CLICKING] 00:00:07.410 --> 00:00:09.647 align:middle line:90% 00:00:09.647 --> 00:00:10.980 align:middle line:90% Zachary Abel: Thank you so much. 00:00:10.980 --> 00:00:13.760 align:middle line:90% Welcome to 6.1200, 6.120. 00:00:13.760 --> 00:00:15.510 align:middle line:84% I don't know how you want to pronounce it. 00:00:15.510 --> 00:00:16.190 align:middle line:90% Anything goes. 00:00:16.190 --> 00:00:18.080 align:middle line:90% Either way, welcome. 00:00:18.080 --> 00:00:20.910 align:middle line:84% Really looking forward to a great semester with all of you. 00:00:20.910 --> 00:00:26.060 align:middle line:84% So a couple salient points, structure-wise, schedule-wise, 00:00:26.060 --> 00:00:29.520 align:middle line:84% we have lectures on Tuesday, Thursday, 2:30 to 4:00, 00:00:29.520 --> 00:00:32.240 align:middle line:84% which I suppose you know if you're here in person. 00:00:32.240 --> 00:00:35.430 align:middle line:84% We have recitations on Wednesdays and Fridays. 00:00:35.430 --> 00:00:37.530 align:middle line:84% We do take attendance at recitations. 00:00:37.530 --> 00:00:38.820 align:middle line:90% It's worth 10% of your grade. 00:00:38.820 --> 00:00:41.490 align:middle line:84% So we highly recommend that you go, both for your grade. 00:00:41.490 --> 00:00:43.520 align:middle line:84% And also, it's some of the most useful time 00:00:43.520 --> 00:00:45.470 align:middle line:90% that you'll have this semester. 00:00:45.470 --> 00:00:47.240 align:middle line:84% You'll be working with your peers 00:00:47.240 --> 00:00:49.260 align:middle line:84% to solve problems in small groups. 00:00:49.260 --> 00:00:52.680 align:middle line:84% You'll have a TA there to offer advice and suggestions. 00:00:52.680 --> 00:00:57.030 align:middle line:84% It's really valuable time, and I highly recommend you go. 00:00:57.030 --> 00:01:00.630 align:middle line:84% I guarantee you'll get a lot out of it. 00:01:00.630 --> 00:01:03.879 align:middle line:84% Since it's such a valuable time and such a limited resource, 00:01:03.879 --> 00:01:04.379 align:middle line:90% Just. 00:01:04.379 --> 00:01:09.450 align:middle line:84% Two hours per week, we have what we call warmup problems, which 00:01:09.450 --> 00:01:13.860 align:middle line:84% are just quick, multiple choice problems right there on Canvas, 00:01:13.860 --> 00:01:17.460 align:middle line:84% to get you thinking about the lecture material a little bit. 00:01:17.460 --> 00:01:19.030 align:middle line:90% They're not meant to be hard. 00:01:19.030 --> 00:01:21.300 align:middle line:84% They're not meant to be stressful. 00:01:21.300 --> 00:01:23.970 align:middle line:84% But we do want you to think about the problems a little bit 00:01:23.970 --> 00:01:26.610 align:middle line:84% and answer them and get the answers right. 00:01:26.610 --> 00:01:29.400 align:middle line:84% Getting the answer right is not as strenuous as it sounds. 00:01:29.400 --> 00:01:32.550 align:middle line:84% Because you get infinite retries and instant feedback up 00:01:32.550 --> 00:01:33.750 align:middle line:90% until the deadline. 00:01:33.750 --> 00:01:35.680 align:middle line:84% So as long as you remember to do it, 00:01:35.680 --> 00:01:38.040 align:middle line:90% you're going to get full points. 00:01:38.040 --> 00:01:41.608 align:middle line:84% You could, if you want, just option A, click. 00:01:41.608 --> 00:01:42.400 align:middle line:90% Nope, that's wrong. 00:01:42.400 --> 00:01:43.228 align:middle line:90% Option B, click. 00:01:43.228 --> 00:01:44.020 align:middle line:90% Nope, that's wrong. 00:01:44.020 --> 00:01:46.090 align:middle line:84% Option C-- we don't recommend that. 00:01:46.090 --> 00:01:48.930 align:middle line:84% Ideally, it's in your favor to actually think 00:01:48.930 --> 00:01:51.850 align:middle line:84% about the problem a little bit, might even be a little faster. 00:01:51.850 --> 00:01:54.390 align:middle line:84% But the point is to be a reminder 00:01:54.390 --> 00:01:56.520 align:middle line:84% to think about the lecture material a little bit 00:01:56.520 --> 00:01:58.400 align:middle line:90% before you come to recitation. 00:01:58.400 --> 00:02:00.878 align:middle line:84% What we want to avoid is everyone shows up to recitation 00:02:00.878 --> 00:02:02.670 align:middle line:84% and says, wait, what are we covering today? 00:02:02.670 --> 00:02:03.450 align:middle line:90% I don't even know. 00:02:03.450 --> 00:02:04.710 align:middle line:90% I wasn't able to go to lecture. 00:02:04.710 --> 00:02:05.960 align:middle line:90% I don't know what was covered. 00:02:05.960 --> 00:02:07.640 align:middle line:90% Can someone fill me in? 00:02:07.640 --> 00:02:11.700 align:middle line:84% I would prefer that we do the catching up before recitation. 00:02:11.700 --> 00:02:13.618 align:middle line:84% So by the time we get there, we know 00:02:13.618 --> 00:02:14.910 align:middle line:90% what we're going to talk about. 00:02:14.910 --> 00:02:16.590 align:middle line:84% We can ask any questions we have. 00:02:16.590 --> 00:02:18.140 align:middle line:84% We can start working on the problems 00:02:18.140 --> 00:02:22.100 align:middle line:84% and use that time as productively as possible. 00:02:22.100 --> 00:02:23.850 align:middle line:84% So that's recitations and warmups. 00:02:23.850 --> 00:02:25.080 align:middle line:90% We have problem sets. 00:02:25.080 --> 00:02:28.140 align:middle line:84% We'll have weekly problem sets, due on Monday nights. 00:02:28.140 --> 00:02:31.100 align:middle line:84% I hope everyone turned it in yesterday. 00:02:31.100 --> 00:02:32.670 align:middle line:90% They'll be released on Tuesdays. 00:02:32.670 --> 00:02:33.750 align:middle line:90% Oh, I got a thumbs up. 00:02:33.750 --> 00:02:37.040 align:middle line:90% Where did you send that? 00:02:37.040 --> 00:02:38.760 align:middle line:84% We release problem sets Tuesdays. 00:02:38.760 --> 00:02:41.550 align:middle line:84% They'll be due on Monday nights, 11:59 PM, 00:02:41.550 --> 00:02:46.130 align:middle line:84% so as Monday night as we can get. 00:02:46.130 --> 00:02:50.420 align:middle line:84% And I want to say a few words about these problem sets, about 00:02:50.420 --> 00:02:52.410 align:middle line:84% the structure and about the collaboration. 00:02:52.410 --> 00:02:53.550 align:middle line:90% We want you to collaborate. 00:02:53.550 --> 00:02:55.140 align:middle line:84% We want you to work with your peers. 00:02:55.140 --> 00:02:57.560 align:middle line:84% We want you to get help from us in office hours 00:02:57.560 --> 00:02:59.280 align:middle line:84% and on Piazza and things like that. 00:02:59.280 --> 00:03:02.010 align:middle line:84% We want you to solve these problem sets collaboratively, 00:03:02.010 --> 00:03:04.910 align:middle line:90% if you so choose. 00:03:04.910 --> 00:03:07.140 align:middle line:84% We are signed up for pset partners. 00:03:07.140 --> 00:03:10.520 align:middle line:84% So once a week, shortly after the problem sets go out, 00:03:10.520 --> 00:03:12.680 align:middle line:84% the pset partner site will match you 00:03:12.680 --> 00:03:16.183 align:middle line:84% with other problem set solvers that have similar schedules. 00:03:16.183 --> 00:03:17.850 align:middle line:84% So if you want to find a group that way, 00:03:17.850 --> 00:03:19.020 align:middle line:90% you're more than welcome to. 00:03:19.020 --> 00:03:21.980 align:middle line:84% We'll set it to go off a couple days after each problem 00:03:21.980 --> 00:03:22.800 align:middle line:90% set releases. 00:03:22.800 --> 00:03:24.360 align:middle line:84% So just be sure to sign up on time. 00:03:24.360 --> 00:03:27.500 align:middle line:84% If you miss the first round, you can sign up in the second round. 00:03:27.500 --> 00:03:29.805 align:middle line:84% Or you can come to office hours and find friends there. 00:03:29.805 --> 00:03:31.430 align:middle line:84% You can work with friends at your dorm. 00:03:31.430 --> 00:03:33.680 align:middle line:84% However you want to find partners, go ahead and do it, 00:03:33.680 --> 00:03:34.580 align:middle line:90% if you want. 00:03:34.580 --> 00:03:37.250 align:middle line:84% And we encourage you to solve these problems correctly-- 00:03:37.250 --> 00:03:39.480 align:middle line:84% or sorry, collaboratively-- also, correctly-- 00:03:39.480 --> 00:03:41.296 align:middle line:90% [LAUGHTER] 00:03:41.296 --> 00:03:43.970 align:middle line:90% --not the worst Freudian slip. 00:03:43.970 --> 00:03:45.840 align:middle line:84% We encourage you to solve collaboratively. 00:03:45.840 --> 00:03:46.735 align:middle line:90% Figure things out. 00:03:46.735 --> 00:03:48.360 align:middle line:84% Some of these problems are challenging. 00:03:48.360 --> 00:03:52.110 align:middle line:84% Some of them have tricks that you might not see immediately, 00:03:52.110 --> 00:03:54.650 align:middle line:84% and you have to figure out what's going on. 00:03:54.650 --> 00:03:58.970 align:middle line:84% However, there's a second step to the problem set process 00:03:58.970 --> 00:04:00.420 align:middle line:84% that I want to call attention to. 00:04:00.420 --> 00:04:02.240 align:middle line:90% And that's the writing step. 00:04:02.240 --> 00:04:05.070 align:middle line:84% One of the main goals of this course, 00:04:05.070 --> 00:04:06.960 align:middle line:84% one of the main exports for this course, 00:04:06.960 --> 00:04:12.350 align:middle line:84% is proof writing, the ability to write precise, concise, correct 00:04:12.350 --> 00:04:14.310 align:middle line:90% mathematical arguments. 00:04:14.310 --> 00:04:17.209 align:middle line:84% And that is a skill that we will be practicing and developing 00:04:17.209 --> 00:04:18.660 align:middle line:90% all semester. 00:04:18.660 --> 00:04:21.390 align:middle line:84% And everyone has a slightly different style. 00:04:21.390 --> 00:04:23.930 align:middle line:84% Everyone thinks about these things differently. 00:04:23.930 --> 00:04:27.530 align:middle line:84% And the act of just constructing a proof, even if it all 00:04:27.530 --> 00:04:30.680 align:middle line:84% makes perfect sense in your head, the act of translating 00:04:30.680 --> 00:04:33.740 align:middle line:84% that into a proof on paper that's 00:04:33.740 --> 00:04:37.070 align:middle line:84% able to be read and consumed and believed by others 00:04:37.070 --> 00:04:39.270 align:middle line:90% is its own special skill. 00:04:39.270 --> 00:04:42.080 align:middle line:84% And we want you to get lots of practice with that, 00:04:42.080 --> 00:04:44.880 align:middle line:84% keyword being "you," not your group. 00:04:44.880 --> 00:04:47.610 align:middle line:84% So go ahead and solve the problems collaboratively. 00:04:47.610 --> 00:04:49.620 align:middle line:84% Whatever notes you've generated, that's fine. 00:04:49.620 --> 00:04:50.280 align:middle line:90% Keep them. 00:04:50.280 --> 00:04:51.430 align:middle line:90% Study from them. 00:04:51.430 --> 00:04:53.430 align:middle line:84% But when you're going to write your problem set, 00:04:53.430 --> 00:04:57.500 align:middle line:84% put those notes aside, please, and piece things together 00:04:57.500 --> 00:05:00.810 align:middle line:84% yourself in your words, showing your understanding. 00:05:00.810 --> 00:05:03.230 align:middle line:90% So solve together, write alone. 00:05:03.230 --> 00:05:05.550 align:middle line:84% Just to emphasize this one more time, 00:05:05.550 --> 00:05:09.590 align:middle line:84% the reason I'm spending all this time belaboring this point 00:05:09.590 --> 00:05:13.970 align:middle line:84% is something I like to call the P versus NP fallacy, which just 00:05:13.970 --> 00:05:16.220 align:middle line:84% says that reading and understanding someone else's 00:05:16.220 --> 00:05:20.540 align:middle line:84% proof is easier than constructing 00:05:20.540 --> 00:05:22.190 align:middle line:90% that proof yourself. 00:05:22.190 --> 00:05:24.710 align:middle line:84% And it's really tempting to think that these two 00:05:24.710 --> 00:05:25.910 align:middle line:90% tasks are equivalent. 00:05:25.910 --> 00:05:28.070 align:middle line:84% When you're studying for a test, you 00:05:28.070 --> 00:05:30.052 align:middle line:84% read through our problem set solutions, 00:05:30.052 --> 00:05:31.260 align:middle line:90% like, yeah, that makes sense. 00:05:31.260 --> 00:05:31.830 align:middle line:90% That makes sense. 00:05:31.830 --> 00:05:32.430 align:middle line:90% That makes sense. 00:05:32.430 --> 00:05:32.930 align:middle line:90% Great. 00:05:32.930 --> 00:05:33.920 align:middle line:90% I'm convinced. 00:05:33.920 --> 00:05:36.290 align:middle line:84% That doesn't immediately translate to, 00:05:36.290 --> 00:05:39.920 align:middle line:84% I therefore could have put it together myself. 00:05:39.920 --> 00:05:42.260 align:middle line:84% So understanding a proof and writing 00:05:42.260 --> 00:05:43.680 align:middle line:90% a proof-- two different skills. 00:05:43.680 --> 00:05:46.070 align:middle line:84% We want you to get practice with both. 00:05:46.070 --> 00:05:48.590 align:middle line:84% Any questions about that, about the intent 00:05:48.590 --> 00:05:51.870 align:middle line:90% behind our collaboration policy? 00:05:51.870 --> 00:05:54.210 align:middle line:90% Awesome. 00:05:54.210 --> 00:05:55.590 align:middle line:90% Let's see. 00:05:55.590 --> 00:05:58.030 align:middle line:84% Late problem sets, last thing I want to talk about, 00:05:58.030 --> 00:06:00.070 align:middle line:90% we do accept late problem sets. 00:06:00.070 --> 00:06:03.060 align:middle line:84% We accept very late problem sets. 00:06:03.060 --> 00:06:06.548 align:middle line:84% So we put a lot of effort into making these problem sets, 00:06:06.548 --> 00:06:08.340 align:middle line:84% at least trying to make these problem sets, 00:06:08.340 --> 00:06:10.540 align:middle line:90% both interesting and useful. 00:06:10.540 --> 00:06:12.540 align:middle line:84% And so we want to incentivize all of you 00:06:12.540 --> 00:06:16.980 align:middle line:84% to work the problem sets as much as you can in as many ways 00:06:16.980 --> 00:06:20.310 align:middle line:84% as you can, even if that means it's late. 00:06:20.310 --> 00:06:21.810 align:middle line:84% If the schedule doesn't line up, you 00:06:21.810 --> 00:06:23.227 align:middle line:84% had to miss some problem sets, We. 00:06:23.227 --> 00:06:26.740 align:middle line:84% Want you to be able to catch up by doing the problem sets. 00:06:26.740 --> 00:06:30.840 align:middle line:84% So we allow late problem sets up to and including 00:06:30.840 --> 00:06:33.420 align:middle line:90% the last day of classes. 00:06:33.420 --> 00:06:36.267 align:middle line:84% If you forget to do problem set one, and you remember, 00:06:36.267 --> 00:06:38.100 align:middle line:84% oh, in the last week, maybe I should go back 00:06:38.100 --> 00:06:40.695 align:middle line:90% and solve those problems-- 00:06:40.695 --> 00:06:42.570 align:middle line:84% probably would have been more useful to do it 00:06:42.570 --> 00:06:46.260 align:middle line:84% before the first quiz, but better late than never. 00:06:46.260 --> 00:06:48.210 align:middle line:84% You can still turn in that problem set late 00:06:48.210 --> 00:06:49.870 align:middle line:90% for half credit. 00:06:49.870 --> 00:06:52.540 align:middle line:84% More precisely, our late problem set 00:06:52.540 --> 00:06:56.020 align:middle line:84% policy is that for the first 50 hours, 00:06:56.020 --> 00:06:57.850 align:middle line:84% the late penalty or the percentage 00:06:57.850 --> 00:06:59.980 align:middle line:84% of points that you will earn from your submission 00:06:59.980 --> 00:07:03.490 align:middle line:84% goes from 100%, if it's on time, down to 50%, 00:07:03.490 --> 00:07:07.850 align:middle line:84% if it's 50 hours late, and 1% each hour for those first 50 00:07:07.850 --> 00:07:10.040 align:middle line:90% hours, those first two-ish days. 00:07:10.040 --> 00:07:13.930 align:middle line:84% And then after that, it stays at 50 for the rest of term. 00:07:13.930 --> 00:07:18.850 align:middle line:84% That also means that if it's 11:45, 00:07:18.850 --> 00:07:21.530 align:middle line:84% and you have one problem left-- you know how to do it. 00:07:21.530 --> 00:07:24.260 align:middle line:84% You just down to the wire for writing it down. 00:07:24.260 --> 00:07:27.640 align:middle line:84% Or you want to take another hour or 30 minutes or something 00:07:27.640 --> 00:07:29.330 align:middle line:90% to think about that problem. 00:07:29.330 --> 00:07:31.220 align:middle line:90% Go ahead and be a little late. 00:07:31.220 --> 00:07:32.428 align:middle line:90% It's not a big deal. 00:07:32.428 --> 00:07:33.970 align:middle line:84% If you're one hour late, you're still 00:07:33.970 --> 00:07:37.280 align:middle line:84% going to get 99% of the points that you would have otherwise. 00:07:37.280 --> 00:07:39.700 align:middle line:84% It's worth it to take that extra time. 00:07:39.700 --> 00:07:43.480 align:middle line:84% It means that there isn't a sharp cutoff 00:07:43.480 --> 00:07:46.040 align:middle line:84% between before the deadline and after the deadline. 00:07:46.040 --> 00:07:47.930 align:middle line:90% It's not a big deal. 00:07:47.930 --> 00:07:49.050 align:middle line:90% Don't worry about it. 00:07:49.050 --> 00:07:51.540 align:middle line:84% We want to reduce stress for everyone involved. 00:07:51.540 --> 00:07:53.065 align:middle line:90% Any questions about that? 00:07:53.065 --> 00:07:56.325 align:middle line:90% 00:07:56.325 --> 00:07:56.825 align:middle line:90% Excellent. 00:07:56.825 --> 00:07:57.900 align:middle line:90% All right. 00:07:57.900 --> 00:08:00.530 align:middle line:84% Like I said, lots more details there on the course information 00:08:00.530 --> 00:08:04.400 align:middle line:90% policies PDF on our Canvas site. 00:08:04.400 --> 00:08:07.970 align:middle line:84% And with that, let's talk about some math. 00:08:07.970 --> 00:08:13.430 align:middle line:84% So as I said earlier, one of the main skills and tools 00:08:13.430 --> 00:08:15.860 align:middle line:84% that we want you to have, coming out of this class, 00:08:15.860 --> 00:08:19.190 align:middle line:84% is the ability to write proofs, and read proofs, and evaluate 00:08:19.190 --> 00:08:20.270 align:middle line:90% proofs. 00:08:20.270 --> 00:08:22.700 align:middle line:84% As one of my colleagues says, this class 00:08:22.700 --> 00:08:26.030 align:middle line:84% could have been called Proofs, Proofs, and More Proofs. 00:08:26.030 --> 00:08:27.900 align:middle line:84% We're going to talk about proofs a lot, 00:08:27.900 --> 00:08:30.410 align:middle line:84% especially today, where we're going to start with, 00:08:30.410 --> 00:08:32.895 align:middle line:90% what is a proof? 00:08:32.895 --> 00:08:35.520 align:middle line:84% And I'm going to write that down because it's important enough. 00:08:35.520 --> 00:08:40.610 align:middle line:90% What is a proof? 00:08:40.610 --> 00:08:42.827 align:middle line:84% And I'm going to give a starting answer. 00:08:42.827 --> 00:08:44.910 align:middle line:84% And we're going to elaborate on this a little bit. 00:08:44.910 --> 00:08:48.390 align:middle line:90% Here's a question and answer. 00:08:48.390 --> 00:08:59.370 align:middle line:84% It's a method of ascertaining truth. 00:08:59.370 --> 00:09:01.910 align:middle line:84% A proof is how you show that something is true. 00:09:01.910 --> 00:09:04.860 align:middle line:90% 00:09:04.860 --> 00:09:07.140 align:middle line:84% And we as a society have lots and lots 00:09:07.140 --> 00:09:10.470 align:middle line:84% of methods for determining what's true, 00:09:10.470 --> 00:09:11.860 align:middle line:90% I'm going to give you a couple. 00:09:11.860 --> 00:09:19.860 align:middle line:84% For example, we have experiments, and observation, 00:09:19.860 --> 00:09:21.900 align:middle line:90% and the scientific method. 00:09:21.900 --> 00:09:25.740 align:middle line:84% That is how we determine what's true in the physical sciences. 00:09:25.740 --> 00:09:27.183 align:middle line:90% We have sampling. 00:09:27.183 --> 00:09:30.300 align:middle line:90% 00:09:30.300 --> 00:09:32.680 align:middle line:84% If you want to know who's likely to win an election, 00:09:32.680 --> 00:09:35.520 align:middle line:84% you pick a random sample, and you ask 00:09:35.520 --> 00:09:37.540 align:middle line:90% who they're going to vote for. 00:09:37.540 --> 00:09:40.320 align:middle line:90% And you generalize. 00:09:40.320 --> 00:09:44.110 align:middle line:90% We have legal proceedings. 00:09:44.110 --> 00:09:46.060 align:middle line:90% You hold a trial-- 00:09:46.060 --> 00:09:48.520 align:middle line:90% two different L's-- who knew? 00:09:48.520 --> 00:09:51.370 align:middle line:84% You hold a trial to determine what happened 00:09:51.370 --> 00:09:52.880 align:middle line:90% and who's culpable. 00:09:52.880 --> 00:09:56.410 align:middle line:84% That is how we determine truth in those cases. 00:09:56.410 --> 00:09:57.650 align:middle line:90% Who can give me some others? 00:09:57.650 --> 00:09:58.920 align:middle line:90% How do we determine truth? 00:09:58.920 --> 00:10:05.290 align:middle line:90% 00:10:05.290 --> 00:10:06.520 align:middle line:90% Yes? 00:10:06.520 --> 00:10:09.710 align:middle line:84% STUDENT: The investigation, like in detective movies. 00:10:09.710 --> 00:10:14.410 align:middle line:84% Zachary Abel: All right, yeah investigation, absolutely, 00:10:14.410 --> 00:10:19.120 align:middle line:84% investigation, looking for evidence, 00:10:19.120 --> 00:10:22.540 align:middle line:84% looking for the most likely explanation that 00:10:22.540 --> 00:10:24.412 align:middle line:84% fits the observation, so actually, 00:10:24.412 --> 00:10:26.120 align:middle line:84% pretty similar to some of the other ones. 00:10:26.120 --> 00:10:26.530 align:middle line:90% Yes? 00:10:26.530 --> 00:10:27.380 align:middle line:90% STUDENT: Asking questions? 00:10:27.380 --> 00:10:28.672 align:middle line:90% Zachary Abel: Asking questions? 00:10:28.672 --> 00:10:29.410 align:middle line:90% Absolutely. 00:10:29.410 --> 00:10:33.450 align:middle line:90% 00:10:33.450 --> 00:10:33.950 align:middle line:90% Cool. 00:10:33.950 --> 00:10:35.760 align:middle line:90% Let me give you a couple more. 00:10:35.760 --> 00:10:40.390 align:middle line:90% 00:10:40.390 --> 00:10:41.650 align:middle line:90% Authority. 00:10:41.650 --> 00:10:45.900 align:middle line:84% If your professor says this is true, then it's true, right? 00:10:45.900 --> 00:10:48.792 align:middle line:90% [LAUGHTER] 00:10:48.792 --> 00:10:49.760 align:middle line:90% 00:10:49.760 --> 00:10:52.490 align:middle line:90% Please don't do that. 00:10:52.490 --> 00:10:54.350 align:middle line:84% We make mistakes all the time, and we want 00:10:54.350 --> 00:10:56.630 align:middle line:90% you to call us out on that. 00:10:56.630 --> 00:10:57.853 align:middle line:90% We have religion. 00:10:57.853 --> 00:11:02.930 align:middle line:90% 00:11:02.930 --> 00:11:04.740 align:middle line:90% This is what our religion says. 00:11:04.740 --> 00:11:06.120 align:middle line:90% This is what our deity says. 00:11:06.120 --> 00:11:09.360 align:middle line:84% Therefore, it is true, and it is how I should live my life. 00:11:09.360 --> 00:11:13.460 align:middle line:84% These are the values I should value. 00:11:13.460 --> 00:11:15.436 align:middle line:90% We have inner conviction. 00:11:15.436 --> 00:11:20.900 align:middle line:90% 00:11:20.900 --> 00:11:23.300 align:middle line:84% I just feel like it should be this way. 00:11:23.300 --> 00:11:25.080 align:middle line:84% Deep in my gut, I know it's true. 00:11:25.080 --> 00:11:27.200 align:middle line:90% It came to me in a dream. 00:11:27.200 --> 00:11:33.110 align:middle line:84% Sometimes, that is how things work; 00:11:33.110 --> 00:11:36.537 align:middle line:84% so lots of different ways of determining truth, 00:11:36.537 --> 00:11:38.120 align:middle line:84% lots of different meanings of the word 00:11:38.120 --> 00:11:41.080 align:middle line:90% truth in those situations. 00:11:41.080 --> 00:11:44.200 align:middle line:84% And as mathematicians, we have our own procedures 00:11:44.200 --> 00:11:46.330 align:middle line:90% for determining truth. 00:11:46.330 --> 00:11:49.360 align:middle line:84% And that is a mathematical proof. 00:11:49.360 --> 00:12:02.080 align:middle line:84% So definition, a mathematical proof 00:12:02.080 --> 00:12:20.284 align:middle line:84% is a verification of a proposition 00:12:20.284 --> 00:12:38.323 align:middle line:84% by a chain of logical deductions from a base set of axioms. 00:12:38.323 --> 00:12:42.590 align:middle line:90% 00:12:42.590 --> 00:12:44.540 align:middle line:84% A mathematical proof is a verification 00:12:44.540 --> 00:12:47.930 align:middle line:84% of a proposition by a chain of logical deductions from a base 00:12:47.930 --> 00:12:49.470 align:middle line:90% set of axioms. 00:12:49.470 --> 00:12:52.190 align:middle line:84% And there are three terms inside of that definition 00:12:52.190 --> 00:12:54.930 align:middle line:84% that we're going to focus on today and tomorrow. 00:12:54.930 --> 00:12:56.660 align:middle line:84% We'll break them down one at a time 00:12:56.660 --> 00:12:59.840 align:middle line:84% and hopefully, make this make more sense. 00:12:59.840 --> 00:13:02.340 align:middle line:90% Let's start with proposition. 00:13:02.340 --> 00:13:03.586 align:middle line:90% What is a proposition? 00:13:03.586 --> 00:13:07.730 align:middle line:90% 00:13:07.730 --> 00:13:25.925 align:middle line:84% Definition-- a proposition is a statement that is true or false. 00:13:25.925 --> 00:13:29.102 align:middle line:90% 00:13:29.102 --> 00:13:30.750 align:middle line:90% Today is Tuesday. 00:13:30.750 --> 00:13:33.050 align:middle line:90% Proposition? 00:13:33.050 --> 00:13:34.760 align:middle line:90% STUDENT: Yeah. 00:13:34.760 --> 00:13:36.800 align:middle line:84% Zachary Abel: Oh, that is true, yeah. 00:13:36.800 --> 00:13:38.880 align:middle line:90% Today is Thursday. 00:13:38.880 --> 00:13:40.067 align:middle line:90% Proposition? 00:13:40.067 --> 00:13:40.650 align:middle line:90% STUDENT: Yeah. 00:13:40.650 --> 00:13:41.470 align:middle line:90% Zachary Abel: Yeah. 00:13:41.470 --> 00:13:43.306 align:middle line:84% It's false, but it's still a proposition. 00:13:43.306 --> 00:13:47.610 align:middle line:90% 00:13:47.610 --> 00:13:49.440 align:middle line:90% Let's see, examples-- 00:13:49.440 --> 00:13:52.290 align:middle line:90% 2 plus 3 equals 5. 00:13:52.290 --> 00:13:54.450 align:middle line:90% Yeah, proposition. 00:13:54.450 --> 00:13:59.760 align:middle line:84% 2 plus 3 equals 4, still a proposition. 00:13:59.760 --> 00:14:01.680 align:middle line:90% Here's one. 00:14:01.680 --> 00:14:07.410 align:middle line:84% For all n in the natural numbers, 00:14:07.410 --> 00:14:11.540 align:middle line:84% n squared plus n plus 41 is prime. 00:14:11.540 --> 00:14:14.910 align:middle line:90% 00:14:14.910 --> 00:14:16.180 align:middle line:90% This is new notation. 00:14:16.180 --> 00:14:17.200 align:middle line:90% Let's break it down. 00:14:17.200 --> 00:14:20.800 align:middle line:90% This says, for all. 00:14:20.800 --> 00:14:24.850 align:middle line:84% So for every n that is in the natural numbers, 00:14:24.850 --> 00:14:28.140 align:middle line:84% this, what's called, blackboard bold capital N 00:14:28.140 --> 00:14:39.910 align:middle line:84% is the natural numbers, also known as the set 0, 1, 2, 3 00:14:39.910 --> 00:14:40.900 align:middle line:90% and all the way up. 00:14:40.900 --> 00:14:43.600 align:middle line:84% It starts at 0, starts counting upwards. 00:14:43.600 --> 00:14:47.050 align:middle line:84% Outside of this class, you might see 00:14:47.050 --> 00:14:51.610 align:middle line:84% people trying to convince you that 0 is not a natural number. 00:14:51.610 --> 00:14:53.050 align:middle line:90% They're wrong. 00:14:53.050 --> 00:14:54.820 align:middle line:84% At least for the purposes of this class, 00:14:54.820 --> 00:14:56.475 align:middle line:90% 0 is always a natural number. 00:14:56.475 --> 00:14:59.240 align:middle line:90% 00:14:59.240 --> 00:15:03.160 align:middle line:84% So every natural number for every natural number n, n 00:15:03.160 --> 00:15:06.340 align:middle line:90% squared plus n plus 41 is prime. 00:15:06.340 --> 00:15:11.080 align:middle line:90% Is this a proposition? 00:15:11.080 --> 00:15:15.190 align:middle line:84% Yeah, it's a statement where it at least makes sense to ask, 00:15:15.190 --> 00:15:18.280 align:middle line:90% is this true or false? 00:15:18.280 --> 00:15:20.660 align:middle line:84% There's a related concept, by the way. 00:15:20.660 --> 00:15:24.920 align:middle line:84% So this thing in here, just n squared plus n plus 41 is prime, 00:15:24.920 --> 00:15:26.096 align:middle line:90% is that a proposition? 00:15:26.096 --> 00:15:29.410 align:middle line:90% 00:15:29.410 --> 00:15:33.340 align:middle line:84% Let me give an even easier example. 00:15:33.340 --> 00:15:36.990 align:middle line:84% Oh, my examples were already on that board. 00:15:36.990 --> 00:15:37.490 align:middle line:90% Sorry. 00:15:37.490 --> 00:15:41.450 align:middle line:90% 00:15:41.450 --> 00:15:41.960 align:middle line:90% Even 00:15:41.960 --> 00:15:46.640 align:middle line:90% Easier example, p is prime. 00:15:46.640 --> 00:15:49.400 align:middle line:90% Is that a proposition? 00:15:49.400 --> 00:15:51.330 align:middle line:90% I'm seeing some thumbs down. 00:15:51.330 --> 00:15:52.770 align:middle line:90% I'm seeing some thumbs down. 00:15:52.770 --> 00:15:57.920 align:middle line:84% Yeah, it's not a proposition, not a proposition. 00:15:57.920 --> 00:16:00.240 align:middle line:84% Because it's incomplete information. 00:16:00.240 --> 00:16:01.980 align:middle line:90% What p are we talking about? 00:16:01.980 --> 00:16:05.870 align:middle line:84% If you tell me a p is 7 prime, that's a proposition; is 00:16:05.870 --> 00:16:08.070 align:middle line:90% 12 prime, that's a proposition. 00:16:08.070 --> 00:16:11.090 align:middle line:84% So this is sort of a proposition with a variable 00:16:11.090 --> 00:16:14.190 align:middle line:84% that we need to fill in before it becomes a proposition. 00:16:14.190 --> 00:16:17.150 align:middle line:84% This is what's called a predicate. 00:16:17.150 --> 00:16:18.846 align:middle line:90% This is a predicate. 00:16:18.846 --> 00:16:22.958 align:middle line:90% 00:16:22.958 --> 00:16:24.500 align:middle line:84% And I was bringing back the old board 00:16:24.500 --> 00:16:26.116 align:middle line:84% so I could write down that definition. 00:16:26.116 --> 00:16:31.280 align:middle line:90% 00:16:31.280 --> 00:16:44.100 align:middle line:84% A predicate is a proposition whose truth 00:16:44.100 --> 00:16:47.776 align:middle line:90% depends on variables. 00:16:47.776 --> 00:16:52.110 align:middle line:90% 00:16:52.110 --> 00:16:54.550 align:middle line:84% So it's a parametrized proposition. 00:16:54.550 --> 00:16:56.650 align:middle line:84% Once you tell me what the variables are, 00:16:56.650 --> 00:16:58.710 align:middle line:90% then it's a proposition. 00:16:58.710 --> 00:17:03.006 align:middle line:84% And we just saw an example up here. 00:17:03.006 --> 00:17:05.339 align:middle line:84% I hope no one's getting dizzy from all the boards moving 00:17:05.339 --> 00:17:07.200 align:middle line:90% around. 00:17:07.200 --> 00:17:11.790 align:middle line:84% We just saw an example here, where this inside is 00:17:11.790 --> 00:17:14.490 align:middle line:90% a predicate that depends on n. 00:17:14.490 --> 00:17:18.780 align:middle line:84% And this for all is able to take that predicate depending on n 00:17:18.780 --> 00:17:22.900 align:middle line:84% and wrap it up into what is now a proposition. 00:17:22.900 --> 00:17:25.349 align:middle line:84% It says that this predicate about n 00:17:25.349 --> 00:17:27.760 align:middle line:84% needs to be true for every natural number. 00:17:27.760 --> 00:17:29.110 align:middle line:90% And now it's a proposition. 00:17:29.110 --> 00:17:32.410 align:middle line:90% It's either true or false. 00:17:32.410 --> 00:17:33.860 align:middle line:90% Is it true or false? 00:17:33.860 --> 00:17:35.540 align:middle line:84% Let's see if we can figure it out. 00:17:35.540 --> 00:17:36.950 align:middle line:90% We know it's a proposition. 00:17:36.950 --> 00:17:37.790 align:middle line:90% But is it true? 00:17:37.790 --> 00:17:39.760 align:middle line:90% Is it false? 00:17:39.760 --> 00:17:44.150 align:middle line:84% Well, let's do the natural thing and try some examples. 00:17:44.150 --> 00:17:46.270 align:middle line:90% OK, well, here's n. 00:17:46.270 --> 00:17:49.460 align:middle line:90% Here's n squared plus n plus 41. 00:17:49.460 --> 00:17:52.820 align:middle line:90% Is it prime? 00:17:52.820 --> 00:17:53.320 align:middle line:90% Great. 00:17:53.320 --> 00:17:55.850 align:middle line:84% So natural numbers, remember, start at 0. 00:17:55.850 --> 00:17:58.840 align:middle line:90% So when n is 0, this is 41-- 00:17:58.840 --> 00:18:00.790 align:middle line:90% absolutely prime. 00:18:00.790 --> 00:18:05.180 align:middle line:84% When n is 1, this is 43, still prime. 00:18:05.180 --> 00:18:06.260 align:middle line:90% Looking good so far. 00:18:06.260 --> 00:18:07.310 align:middle line:90% How about 2? 00:18:07.310 --> 00:18:11.710 align:middle line:90% I think we get 47, also prime. 00:18:11.710 --> 00:18:15.940 align:middle line:84% And if you check 3, 4, 5, 6, prime, prime, prime, prime. 00:18:15.940 --> 00:18:18.490 align:middle line:90% It keeps being prime. 00:18:18.490 --> 00:18:21.590 align:middle line:84% What if we go all the way up to 39? 00:18:21.590 --> 00:18:25.900 align:middle line:84% We get 1,601, if my notes are to be believed, 00:18:25.900 --> 00:18:28.630 align:middle line:90% which is also prime. 00:18:28.630 --> 00:18:31.650 align:middle line:84% This is looking pretty convincing, right? 00:18:31.650 --> 00:18:33.450 align:middle line:90% There we go. 00:18:33.450 --> 00:18:36.780 align:middle line:84% And in some fields of study, just doing a sufficient number 00:18:36.780 --> 00:18:40.680 align:middle line:84% of examples would be enough evidence for it to be true, 00:18:40.680 --> 00:18:43.450 align:middle line:84% or effectively true, or as good as true. 00:18:43.450 --> 00:18:45.550 align:middle line:84% Unfortunately, for us mathematicians, 00:18:45.550 --> 00:18:49.450 align:middle line:84% if we say for all n, we need all of the n. 00:18:49.450 --> 00:18:51.725 align:middle line:84% Just these 40 examples isn't enough. 00:18:51.725 --> 00:18:55.260 align:middle line:90% 00:18:55.260 --> 00:18:59.430 align:middle line:84% And of course, if you go one more-- 00:18:59.430 --> 00:19:00.820 align:middle line:90% did I write this down? 00:19:00.820 --> 00:19:02.790 align:middle line:90% Oh, here. 00:19:02.790 --> 00:19:04.660 align:middle line:90% Let's jump to 41. 00:19:04.660 --> 00:19:07.530 align:middle line:84% 41 looks a good example because that's 00:19:07.530 --> 00:19:12.060 align:middle line:84% going to be a multiple of 41, multiple of 41, multiple of 41. 00:19:12.060 --> 00:19:21.420 align:middle line:84% It turns out, what you get is 41 times 43, which is not prime. 00:19:21.420 --> 00:19:25.000 align:middle line:84% And so we found an example where this is not prime. 00:19:25.000 --> 00:19:31.040 align:middle line:84% And therefore, this proposition is false. 00:19:31.040 --> 00:19:34.200 align:middle line:84% If it's supposed to be true for every natural number, 00:19:34.200 --> 00:19:35.990 align:middle line:84% to disprove it, all we need to find 00:19:35.990 --> 00:19:39.620 align:middle line:84% is a single counterexample, a single number n 00:19:39.620 --> 00:19:43.100 align:middle line:84% that doesn't satisfy the required conditions. 00:19:43.100 --> 00:19:45.020 align:middle line:90% Does that make sense? 00:19:45.020 --> 00:19:45.620 align:middle line:90% Cool. 00:19:45.620 --> 00:19:48.020 align:middle line:84% Turns out, if you go to 40, I'm pretty 00:19:48.020 --> 00:19:52.130 align:middle line:84% sure you get 41 squared, which is also not prime. 00:19:52.130 --> 00:19:55.490 align:middle line:84% So 0 through 39 are really the most 00:19:55.490 --> 00:19:59.100 align:middle line:90% you're going to get in that run. 00:19:59.100 --> 00:20:05.210 align:middle line:84% Let's look at another example of a fun proposition, 00:20:05.210 --> 00:20:07.890 align:middle line:84% just to get more familiar with this. 00:20:07.890 --> 00:20:09.636 align:middle line:90% Let's see Goldbach's Conjecture. 00:20:09.636 --> 00:20:19.250 align:middle line:90% 00:20:19.250 --> 00:20:22.370 align:middle line:90% Let's see. 00:20:22.370 --> 00:20:33.730 align:middle line:84% Every even number greater than 2 is the sum of two primes. 00:20:33.730 --> 00:20:38.720 align:middle line:90% 00:20:38.720 --> 00:20:41.580 align:middle line:84% Every even number greater than 2 is the sum of two primes. 00:20:41.580 --> 00:20:43.405 align:middle line:90% For example, can we do 12? 00:20:43.405 --> 00:20:47.210 align:middle line:90% 00:20:47.210 --> 00:20:48.470 align:middle line:90% Who can tell me? 00:20:48.470 --> 00:20:49.050 align:middle line:90% Let's see. 00:20:49.050 --> 00:20:49.330 align:middle line:90% Yes? 00:20:49.330 --> 00:20:50.080 align:middle line:90% STUDENT: 7 plus 5. 00:20:50.080 --> 00:20:51.500 align:middle line:84% Zachary Abel: 7 plus 5, excellent. 00:20:51.500 --> 00:20:55.800 align:middle line:84% 7 plus 5, prime plus prime, so 12 is a sum of two primes. 00:20:55.800 --> 00:20:58.850 align:middle line:84% The claim is that every even number starting at 4 00:20:58.850 --> 00:21:02.630 align:middle line:84% can be written as the sum of two primes. 00:21:02.630 --> 00:21:05.240 align:middle line:84% And it's called a conjecture because we 00:21:05.240 --> 00:21:08.030 align:middle line:90% don't know if it's true. 00:21:08.030 --> 00:21:12.900 align:middle line:84% I love, in math, and often, especially in number theory, 00:21:12.900 --> 00:21:17.880 align:middle line:84% you can state these very simply stated conjectures, theorems, 00:21:17.880 --> 00:21:22.680 align:middle line:84% questions, that are just beyond human understanding. 00:21:22.680 --> 00:21:25.950 align:middle line:84% They're beyond the scope of what modern mathematical tools are 00:21:25.950 --> 00:21:27.820 align:middle line:90% able to tackle. 00:21:27.820 --> 00:21:29.850 align:middle line:84% And it's really pretty, that right here 00:21:29.850 --> 00:21:31.510 align:middle line:84% is beyond the cutting edge of math. 00:21:31.510 --> 00:21:37.110 align:middle line:84% We've already gotten their first lecture, yeah, really pretty 00:21:37.110 --> 00:21:37.990 align:middle line:90% problem. 00:21:37.990 --> 00:21:44.070 align:middle line:84% And in fact, in, I think, 1995, it was highlighted in The Globe 00:21:44.070 --> 00:21:48.155 align:middle line:84% as one of the biggest unknown questions in math, 00:21:48.155 --> 00:21:52.170 align:middle line:84% right here, Goldbach's Conjecture. 00:21:52.170 --> 00:21:57.540 align:middle line:84% For example, 20 can be expressed as 9 plus 11. 00:21:57.540 --> 00:21:59.105 align:middle line:90% Oops, 9 isn't prime. 00:21:59.105 --> 00:22:04.180 align:middle line:90% 00:22:04.180 --> 00:22:07.620 align:middle line:84% So have we disproven Goldbach's conjecture? 00:22:07.620 --> 00:22:08.280 align:middle line:90% No. 00:22:08.280 --> 00:22:08.920 align:middle line:90% Why not? 00:22:08.920 --> 00:22:09.992 align:middle line:90% Yes? 00:22:09.992 --> 00:22:12.120 align:middle line:90% Yeah, it's 17 plus 3. 00:22:12.120 --> 00:22:19.450 align:middle line:84% 20 is 17 plus 3; so really pretty question, 00:22:19.450 --> 00:22:21.600 align:middle line:84% wish the reporting had been better. 00:22:21.600 --> 00:22:24.880 align:middle line:90% But we can forgive them. 00:22:24.880 --> 00:22:27.820 align:middle line:84% Are there any questions about propositions and predicates 00:22:27.820 --> 00:22:28.940 align:middle line:90% so far? 00:22:28.940 --> 00:22:30.370 align:middle line:90% Yes? 00:22:30.370 --> 00:22:33.310 align:middle line:84% STUDENT: The p is prime, which is not a proposition. 00:22:33.310 --> 00:22:37.660 align:middle line:84% And define predicate as a proposition? 00:22:37.660 --> 00:22:39.920 align:middle line:84% Zachary Abel: Yeah, that's a very good point. 00:22:39.920 --> 00:22:41.845 align:middle line:84% That wording is maybe not great-- 00:22:41.845 --> 00:22:45.526 align:middle line:90% 00:22:45.526 --> 00:22:54.345 align:middle line:84% is a proposition that depends on variables except that-- 00:22:54.345 --> 00:22:56.890 align:middle line:90% 00:22:56.890 --> 00:22:59.770 align:middle line:90% it depends on variables. 00:22:59.770 --> 00:23:02.170 align:middle line:84% It becomes a proposition after you give values 00:23:02.170 --> 00:23:03.290 align:middle line:90% to the variables. 00:23:03.290 --> 00:23:06.165 align:middle line:84% Yeah, I agree we should fix that for the next go around. 00:23:06.165 --> 00:23:08.800 align:middle line:90% 00:23:08.800 --> 00:23:10.150 align:middle line:90% All right. 00:23:10.150 --> 00:23:14.020 align:middle line:84% Now that we have propositions and predicates, 00:23:14.020 --> 00:23:17.140 align:middle line:84% really useful thing is to be able to combine them 00:23:17.140 --> 00:23:19.870 align:middle line:84% into bigger and more complicated and new 00:23:19.870 --> 00:23:22.110 align:middle line:90% propositions and predicates. 00:23:22.110 --> 00:23:23.850 align:middle line:84% And we have a couple, what are called. 00:23:23.850 --> 00:23:26.810 align:middle line:84% Boolean operators, that help us do that. 00:23:26.810 --> 00:23:37.070 align:middle line:84% For example, if A and B are propositions, propositions, 00:23:37.070 --> 00:23:43.670 align:middle line:84% we have what's called not A. This is a new proposition. 00:23:43.670 --> 00:23:49.200 align:middle line:84% It's sometimes written instead as A with a bar on top. 00:23:49.200 --> 00:23:53.270 align:middle line:84% It's sometimes also written as that weird right angle 00:23:53.270 --> 00:23:56.060 align:middle line:90% symbol next to the A. 00:23:56.060 --> 00:23:57.630 align:middle line:90% These all mean the same thing. 00:23:57.630 --> 00:24:00.500 align:middle line:84% They're just different notations for the same thing. 00:24:00.500 --> 00:24:06.290 align:middle line:84% And this is just the proposition that A is false. 00:24:06.290 --> 00:24:09.590 align:middle line:90% A is false. 00:24:09.590 --> 00:24:13.290 align:middle line:84% So you read it as not A. And it means A is false. 00:24:13.290 --> 00:24:17.600 align:middle line:84% We can clarify this with what's called a truth table. 00:24:17.600 --> 00:24:20.360 align:middle line:84% So here's A. And A is a proposition, 00:24:20.360 --> 00:24:23.240 align:middle line:84% which means there are really only two possibilities. 00:24:23.240 --> 00:24:25.840 align:middle line:90% It's either true or it's false. 00:24:25.840 --> 00:24:28.850 align:middle line:84% And so let's write down both of those options. 00:24:28.850 --> 00:24:31.100 align:middle line:90% It can be either true or false. 00:24:31.100 --> 00:24:35.140 align:middle line:84% And then we can look at not A and ask, 00:24:35.140 --> 00:24:37.640 align:middle line:90% what is not A when A is true? 00:24:37.640 --> 00:24:40.630 align:middle line:84% Well not A is supposed to flip it, so not A is false. 00:24:40.630 --> 00:24:44.800 align:middle line:84% And when A is false, not A is true. 00:24:44.800 --> 00:24:49.240 align:middle line:84% So this truth table can be taken as the definition 00:24:49.240 --> 00:24:59.030 align:middle line:90% of not, so truth table. 00:24:59.030 --> 00:25:01.395 align:middle line:84% And let's do some more interesting truth tables. 00:25:01.395 --> 00:25:06.730 align:middle line:90% 00:25:06.730 --> 00:25:08.560 align:middle line:90% Let's use two propositions now. 00:25:08.560 --> 00:25:16.256 align:middle line:84% If we have A and B, well, then there are four possibilities. 00:25:16.256 --> 00:25:20.960 align:middle line:90% 00:25:20.960 --> 00:25:22.770 align:middle line:90% They might both be true. 00:25:22.770 --> 00:25:25.520 align:middle line:84% A might be true, and B is false, or false 00:25:25.520 --> 00:25:26.970 align:middle line:90% and true or false and false. 00:25:26.970 --> 00:25:29.610 align:middle line:84% And our truth table is supposed to cover all the possibilities. 00:25:29.610 --> 00:25:31.950 align:middle line:84% So we're going to have four rows this time. 00:25:31.950 --> 00:25:38.600 align:middle line:84% And let's look at A and B. This is a new operator. 00:25:38.600 --> 00:25:41.120 align:middle line:84% And it means what you think it means because it's also 00:25:41.120 --> 00:25:41.840 align:middle line:90% an English word. 00:25:41.840 --> 00:25:45.740 align:middle line:84% A and B means that both A and B are supposed to be true. 00:25:45.740 --> 00:25:51.290 align:middle line:84% So if A and B are both true, then A and B is true. 00:25:51.290 --> 00:25:53.185 align:middle line:90% Otherwise, it's false. 00:25:53.185 --> 00:25:55.825 align:middle line:90% 00:25:55.825 --> 00:26:00.200 align:middle line:84% A and B means both A and B have to be true. 00:26:00.200 --> 00:26:08.090 align:middle line:84% Likewise, we have A or B, which means at least one of A and B 00:26:08.090 --> 00:26:09.950 align:middle line:90% has to be true. 00:26:09.950 --> 00:26:12.470 align:middle line:84% So if they're both true, that's fine. 00:26:12.470 --> 00:26:15.630 align:middle line:90% If only A is true, that's fine. 00:26:15.630 --> 00:26:17.500 align:middle line:90% If only B is true, that's fine. 00:26:17.500 --> 00:26:22.920 align:middle line:84% The only way for this to fail is if both A and B are false. 00:26:22.920 --> 00:26:27.310 align:middle line:84% So I want to emphasize this is an inclusive OR. 00:26:27.310 --> 00:26:29.560 align:middle line:84% It's fine if both of them are true. 00:26:29.560 --> 00:26:34.800 align:middle line:84% It just means that at least one of them has to be true. 00:26:34.800 --> 00:26:41.400 align:middle line:84% Notation-wise, A and B is also sometimes denoted 00:26:41.400 --> 00:26:45.870 align:middle line:84% A and this little wedge symbol, B. 00:26:45.870 --> 00:26:48.660 align:middle line:84% You can remember it because that looks kind of like a capital A 00:26:48.660 --> 00:26:51.010 align:middle line:84% for And, but you don't put the cross in. 00:26:51.010 --> 00:26:53.110 align:middle line:84% But honestly, if we're looking for clarity, 00:26:53.110 --> 00:26:55.590 align:middle line:84% I think that's a lot clearer, especially 00:26:55.590 --> 00:27:01.350 align:middle line:84% because A or B is also often denoted with the same symbol 00:27:01.350 --> 00:27:02.490 align:middle line:90% upside down. 00:27:02.490 --> 00:27:05.290 align:middle line:84% And I don't have a clever mnemonic for that one. 00:27:05.290 --> 00:27:08.440 align:middle line:84% So if you're trying to remember which one to use, 00:27:08.440 --> 00:27:09.700 align:middle line:90% just use these. 00:27:09.700 --> 00:27:13.020 align:middle line:90% That's my advice. 00:27:13.020 --> 00:27:15.940 align:middle line:84% But back to this OR, I said this is an inclusive OR. 00:27:15.940 --> 00:27:19.780 align:middle line:84% So it's totally fine if A and B are both true. 00:27:19.780 --> 00:27:23.387 align:middle line:84% In English, that's not always what we mean. 00:27:23.387 --> 00:27:28.810 align:middle line:84% "And," "or," "not," these are words in spoken English as well. 00:27:28.810 --> 00:27:30.710 align:middle line:84% And when we're talking about math, 00:27:30.710 --> 00:27:34.000 align:middle line:84% when we're using these words in a math formula, 00:27:34.000 --> 00:27:37.120 align:middle line:84% we mean exactly what's written here. 00:27:37.120 --> 00:27:38.930 align:middle line:84% But when we use these words in English, 00:27:38.930 --> 00:27:40.870 align:middle line:90% we don't always mean that. 00:27:40.870 --> 00:27:44.440 align:middle line:84% Spoken colloquial English or other spoken languages 00:27:44.440 --> 00:27:47.390 align:middle line:84% are imprecise and context-dependent. 00:27:47.390 --> 00:27:51.220 align:middle line:84% And that really makes it difficult to communicate 00:27:51.220 --> 00:27:58.310 align:middle line:84% precise things, especially given the fact that as humans, 00:27:58.310 --> 00:28:00.950 align:middle line:84% we communicate math using spoken language. 00:28:00.950 --> 00:28:03.100 align:middle line:84% We're trying to communicate precise things 00:28:03.100 --> 00:28:06.680 align:middle line:84% with an imprecise channel, with an imprecise language. 00:28:06.680 --> 00:28:10.810 align:middle line:84% So as an example of how bad this can get, 00:28:10.810 --> 00:28:14.420 align:middle line:90% let's look at this word "or." 00:28:14.420 --> 00:28:17.270 align:middle line:90% Say you're at a nice wedding. 00:28:17.270 --> 00:28:19.650 align:middle line:84% The waiter comes around and says, 00:28:19.650 --> 00:28:23.620 align:middle line:84% did you order the chicken or the pasta? 00:28:23.620 --> 00:28:27.200 align:middle line:90% 00:28:27.200 --> 00:28:28.920 align:middle line:84% And when the waiter asks this question, 00:28:28.920 --> 00:28:32.506 align:middle line:84% they're trying to communicate some set of allowable responses. 00:28:32.506 --> 00:28:35.970 align:middle line:90% 00:28:35.970 --> 00:28:39.200 align:middle line:84% So are you allowed to say, I ordered the chicken, just 00:28:39.200 --> 00:28:40.430 align:middle line:90% the chicken? 00:28:40.430 --> 00:28:41.976 align:middle line:90% Yeah, seems reasonable. 00:28:41.976 --> 00:28:45.770 align:middle line:90% 00:28:45.770 --> 00:28:49.400 align:middle line:84% Pasta, you can say I ordered the pasta. 00:28:49.400 --> 00:28:53.080 align:middle line:90% Can you say I ordered both? 00:28:53.080 --> 00:28:56.150 align:middle line:90% 00:28:56.150 --> 00:28:58.910 align:middle line:90% Probably not. 00:28:58.910 --> 00:29:03.150 align:middle line:84% The catering company got all the RSVP cards in advance. 00:29:03.150 --> 00:29:05.580 align:middle line:84% And everyone was told to pick just one. 00:29:05.580 --> 00:29:06.630 align:middle line:90% So they did. 00:29:06.630 --> 00:29:09.958 align:middle line:84% So if you tell the waiter you picked both, first of all, 00:29:09.958 --> 00:29:10.500 align:middle line:90% you're lying. 00:29:10.500 --> 00:29:12.920 align:middle line:84% Second of all, stop trying to make their jobs harder. 00:29:12.920 --> 00:29:13.700 align:middle line:90% It's hard enough. 00:29:13.700 --> 00:29:15.820 align:middle line:84% They have hundreds of people to serve. 00:29:15.820 --> 00:29:19.870 align:middle line:84% So you're probably not expected to say that one. 00:29:19.870 --> 00:29:24.790 align:middle line:84% Likewise, you're probably not expected to say neither. 00:29:24.790 --> 00:29:26.990 align:middle line:84% I mean, I could go back and forth on that one. 00:29:26.990 --> 00:29:30.190 align:middle line:84% But let's say everyone filled out the RSVP card correctly, 00:29:30.190 --> 00:29:33.250 align:middle line:84% and so the waiter isn't expecting a Neither. 00:29:33.250 --> 00:29:37.540 align:middle line:84% So here, chicken or pasta, they were 00:29:37.540 --> 00:29:39.530 align:middle line:84% allowing this response and this response, 00:29:39.530 --> 00:29:41.680 align:middle line:84% but not these other two responses. 00:29:41.680 --> 00:29:45.610 align:middle line:84% This is not how we defined "or" in a math sense. 00:29:45.610 --> 00:29:48.340 align:middle line:84% This is what's called exclusive OR. 00:29:48.340 --> 00:29:50.210 align:middle line:84% You have to pick one or the other, 00:29:50.210 --> 00:29:52.570 align:middle line:90% but not both and not neither. 00:29:52.570 --> 00:29:56.590 align:middle line:84% So this is what mathematicians would call XOR, 00:29:56.590 --> 00:30:00.170 align:middle line:90% for exclusive OR. 00:30:00.170 --> 00:30:03.850 align:middle line:84% So after dinner, the waiter comes around again and says, 00:30:03.850 --> 00:30:05.085 align:middle line:90% you can have coffee or tea. 00:30:05.085 --> 00:30:09.580 align:middle line:90% 00:30:09.580 --> 00:30:10.540 align:middle line:90% Excellent. 00:30:10.540 --> 00:30:13.220 align:middle line:84% What responses are they expecting this time? 00:30:13.220 --> 00:30:14.340 align:middle line:90% Can you say just coffee? 00:30:14.340 --> 00:30:16.960 align:middle line:90% 00:30:16.960 --> 00:30:17.800 align:middle line:90% Yeah. 00:30:17.800 --> 00:30:19.820 align:middle line:90% Can you say, I want tea? 00:30:19.820 --> 00:30:22.420 align:middle line:90% Yeah, absolutely. 00:30:22.420 --> 00:30:25.540 align:middle line:90% Can you say I want both? 00:30:25.540 --> 00:30:28.400 align:middle line:84% Waiter is going to look at you side-eyed and say, 00:30:28.400 --> 00:30:31.040 align:middle line:84% there's only one mug in front of you, and I am not mixing them. 00:30:31.040 --> 00:30:32.470 align:middle line:90% This is a fancy venue. 00:30:32.470 --> 00:30:36.710 align:middle line:84% So, no, they're probably not expecting an answer of both. 00:30:36.710 --> 00:30:37.880 align:middle line:90% Can you say neither? 00:30:37.880 --> 00:30:40.270 align:middle line:90% I don't want coffee or tea. 00:30:40.270 --> 00:30:41.060 align:middle line:90% Yeah, that's fine. 00:30:41.060 --> 00:30:42.500 align:middle line:90% Not everyone wants a hot drink. 00:30:42.500 --> 00:30:44.500 align:middle line:84% Even at a fancy restaurant, not everyone wants-- 00:30:44.500 --> 00:30:45.530 align:middle line:90% a fancy wedding-- 00:30:45.530 --> 00:30:48.130 align:middle line:90% I apologize. 00:30:48.130 --> 00:30:52.160 align:middle line:84% Not everyone wants a hot drink after dinner. 00:30:52.160 --> 00:30:57.010 align:middle line:84% So now you can ask for coffee or tea or neither, but not both. 00:30:57.010 --> 00:31:00.035 align:middle line:84% Surely, that's how we defined OR in a math sense, right? 00:31:00.035 --> 00:31:02.890 align:middle line:90% 00:31:02.890 --> 00:31:05.200 align:middle line:90% No. 00:31:05.200 --> 00:31:06.770 align:middle line:90% These two are swapped. 00:31:06.770 --> 00:31:07.580 align:middle line:90% This isn't OR. 00:31:07.580 --> 00:31:10.590 align:middle line:84% This is what mathematicians call NAND. 00:31:10.590 --> 00:31:16.710 align:middle line:90% It's the Not of the And, NAND. 00:31:16.710 --> 00:31:20.060 align:middle line:84% Well, waiter used the word "or" twice. 00:31:20.060 --> 00:31:21.560 align:middle line:90% Let's try one more. 00:31:21.560 --> 00:31:23.702 align:middle line:84% Maybe they'll get it right this time. 00:31:23.702 --> 00:31:26.410 align:middle line:84% You ask for the coffee, and they say you can have cream or sugar. 00:31:26.410 --> 00:31:32.240 align:middle line:90% 00:31:32.240 --> 00:31:33.920 align:middle line:90% Can you ask for cream? 00:31:33.920 --> 00:31:36.830 align:middle line:90% Yeah, absolutely. 00:31:36.830 --> 00:31:40.760 align:middle line:84% Can you ask for sugar, just sugar? 00:31:40.760 --> 00:31:42.020 align:middle line:90% Yeah. 00:31:42.020 --> 00:31:44.600 align:middle line:90% Can you ask for both? 00:31:44.600 --> 00:31:50.150 align:middle line:84% Yeah, some people like a really sweet and creamy drink, both. 00:31:50.150 --> 00:31:51.325 align:middle line:90% Can you ask for neither? 00:31:51.325 --> 00:31:53.990 align:middle line:90% 00:31:53.990 --> 00:31:59.300 align:middle line:84% Yeah, some people like black, bitter coffee, inexplicably. 00:31:59.300 --> 00:32:02.220 align:middle line:84% So all four of these options are valid. 00:32:02.220 --> 00:32:05.390 align:middle line:84% This use of the word "or" also doesn't mean what mathematicians 00:32:05.390 --> 00:32:07.320 align:middle line:90% mean by the word "or." 00:32:07.320 --> 00:32:10.210 align:middle line:90% This one means anything goes. 00:32:10.210 --> 00:32:12.240 align:middle line:84% I'm just telling you what your options are. 00:32:12.240 --> 00:32:14.736 align:middle line:84% I don't have a math term for that. 00:32:14.736 --> 00:32:15.690 align:middle line:90% STUDENT: True. 00:32:15.690 --> 00:32:18.180 align:middle line:90% Zachary Abel: True, yes. 00:32:18.180 --> 00:32:21.270 align:middle line:84% So three uses of the word "or," none of them mean what 00:32:21.270 --> 00:32:23.640 align:middle line:90% "or" means. 00:32:23.640 --> 00:32:25.890 align:middle line:90% Isn't language fun? 00:32:25.890 --> 00:32:29.250 align:middle line:84% The moral here is, first of all, when 00:32:29.250 --> 00:32:32.370 align:middle line:84% you see these in an actual formula with other predicates 00:32:32.370 --> 00:32:35.760 align:middle line:84% and quantifiers and whatnot, we mean exactly what's 00:32:35.760 --> 00:32:36.970 align:middle line:90% written here. 00:32:36.970 --> 00:32:40.830 align:middle line:84% If we use it in a sentence, ideally, we also 00:32:40.830 --> 00:32:42.610 align:middle line:90% mean what's written here. 00:32:42.610 --> 00:32:45.180 align:middle line:84% But we might mess up, as well, and slip 00:32:45.180 --> 00:32:47.740 align:middle line:84% into colloquial, context-dependent English. 00:32:47.740 --> 00:32:51.330 align:middle line:84% And if you're ever confused, ask. 00:32:51.330 --> 00:32:52.470 align:middle line:90% If you're right, And. 00:32:52.470 --> 00:32:55.710 align:middle line:84% We messed up the wording, we will thank you and fix it 00:32:55.710 --> 00:32:59.730 align:middle line:84% if you're wrong, and we meant what we wrote, 00:32:59.730 --> 00:33:01.470 align:middle line:84% well, now you have confidence that that 00:33:01.470 --> 00:33:03.270 align:middle line:90% is what we meant to write. 00:33:03.270 --> 00:33:06.540 align:middle line:84% And both parties are happier because of it. 00:33:06.540 --> 00:33:08.330 align:middle line:84% So please ask for clarity if you're 00:33:08.330 --> 00:33:10.980 align:middle line:84% ever unsure of the wording we use, 00:33:10.980 --> 00:33:14.400 align:middle line:84% whether it's these specific words or anything else. 00:33:14.400 --> 00:33:17.090 align:middle line:90% Any questions? 00:33:17.090 --> 00:33:18.800 align:middle line:90% Wonderful. 00:33:18.800 --> 00:33:22.250 align:middle line:84% Let's look at one other operator. 00:33:22.250 --> 00:33:24.620 align:middle line:84% We've looked at AND and OR and NOT. 00:33:24.620 --> 00:33:28.197 align:middle line:84% XOR and NAND, I mean, they're there. 00:33:28.197 --> 00:33:29.370 align:middle line:90% They're less important. 00:33:29.370 --> 00:33:31.470 align:middle line:84% I would master those three first. 00:33:31.470 --> 00:33:33.380 align:middle line:84% I want to look at one more that's 00:33:33.380 --> 00:33:37.980 align:middle line:84% really important, both for this class and for proof writing. 00:33:37.980 --> 00:33:42.596 align:middle line:84% And that is Implies, implication. 00:33:42.596 --> 00:33:46.730 align:middle line:90% 00:33:46.730 --> 00:33:55.460 align:middle line:84% We're going to write it as A implies B. Sometimes, 00:33:55.460 --> 00:33:59.570 align:middle line:84% you'll see it written as A arrow B or A double arrow 00:33:59.570 --> 00:34:01.380 align:middle line:90% B. There's no difference. 00:34:01.380 --> 00:34:04.240 align:middle line:84% These are all notations for the same thing for this concept 00:34:04.240 --> 00:34:05.000 align:middle line:90% of implication. 00:34:05.000 --> 00:34:07.900 align:middle line:90% A implies B 00:34:07.900 --> 00:34:13.600 align:middle line:84% And when read aloud, I recommend reading it as A implies B, 00:34:13.600 --> 00:34:24.790 align:middle line:84% or equivalently, if A, then B. If A, then B. This is already 00:34:24.790 --> 00:34:27.670 align:middle line:84% starting to look like a programming language. 00:34:27.670 --> 00:34:29.889 align:middle line:84% And IF statements in programming languages 00:34:29.889 --> 00:34:33.290 align:middle line:84% are exactly doing what implication in math is doing. 00:34:33.290 --> 00:34:36.159 align:middle line:84% So let's talk about what that is. 00:34:36.159 --> 00:34:39.489 align:middle line:84% As before, we can figure out what these terms mean 00:34:39.489 --> 00:34:42.190 align:middle line:90% by drawing the truth table. 00:34:42.190 --> 00:34:52.130 align:middle line:84% So we have A and B, true, true, false, false, true, false, true, 00:34:52.130 --> 00:34:53.540 align:middle line:90% false. 00:34:53.540 --> 00:35:02.020 align:middle line:84% A implies B. If A and B are both true, does A imply B? 00:35:02.020 --> 00:35:04.730 align:middle line:84% Yeah, seems pretty intuitive, yeah. 00:35:04.730 --> 00:35:07.190 align:middle line:84% If A and B are both false, if false, 00:35:07.190 --> 00:35:10.185 align:middle line:84% then false, again, seems OK to me. 00:35:10.185 --> 00:35:13.220 align:middle line:90% 00:35:13.220 --> 00:35:18.490 align:middle line:84% If A is true, and B is false, if true, then false, 00:35:18.490 --> 00:35:21.010 align:middle line:90% does that sound good to you? 00:35:21.010 --> 00:35:24.780 align:middle line:84% I saw a couple head shakes, a couple nods, more head shakes, 00:35:24.780 --> 00:35:25.280 align:middle line:90% though. 00:35:25.280 --> 00:35:27.168 align:middle line:90% Yeah, if true, then false. 00:35:27.168 --> 00:35:29.210 align:middle line:84% That's kind of the whole purpose of this wording. 00:35:29.210 --> 00:35:32.950 align:middle line:90% If A is true, then B is true. 00:35:32.950 --> 00:35:35.720 align:middle line:84% So if A is true, then B better be true. 00:35:35.720 --> 00:35:36.945 align:middle line:90% And if it's false, we're sad. 00:35:36.945 --> 00:35:40.250 align:middle line:90% 00:35:40.250 --> 00:35:43.690 align:middle line:84% Last case, if A is false and B is true, 00:35:43.690 --> 00:35:46.720 align:middle line:90% should this be true or false? 00:35:46.720 --> 00:35:49.420 align:middle line:90% Think about it for a second. 00:35:49.420 --> 00:35:51.280 align:middle line:90% And let's take a poll. 00:35:51.280 --> 00:35:54.500 align:middle line:84% Who thinks False Implies True should be true? 00:35:54.500 --> 00:35:56.470 align:middle line:90% Raise your hand. 00:35:56.470 --> 00:35:58.540 align:middle line:84% Who thinks False Implies True Should. 00:35:58.540 --> 00:35:59.240 align:middle line:90% Be false? 00:35:59.240 --> 00:36:01.910 align:middle line:90% Raise your hand. 00:36:01.910 --> 00:36:04.890 align:middle line:84% Who just wants the answer and doesn't want to think about it? 00:36:04.890 --> 00:36:06.890 align:middle line:90% Raise both hands. 00:36:06.890 --> 00:36:10.310 align:middle line:90% OK, the both hands win. 00:36:10.310 --> 00:36:13.190 align:middle line:90% It was about an even split. 00:36:13.190 --> 00:36:19.170 align:middle line:84% This answer is not so intuitive in this abstract setting. 00:36:19.170 --> 00:36:22.530 align:middle line:84% If A then B, well, what does it mean? 00:36:22.530 --> 00:36:24.650 align:middle line:90% What do we want it to mean? 00:36:24.650 --> 00:36:26.610 align:middle line:90% I claim, However, this-- 00:36:26.610 --> 00:36:27.230 align:middle line:90% what? 00:36:27.230 --> 00:36:30.460 align:middle line:90% STUDENT: Last line. 00:36:30.460 --> 00:36:32.210 align:middle line:84% Zachary Abel: Last line-- oh, yeah, sorry. 00:36:32.210 --> 00:36:32.760 align:middle line:90% Thank you. 00:36:32.760 --> 00:36:35.300 align:middle line:90% This should be true. 00:36:35.300 --> 00:36:36.000 align:middle line:90% Thank you. 00:36:36.000 --> 00:36:38.840 align:middle line:90% See, I do make mistakes. 00:36:38.840 --> 00:36:42.590 align:middle line:84% It's not so clear, but I claim this is a task that you are all 00:36:42.590 --> 00:36:44.310 align:middle line:84% familiar with in a different context, 00:36:44.310 --> 00:36:45.930 align:middle line:90% in many different contexts. 00:36:45.930 --> 00:36:47.880 align:middle line:90% So let's make it concrete. 00:36:47.880 --> 00:36:49.700 align:middle line:90% Let's pick an example. 00:36:49.700 --> 00:36:50.545 align:middle line:90% Let's see. 00:36:50.545 --> 00:36:59.970 align:middle line:90% 00:36:59.970 --> 00:37:03.330 align:middle line:90% On Wednesdays, we wear pink. 00:37:03.330 --> 00:37:08.340 align:middle line:84% Said differently, if it's Wednesday, then wear pink. 00:37:08.340 --> 00:37:12.570 align:middle line:90% Wednesday implies pink. 00:37:12.570 --> 00:37:16.590 align:middle line:90% Wednesday implies pink. 00:37:16.590 --> 00:37:18.070 align:middle line:90% Now it's an implication. 00:37:18.070 --> 00:37:20.590 align:middle line:84% And let's go through the cases and see what's happening. 00:37:20.590 --> 00:37:24.280 align:middle line:84% So if it's Wednesday, and I'm wearing pink, 00:37:24.280 --> 00:37:26.650 align:middle line:84% am I obeying the rule or violating it? 00:37:26.650 --> 00:37:28.420 align:middle line:90% On Wednesdays we wear pink. 00:37:28.420 --> 00:37:29.920 align:middle line:90% Yeah, I'm doing what it says. 00:37:29.920 --> 00:37:31.090 align:middle line:90% On Wednesdays, we wear pink. 00:37:31.090 --> 00:37:31.870 align:middle line:90% Great. 00:37:31.870 --> 00:37:33.360 align:middle line:90% So that one's fine. 00:37:33.360 --> 00:37:38.050 align:middle line:84% If it's Tuesday, and I'm wearing that color, 00:37:38.050 --> 00:37:41.260 align:middle line:90% whatever it is, and not pink-- 00:37:41.260 --> 00:37:46.110 align:middle line:84% let's hide this- if it's not Wednesday, 00:37:46.110 --> 00:37:49.410 align:middle line:84% and I'm not wearing pink, am I obeying the rule 00:37:49.410 --> 00:37:51.300 align:middle line:90% or violating it? 00:37:51.300 --> 00:37:53.020 align:middle line:90% Yeah, I'm obeying that rule. 00:37:53.020 --> 00:37:54.450 align:middle line:90% That's fine. 00:37:54.450 --> 00:37:58.900 align:middle line:84% If it's Wednesday, and I'm wearing blue, 00:37:58.900 --> 00:38:01.510 align:middle line:84% am I obeying the rule or violating it? 00:38:01.510 --> 00:38:02.480 align:middle line:90% I'm violating it. 00:38:02.480 --> 00:38:03.947 align:middle line:90% On Wednesdays, we wear pink. 00:38:03.947 --> 00:38:05.030 align:middle line:90% I'm supposed to wear pink. 00:38:05.030 --> 00:38:08.170 align:middle line:84% And if I'm not wearing pink on a Wednesday, 00:38:08.170 --> 00:38:10.070 align:middle line:84% I'm not doing what the rule says. 00:38:10.070 --> 00:38:12.230 align:middle line:84% And now we get to the confusing case. 00:38:12.230 --> 00:38:19.630 align:middle line:84% If it's Tuesday, not Wednesday, and I am wearing pink, 00:38:19.630 --> 00:38:21.720 align:middle line:84% Am I obeying the rule or violating it? 00:38:21.720 --> 00:38:25.930 align:middle line:90% 00:38:25.930 --> 00:38:28.420 align:middle line:90% Thumbs up. 00:38:28.420 --> 00:38:30.880 align:middle line:90% Anyone think I'm violating it? 00:38:30.880 --> 00:38:31.480 align:middle line:90% No. 00:38:31.480 --> 00:38:33.980 align:middle line:84% In this way, in this concrete setting, 00:38:33.980 --> 00:38:37.060 align:middle line:84% it makes total sense that on Wednesdays we wear pink. 00:38:37.060 --> 00:38:39.380 align:middle line:90% On all other days, I don't care. 00:38:39.380 --> 00:38:40.910 align:middle line:90% Anything goes. 00:38:40.910 --> 00:38:44.980 align:middle line:84% I only have an opinion on what you're wearing on Wednesday. 00:38:44.980 --> 00:38:48.410 align:middle line:84% So we're going to define this to be true. 00:38:48.410 --> 00:38:51.640 align:middle line:90% False implies true is true. 00:38:51.640 --> 00:38:54.800 align:middle line:84% As I said, confusing in an abstract setting 00:38:54.800 --> 00:38:56.660 align:middle line:84% makes more sense if you think about it 00:38:56.660 --> 00:38:59.660 align:middle line:90% as obeying a certain rule. 00:38:59.660 --> 00:39:02.910 align:middle line:84% If it's Wednesday, you have to do something specific. 00:39:02.910 --> 00:39:06.350 align:middle line:84% If it's not Wednesday, anything is fine. 00:39:06.350 --> 00:39:07.910 align:middle line:90% Yes, question? 00:39:07.910 --> 00:39:10.610 align:middle line:84% STUDENT: So with the example of, say, if it's Wednesday, 00:39:10.610 --> 00:39:14.180 align:middle line:84% and you're wearing-- you're violating the rule, 00:39:14.180 --> 00:39:19.850 align:middle line:84% what if you're wearing pink as well? 00:39:19.850 --> 00:39:21.330 align:middle line:84% Zachary Abel: Yeah, good question. 00:39:21.330 --> 00:39:23.370 align:middle line:84% What if it's Wednesday, and you're wearing blue? 00:39:23.370 --> 00:39:26.310 align:middle line:84% So that's what we called the true and false case. 00:39:26.310 --> 00:39:29.180 align:middle line:84% But what if you're wearing blue and pink? 00:39:29.180 --> 00:39:29.810 align:middle line:90% Yeah. 00:39:29.810 --> 00:39:33.120 align:middle line:84% That was a less precise phrasing on my part. 00:39:33.120 --> 00:39:36.110 align:middle line:84% I should have said, you're wearing only blue. 00:39:36.110 --> 00:39:38.850 align:middle line:84% I assume if you're wearing pink and some other colors, 00:39:38.850 --> 00:39:41.160 align:middle line:90% I think that is OK here. 00:39:41.160 --> 00:39:41.730 align:middle line:90% I don't know. 00:39:41.730 --> 00:39:43.147 align:middle line:84% I'd have to watch the movie again. 00:39:43.147 --> 00:39:45.680 align:middle line:90% 00:39:45.680 --> 00:39:48.480 align:middle line:84% Anyway, implication is confusing. 00:39:48.480 --> 00:39:51.050 align:middle line:84% Let's talk a little bit more about implication. 00:39:51.050 --> 00:39:55.180 align:middle line:84% In English, when we say A implies B, 00:39:55.180 --> 00:39:58.580 align:middle line:84% often there's a connotation of causality. 00:39:58.580 --> 00:40:06.980 align:middle line:84% A causes B. And often, there's a connotation of time. 00:40:06.980 --> 00:40:12.040 align:middle line:84% A happens, and then after that, B happens because of A. 00:40:12.040 --> 00:40:16.450 align:middle line:90% So B is after is a consequence. 00:40:16.450 --> 00:40:20.210 align:middle line:84% And neither of those is what we mean. 00:40:20.210 --> 00:40:21.230 align:middle line:90% There's no causality. 00:40:21.230 --> 00:40:24.340 align:middle line:90% There's no time. 00:40:24.340 --> 00:40:28.420 align:middle line:84% How do you put a time on Goldbach's Conjecture. 00:40:28.420 --> 00:40:30.040 align:middle line:90% When it was conjecture maybe. 00:40:30.040 --> 00:40:32.560 align:middle line:90% But that's not what we mean. 00:40:32.560 --> 00:40:33.970 align:middle line:84% When we say Goldbach's Conjecture 00:40:33.970 --> 00:40:38.590 align:middle line:84% implies that I'm wearing pink on a Wednesday. 00:40:38.590 --> 00:40:42.280 align:middle line:84% As mathematicians, when we say the word "implies," 00:40:42.280 --> 00:40:47.270 align:middle line:84% we mean this truth table and only this truth table. 00:40:47.270 --> 00:40:49.100 align:middle line:84% There's no additional connotation. 00:40:49.100 --> 00:40:50.025 align:middle line:90% There's no causality. 00:40:50.025 --> 00:40:50.900 align:middle line:90% There's nothing else. 00:40:50.900 --> 00:40:54.710 align:middle line:90% It's just this truth table. 00:40:54.710 --> 00:40:56.100 align:middle line:90% Does that make sense? 00:40:56.100 --> 00:40:57.740 align:middle line:84% It can be hard to flip that switch 00:40:57.740 --> 00:40:59.270 align:middle line:84% and redefine this term that we're 00:40:59.270 --> 00:41:01.500 align:middle line:84% familiar with from somewhere else. 00:41:01.500 --> 00:41:03.390 align:middle line:84% Because it does mean something different now. 00:41:03.390 --> 00:41:05.120 align:middle line:90% It means that. 00:41:05.120 --> 00:41:07.190 align:middle line:84% And if you see this in class, even 00:41:07.190 --> 00:41:09.050 align:middle line:84% if it's written in a paragraph, you 00:41:09.050 --> 00:41:12.440 align:middle line:84% should assume we mean that first. 00:41:12.440 --> 00:41:13.770 align:middle line:90% Was that your question? 00:41:13.770 --> 00:41:14.870 align:middle line:90% Please? 00:41:14.870 --> 00:41:16.228 align:middle line:90% STUDENT: So [INAUDIBLE]. 00:41:16.228 --> 00:41:30.050 align:middle line:90% 00:41:30.050 --> 00:41:32.600 align:middle line:84% Zachary Abel: So clarification on the concept 00:41:32.600 --> 00:41:36.080 align:middle line:84% of time, if A then B, do we need to know whether A is true 00:41:36.080 --> 00:41:40.430 align:middle line:90% or not before we decide about B? 00:41:40.430 --> 00:41:42.260 align:middle line:84% I would recommend thinking about and B 00:41:42.260 --> 00:41:44.400 align:middle line:90% as two separate propositions. 00:41:44.400 --> 00:41:46.380 align:middle line:84% They each have their own truth value. 00:41:46.380 --> 00:41:47.730 align:middle line:90% A is either true or false. 00:41:47.730 --> 00:41:49.570 align:middle line:90% B is either true or false. 00:41:49.570 --> 00:41:54.270 align:middle line:84% We don't necessarily know what they are, now or in the future. 00:41:54.270 --> 00:41:56.267 align:middle line:90% But they're still propositions. 00:41:56.267 --> 00:41:58.350 align:middle line:84% A might be the Goldbach Conjecture, where we don't 00:41:58.350 --> 00:41:59.650 align:middle line:90% know if it's true or false. 00:41:59.650 --> 00:42:01.450 align:middle line:90% But it's still a proposition. 00:42:01.450 --> 00:42:04.150 align:middle line:84% It's still going to be true or false. 00:42:04.150 --> 00:42:05.820 align:middle line:90% We just don't know yet. 00:42:05.820 --> 00:42:09.150 align:middle line:84% Likewise, we can always combine propositions A and B 00:42:09.150 --> 00:42:11.130 align:middle line:84% into this bigger proposition, if A, 00:42:11.130 --> 00:42:16.170 align:middle line:84% then B. A implies B. It's just a new proposition where 00:42:16.170 --> 00:42:18.220 align:middle line:84% once we know whether A and B are true, 00:42:18.220 --> 00:42:21.870 align:middle line:84% we'll know whether the implication is true. 00:42:21.870 --> 00:42:23.410 align:middle line:90% Does that make sense? 00:42:23.410 --> 00:42:23.910 align:middle line:90% Cool. 00:42:23.910 --> 00:42:26.950 align:middle line:90% 00:42:26.950 --> 00:42:28.290 align:middle line:90% Let's see. 00:42:28.290 --> 00:42:35.610 align:middle line:84% So another example of implication, 00:42:35.610 --> 00:42:41.130 align:middle line:84% you might be familiar with less than 3. 00:42:41.130 --> 00:42:46.140 align:middle line:84% This is a preemoji heart, means I love you. 00:42:46.140 --> 00:42:46.890 align:middle line:90% Got some chuckles. 00:42:46.890 --> 00:42:48.400 align:middle line:90% I don't know why. 00:42:48.400 --> 00:42:51.430 align:middle line:90% I'm showing my age maybe. 00:42:51.430 --> 00:42:53.120 align:middle line:90% So less than 3 means I love you. 00:42:53.120 --> 00:42:59.350 align:middle line:84% What I have seen, I have seen occasionally less than 4 00:42:59.350 --> 00:43:02.390 align:middle line:84% as an intensifier, like I really love you. 00:43:02.390 --> 00:43:06.550 align:middle line:84% I love you more than 3, I love you 4. 00:43:06.550 --> 00:43:08.560 align:middle line:90% And this irks me. 00:43:08.560 --> 00:43:13.630 align:middle line:84% I hate this, not because I'm against love, 00:43:13.630 --> 00:43:17.380 align:middle line:84% but because this is the stronger mathematical statement. 00:43:17.380 --> 00:43:20.120 align:middle line:90% Less than 3 implies less than 4. 00:43:20.120 --> 00:43:23.380 align:middle line:84% If we know x is less than 3, some number x, well, 00:43:23.380 --> 00:43:26.200 align:middle line:84% then we know that x is less than 4. 00:43:26.200 --> 00:43:28.670 align:middle line:84% And in fact, we know more information than that. 00:43:28.670 --> 00:43:31.120 align:middle line:90% We know it's not pi. 00:43:31.120 --> 00:43:32.620 align:middle line:90% Right? 00:43:32.620 --> 00:43:35.030 align:middle line:90% So this implies that. 00:43:35.030 --> 00:43:37.310 align:middle line:84% So this is the stronger statement. 00:43:37.310 --> 00:43:38.570 align:middle line:90% So you've weakened it. 00:43:38.570 --> 00:43:39.920 align:middle line:90% You love me less. 00:43:39.920 --> 00:43:41.590 align:middle line:90% [LAUGHTER] 00:43:41.590 --> 00:43:44.420 align:middle line:84% It should be less than 2, or something. 00:43:44.420 --> 00:43:46.710 align:middle line:84% But only mathematicians would get that. 00:43:46.710 --> 00:43:49.700 align:middle line:84% So I don't actually recommend sending that to your Valentine. 00:43:49.700 --> 00:43:53.720 align:middle line:90% But that's always bothered me. 00:43:53.720 --> 00:43:56.330 align:middle line:84% Let's do one other quick example. 00:43:56.330 --> 00:44:00.290 align:middle line:84% Who's familiar with Descartes, I think, therefore I am? 00:44:00.290 --> 00:44:03.500 align:middle line:90% Excellent. 00:44:03.500 --> 00:44:07.340 align:middle line:90% So, I think, therefore I am. 00:44:07.340 --> 00:44:11.690 align:middle line:90% I think, therefore I am. 00:44:11.690 --> 00:44:14.580 align:middle line:84% And this one has two claims to it. 00:44:14.580 --> 00:44:17.900 align:middle line:84% It's not only saying that thinking implies am-ing. 00:44:17.900 --> 00:44:20.900 align:middle line:84% But it's also saying that I think. 00:44:20.900 --> 00:44:27.860 align:middle line:84% So it's asserting that T is true, and T implies am. 00:44:27.860 --> 00:44:28.490 align:middle line:90% Yeah? 00:44:28.490 --> 00:44:30.110 align:middle line:90% So that's Descartes. 00:44:30.110 --> 00:44:33.290 align:middle line:84% What I have seen that bothers me is a shirt 00:44:33.290 --> 00:44:34.680 align:middle line:90% and a pillow and a meme. 00:44:34.680 --> 00:44:36.360 align:middle line:84% And you can go off and buy all of this. 00:44:36.360 --> 00:44:38.630 align:middle line:84% I don't recommend giving it money. 00:44:38.630 --> 00:44:43.860 align:middle line:84% "I do not think, therefore I do not am," which I find hilarious, 00:44:43.860 --> 00:44:46.050 align:middle line:90% but also frustrating. 00:44:46.050 --> 00:44:54.450 align:middle line:84% So "I do not think" implies "I do not am." 00:44:54.450 --> 00:44:56.100 align:middle line:84% So it's also claiming I don't think. 00:44:56.100 --> 00:44:57.810 align:middle line:90% I don't have an opinion on that. 00:44:57.810 --> 00:45:00.190 align:middle line:84% But it's saying this implication, 00:45:00.190 --> 00:45:03.390 align:middle line:84% not thinking implies, not aiming. 00:45:03.390 --> 00:45:06.100 align:middle line:84% And it irks me because that's not equivalent. 00:45:06.100 --> 00:45:09.390 align:middle line:84% It's not equivalent to Descartes. 00:45:09.390 --> 00:45:13.530 align:middle line:84% Let's go out on a limb here, and let's assume-- 00:45:13.530 --> 00:45:14.490 align:middle line:90% big ask, I know. 00:45:14.490 --> 00:45:18.240 align:middle line:84% Let's assume that my shoe doesn't think, my right shoe 00:45:18.240 --> 00:45:20.250 align:middle line:90% specifically. 00:45:20.250 --> 00:45:23.760 align:middle line:84% But it still exists, let's assume. 00:45:23.760 --> 00:45:25.950 align:middle line:90% If my shoe thinks and doesn't-- 00:45:25.950 --> 00:45:28.270 align:middle line:84% sorry, doesn't think, but still exists, 00:45:28.270 --> 00:45:30.540 align:middle line:90% Descartes is fine with that. 00:45:30.540 --> 00:45:33.720 align:middle line:90% False implies true is fine. 00:45:33.720 --> 00:45:35.710 align:middle line:90% But the meme isn't. 00:45:35.710 --> 00:45:38.680 align:middle line:84% Not thinking is supposed to imply not am-ing. 00:45:38.680 --> 00:45:42.340 align:middle line:84% But my shoe ams even if it doesn't think . 00:45:42.340 --> 00:45:45.640 align:middle line:84% So these two statements are not equivalent. 00:45:45.640 --> 00:45:47.240 align:middle line:90% It should be, I do not am. 00:45:47.240 --> 00:45:48.640 align:middle line:90% Therefore, I do not think. 00:45:48.640 --> 00:45:51.310 align:middle line:90% Which gets zero hits on Google. 00:45:51.310 --> 00:45:52.540 align:middle line:90% We need to fix this. 00:45:52.540 --> 00:45:54.235 align:middle line:90% Question? 00:45:54.235 --> 00:45:55.735 align:middle line:84% STUDENT: Looking at the truth table, 00:45:55.735 --> 00:46:00.010 align:middle line:84% right implementation has certain [INAUDIBLE] 00:46:00.010 --> 00:46:01.940 align:middle line:90% 00:46:01.940 --> 00:46:02.740 align:middle line:90% Zachary Abel: Ooh. 00:46:02.740 --> 00:46:04.880 align:middle line:84% All right, that's a great comment. 00:46:04.880 --> 00:46:05.800 align:middle line:90% Let's see. 00:46:05.800 --> 00:46:07.065 align:middle line:90% Let's go back to the front. 00:46:07.065 --> 00:46:10.390 align:middle line:90% 00:46:10.390 --> 00:46:12.730 align:middle line:84% What your colleague suggested was 00:46:12.730 --> 00:46:19.870 align:middle line:84% we can write A implies B by combining the earlier operators. 00:46:19.870 --> 00:46:21.010 align:middle line:90% Oops. 00:46:21.010 --> 00:46:24.510 align:middle line:90% A implies B is equivalent to-- 00:46:24.510 --> 00:46:29.530 align:middle line:90% 00:46:29.530 --> 00:46:31.690 align:middle line:84% and can you remind me what you said? 00:46:31.690 --> 00:46:34.840 align:middle line:84% STUDENT: It was B or NOT A and NOT B. 00:46:34.840 --> 00:46:43.650 align:middle line:84% Zachary Abel: OK, B or Not A and Not B-- 00:46:43.650 --> 00:46:46.980 align:middle line:84% and the claim is that this is equivalent to A implies B, 00:46:46.980 --> 00:46:50.830 align:middle line:84% has the same truth table as A implies B. In fact, 00:46:50.830 --> 00:46:53.410 align:middle line:84% if you check it, we can simplify it even further. 00:46:53.410 --> 00:46:58.380 align:middle line:84% We don't need this extra clause on B. B 00:46:58.380 --> 00:47:06.810 align:middle line:84% or not A. This is equivalent to A implies B. 00:47:06.810 --> 00:47:08.910 align:middle line:84% What this is really saying is that the only way 00:47:08.910 --> 00:47:12.870 align:middle line:84% to violate this, the only way to violate A implies B, 00:47:12.870 --> 00:47:15.660 align:middle line:84% is for A to be true and B to be false. 00:47:15.660 --> 00:47:21.000 align:middle line:84% And so if either of these is true, then you're fine. 00:47:21.000 --> 00:47:23.190 align:middle line:84% Yeah, absolutely, there are relationships 00:47:23.190 --> 00:47:27.750 align:middle line:84% between these operators, really nice observation. 00:47:27.750 --> 00:47:30.750 align:middle line:84% But going back to Descartes for just a second, 00:47:30.750 --> 00:47:35.490 align:middle line:84% we looked at a implies B and we also 00:47:35.490 --> 00:47:40.250 align:middle line:84% looked at not a implies B. Let's look at some others. 00:47:40.250 --> 00:47:48.190 align:middle line:84% B implies A. Not B implies Not A. If this is our statement, 00:47:48.190 --> 00:47:50.836 align:middle line:84% A implies B, if we flip them around, 00:47:50.836 --> 00:47:52.503 align:middle line:84% B implies A, that's called the converse. 00:47:52.503 --> 00:47:55.510 align:middle line:90% 00:47:55.510 --> 00:47:58.570 align:middle line:84% That's the converse of the original statement. 00:47:58.570 --> 00:48:01.670 align:middle line:84% And this one here, Not B implies Not A, 00:48:01.670 --> 00:48:03.516 align:middle line:84% that's what's called the contrapositive. 00:48:03.516 --> 00:48:09.370 align:middle line:90% 00:48:09.370 --> 00:48:12.490 align:middle line:84% And the one we saw over there with Not A 00:48:12.490 --> 00:48:18.070 align:middle line:84% implies Not B, that's called the inverse. 00:48:18.070 --> 00:48:20.390 align:middle line:84% You don't really need to know all of these terms. 00:48:20.390 --> 00:48:21.932 align:middle line:84% It's just nice to know they're there. 00:48:21.932 --> 00:48:28.360 align:middle line:84% Because A implies B is equivalent to one of these two. 00:48:28.360 --> 00:48:31.660 align:middle line:90% And it's not the nice one. 00:48:31.660 --> 00:48:34.400 align:middle line:84% A implies B is equivalent to its contrapositive. 00:48:34.400 --> 00:48:38.880 align:middle line:84% Not B implies not A Wednesday implies pink. 00:48:38.880 --> 00:48:43.670 align:middle line:84% So if you're not wearing pink, it better not be Wednesday. 00:48:43.670 --> 00:48:48.380 align:middle line:84% A implies B. Not B implies Not A. So these two are equivalent. 00:48:48.380 --> 00:48:50.570 align:middle line:84% And also, the converse and the inverse 00:48:50.570 --> 00:48:52.070 align:middle line:84% are equivalent for the same reason. 00:48:52.070 --> 00:48:56.120 align:middle line:84% They're contrapositives of each other. 00:48:56.120 --> 00:49:00.620 align:middle line:84% So that Descartes meme was using the inverse, when they probably 00:49:00.620 --> 00:49:03.830 align:middle line:84% meant to use the contrapositive, I assume, 00:49:03.830 --> 00:49:07.880 align:middle line:84% because I assume they thought that deeply about it. 00:49:07.880 --> 00:49:10.620 align:middle line:84% That is everything I wanted to say about implication. 00:49:10.620 --> 00:49:12.985 align:middle line:84% Does anyone have more questions before we move on? 00:49:12.985 --> 00:49:16.250 align:middle line:90% 00:49:16.250 --> 00:49:16.990 align:middle line:90% All right. 00:49:16.990 --> 00:49:23.180 align:middle line:90% 00:49:23.180 --> 00:49:27.170 align:middle line:84% Now, very briefly, I've mentioned sets a few times. 00:49:27.170 --> 00:49:30.830 align:middle line:84% And I've used set notation a few times. 00:49:30.830 --> 00:49:33.890 align:middle line:84% So let's go ahead and make that more precise. 00:49:33.890 --> 00:49:36.280 align:middle line:84% Because we're going to be talking about sets all term, 00:49:36.280 --> 00:49:39.850 align:middle line:84% I just want us to have that common language to fall back on. 00:49:39.850 --> 00:49:48.043 align:middle line:84% So a set is a collection of objects. 00:49:48.043 --> 00:49:55.140 align:middle line:90% 00:49:55.140 --> 00:49:58.330 align:middle line:84% I'm not going to be more precise than this. 00:49:58.330 --> 00:50:01.350 align:middle line:84% If you want to dig into set theory, 00:50:01.350 --> 00:50:03.310 align:middle line:84% into the foundations of mathematics, 00:50:03.310 --> 00:50:04.450 align:middle line:90% it's really interesting. 00:50:04.450 --> 00:50:05.500 align:middle line:90% There's a lot of stuff there. 00:50:05.500 --> 00:50:07.458 align:middle line:84% But that's not what we're doing in this course. 00:50:07.458 --> 00:50:09.080 align:middle line:84% A set is a collection of objects. 00:50:09.080 --> 00:50:11.790 align:middle line:90% 00:50:11.790 --> 00:50:13.480 align:middle line:90% Let me give lots of examples. 00:50:13.480 --> 00:50:17.950 align:middle line:84% An example is the set that contains 6, 1, 2, and 0. 00:50:17.950 --> 00:50:20.790 align:middle line:84% And we're going to write sets with these curly braces. 00:50:20.790 --> 00:50:25.740 align:middle line:84% Crucially, when we're talking about a set, 00:50:25.740 --> 00:50:27.125 align:middle line:90% the order doesn't matter. 00:50:27.125 --> 00:50:33.570 align:middle line:90% 00:50:33.570 --> 00:50:35.475 align:middle line:84% And there's no such thing as a repeat. 00:50:35.475 --> 00:50:38.680 align:middle line:90% 00:50:38.680 --> 00:50:42.460 align:middle line:84% All I care about is, which things are in your set and which 00:50:42.460 --> 00:50:43.430 align:middle line:90% ones aren't? 00:50:43.430 --> 00:50:51.100 align:middle line:84% So this set is exactly equal to 2160 00:50:51.100 --> 00:50:53.350 align:middle line:84% because they have the same four elements. 00:50:53.350 --> 00:50:57.310 align:middle line:90% This set is exactly equal to 6-- 00:50:57.310 --> 00:51:01.340 align:middle line:90% yeah, 6, 1, 2, 0, 0. 00:51:01.340 --> 00:51:03.920 align:middle line:84% It's kind of weird to write 0 in there twice, 00:51:03.920 --> 00:51:05.750 align:middle line:90% but technically, it's OK. 00:51:05.750 --> 00:51:08.050 align:middle line:90% I don't recommend doing that. 00:51:08.050 --> 00:51:12.680 align:middle line:84% But this has the four numbers 0, 1, 2, and 6 in it. 00:51:12.680 --> 00:51:16.900 align:middle line:84% And so these other sets, they're exactly the same set. 00:51:16.900 --> 00:51:20.290 align:middle line:84% We already saw, N, the natural numbers, this 00:51:20.290 --> 00:51:25.035 align:middle line:84% is the set 0, 1, 2, 3, and so on, all the way up. 00:51:25.035 --> 00:51:28.210 align:middle line:90% 00:51:28.210 --> 00:51:31.120 align:middle line:90% We also have the set-- 00:51:31.120 --> 00:51:35.210 align:middle line:84% that's a Blackboard bold z, which 00:51:35.210 --> 00:51:39.800 align:middle line:84% includes the integers, natural numbers, and their negatives, 1 00:51:39.800 --> 00:51:48.080 align:middle line:84% and minus 1 and 2 and minus 2, and so on, all the integers, 00:51:48.080 --> 00:51:50.150 align:middle line:90% integers. 00:51:50.150 --> 00:51:57.290 align:middle line:84% A couple more useful ones, we have Q is the set of rationals, 00:51:57.290 --> 00:51:59.990 align:middle line:84% so fractions you can get by dividing some integer 00:51:59.990 --> 00:52:02.390 align:middle line:90% by some non-zero integer. 00:52:02.390 --> 00:52:05.540 align:middle line:90% We have the real numbers. 00:52:05.540 --> 00:52:09.830 align:middle line:84% That's a bold R. Blackboard bold R. 00:52:09.830 --> 00:52:17.283 align:middle line:84% And we also have C, the complex numbers, complex numbers. 00:52:17.283 --> 00:52:21.230 align:middle line:90% 00:52:21.230 --> 00:52:26.720 align:middle line:84% So those are five sets that we will very frequently talk about. 00:52:26.720 --> 00:52:28.910 align:middle line:84% Let's give some more examples of sets. 00:52:28.910 --> 00:52:34.710 align:middle line:90% There's the empty set, Which? 00:52:34.710 --> 00:52:38.560 align:middle line:84% Is written as a circle with a through it. 00:52:38.560 --> 00:52:41.880 align:middle line:84% It's also sometimes written as an open curly brace 00:52:41.880 --> 00:52:46.230 align:middle line:84% and a closed curly brace with nothing inside. 00:52:46.230 --> 00:52:47.853 align:middle line:90% These two mean the same thing. 00:52:47.853 --> 00:52:50.020 align:middle line:84% They're just different notations for the same thing. 00:52:50.020 --> 00:52:53.520 align:middle line:84% The empty set is a set that contains nothing. 00:52:53.520 --> 00:52:54.550 align:middle line:90% It is a set. 00:52:54.550 --> 00:52:55.300 align:middle line:90% It is a thing. 00:52:55.300 --> 00:52:56.140 align:middle line:90% It does exist. 00:52:56.140 --> 00:52:57.910 align:middle line:90% It just has zero elements. 00:52:57.910 --> 00:53:00.670 align:middle line:84% Whereas, this set over here has four elements. 00:53:00.670 --> 00:53:03.820 align:middle line:84% This set over here has infinitely many elements. 00:53:03.820 --> 00:53:06.870 align:middle line:84% These other sets have infinitely many elements as well. 00:53:06.870 --> 00:53:09.762 align:middle line:90% But is it the same infinity? 00:53:09.762 --> 00:53:11.890 align:middle line:90% No, not always. 00:53:11.890 --> 00:53:13.120 align:middle line:90% And that's a confusing thing. 00:53:13.120 --> 00:53:16.050 align:middle line:90% Come ask me if you're curious. 00:53:16.050 --> 00:53:19.750 align:middle line:84% Let's see, one other set I wanted to give an example of. 00:53:19.750 --> 00:53:24.210 align:middle line:84% Let's call this one A, just so we have a name for it. 00:53:24.210 --> 00:53:26.148 align:middle line:90% Let's define B is-- 00:53:26.148 --> 00:53:27.190 align:middle line:90% what did I want to write? 00:53:27.190 --> 00:53:34.240 align:middle line:90% Yeah, 2, 3, 4, and empty. set. 00:53:34.240 --> 00:53:37.510 align:middle line:90% B is a set with three elements. 00:53:37.510 --> 00:53:40.130 align:middle line:84% It has the number 2 as an element. 00:53:40.130 --> 00:53:42.350 align:middle line:90% It has this set as an element. 00:53:42.350 --> 00:53:45.760 align:middle line:84% And it has the empty set as an element. 00:53:45.760 --> 00:53:53.590 align:middle line:84% Is 3 an element of / We would write this as 3 00:53:53.590 --> 00:53:57.920 align:middle line:84% is an element of B. There's that notation again. 00:53:57.920 --> 00:53:59.830 align:middle line:90% Is three an element of B? 00:53:59.830 --> 00:54:02.900 align:middle line:84% No it's inside one of the elements of B, 00:54:02.900 --> 00:54:08.620 align:middle line:84% but it's not one of the elements itself, so no. 00:54:08.620 --> 00:54:10.840 align:middle line:90% So we have this notation. 00:54:10.840 --> 00:54:12.350 align:middle line:90% Oh no, I just used that. 00:54:12.350 --> 00:54:15.250 align:middle line:90% Let's move over here. 00:54:15.250 --> 00:54:17.170 align:middle line:90% Eraser, help. 00:54:17.170 --> 00:54:20.200 align:middle line:90% Found it. 00:54:20.200 --> 00:54:23.170 align:middle line:84% We have this notion of being an element a set 00:54:23.170 --> 00:54:26.560 align:middle line:84% that we have already talked about a few times. 00:54:26.560 --> 00:54:30.730 align:middle line:84% You can also ask a set to be a subset of another one. 00:54:30.730 --> 00:54:44.730 align:middle line:84% So x in A means x is an element of A. A 00:54:44.730 --> 00:54:48.420 align:middle line:84% and then this new symbol-- this is a sideways-- or this 00:54:48.420 --> 00:54:52.410 align:middle line:84% is a curly C with a line underneath it. 00:54:52.410 --> 00:55:03.390 align:middle line:84% A is a subset or equal to B. A is a subset or equal to B. 00:55:03.390 --> 00:55:05.100 align:middle line:90% Note the similarity of two A's. 00:55:05.100 --> 00:55:07.710 align:middle line:84% Less than or equal to B, less equal 00:55:07.710 --> 00:55:10.090 align:middle line:84% doesn't really make sense for sets, but this one does. 00:55:10.090 --> 00:55:14.640 align:middle line:84% Subset means everything that's in A is also in B. 00:55:14.640 --> 00:55:17.910 align:middle line:84% And B might contain more besides. 00:55:17.910 --> 00:55:23.370 align:middle line:84% For example, the set that contains 2 and 1, 00:55:23.370 --> 00:55:27.010 align:middle line:90% is that a subset of A? 00:55:27.010 --> 00:55:30.250 align:middle line:90% A was over there. 00:55:30.250 --> 00:55:34.040 align:middle line:84% Yeah, everything that's in here is also in here-- 00:55:34.040 --> 00:55:34.935 align:middle line:90% totally fine. 00:55:34.935 --> 00:55:38.860 align:middle line:90% 00:55:38.860 --> 00:55:39.560 align:middle line:90% Let's see. 00:55:39.560 --> 00:55:43.875 align:middle line:90% Is the empty set a subset of A? 00:55:43.875 --> 00:55:56.930 align:middle line:90% 00:55:56.930 --> 00:55:57.430 align:middle line:90% Cool. 00:55:57.430 --> 00:55:59.720 align:middle line:90% All right, I got some thumbs up. 00:55:59.720 --> 00:56:01.210 align:middle line:90% I got no thumbs down. 00:56:01.210 --> 00:56:03.140 align:middle line:90% I got a lot of blank faces. 00:56:03.140 --> 00:56:04.610 align:middle line:84% I'm going to assume that means Yes. 00:56:04.610 --> 00:56:08.440 align:middle line:84% Yeah, everything that's an element of the empty set 00:56:08.440 --> 00:56:12.500 align:middle line:84% should also be an element of A. Well, it's kind of weird, right? 00:56:12.500 --> 00:56:15.075 align:middle line:84% Because there are no elements of the empty set. 00:56:15.075 --> 00:56:18.910 align:middle line:84% And we can't really check this rule 00:56:18.910 --> 00:56:21.100 align:middle line:90% for any elements of the left. 00:56:21.100 --> 00:56:23.770 align:middle line:84% So it kind of feels like this shouldn't be true, 00:56:23.770 --> 00:56:26.270 align:middle line:84% because there's nothing to instantiate. 00:56:26.270 --> 00:56:29.480 align:middle line:84% There are no elements here to talk about. 00:56:29.480 --> 00:56:34.850 align:middle line:84% But a better way to think of it is prove me wrong. 00:56:34.850 --> 00:56:36.560 align:middle line:84% Show me an element on the left that 00:56:36.560 --> 00:56:38.250 align:middle line:90% isn't an element on the right. 00:56:38.250 --> 00:56:40.730 align:middle line:90% You can't. 00:56:40.730 --> 00:56:43.310 align:middle line:84% You can't violate the rule, and therefore, the rule 00:56:43.310 --> 00:56:44.086 align:middle line:90% is satisfied. 00:56:44.086 --> 00:56:47.810 align:middle line:90% 00:56:47.810 --> 00:56:51.240 align:middle line:84% A little bit confusing, but when you look at it the right way, 00:56:51.240 --> 00:56:54.470 align:middle line:90% hopefully it makes sense. 00:56:54.470 --> 00:56:59.240 align:middle line:84% Just like we were able to combine propositions- A and B, 00:56:59.240 --> 00:57:00.740 align:middle line:90% A or B-- 00:57:00.740 --> 00:57:02.840 align:middle line:90% we can combine sets. 00:57:02.840 --> 00:57:06.170 align:middle line:90% We can look at A union B-- 00:57:06.170 --> 00:57:09.890 align:middle line:90% this is the union-- 00:57:09.890 --> 00:57:15.680 align:middle line:84% which is the set of elements that are in A or B or both. 00:57:15.680 --> 00:57:20.090 align:middle line:84% So in this case, A union B for our example, 00:57:20.090 --> 00:57:23.820 align:middle line:84% it certainly includes 6 and 1 and 2 and 0. 00:57:23.820 --> 00:57:25.978 align:middle line:84% And also-- well, it already has 2, 00:57:25.978 --> 00:57:27.520 align:middle line:84% so we don't need to write that again. 00:57:27.520 --> 00:57:34.110 align:middle line:84% And it has the set 3, 4, and it has the empty set, so everything 00:57:34.110 --> 00:57:36.840 align:middle line:90% in A or B or both. 00:57:36.840 --> 00:57:38.880 align:middle line:84% And remember, I don't have to say, the Or Both 00:57:38.880 --> 00:57:43.240 align:middle line:84% because OR is inclusive by default. 00:57:43.240 --> 00:57:47.010 align:middle line:84% We can likewise do a intersect B, 00:57:47.010 --> 00:57:52.376 align:middle line:84% which is the elements that are in both, so the intersection. 00:57:52.376 --> 00:57:55.410 align:middle line:90% 00:57:55.410 --> 00:58:01.330 align:middle line:84% In this case, A intersect B is-- well, what's in both A and B? 00:58:01.330 --> 00:58:03.000 align:middle line:90% I think it's just the two. 00:58:03.000 --> 00:58:05.590 align:middle line:84% There's a 2 in both, and nothing else is in common. 00:58:05.590 --> 00:58:09.720 align:middle line:84% So that's going to be just the set containing 2. 00:58:09.720 --> 00:58:14.460 align:middle line:84% There's also the set difference, A minus B, 00:58:14.460 --> 00:58:19.920 align:middle line:84% sometimes written as A backslash B. 00:58:19.920 --> 00:58:21.436 align:middle line:90% And that's the set difference. 00:58:21.436 --> 00:58:25.720 align:middle line:90% 00:58:25.720 --> 00:58:29.110 align:middle line:84% It's the things in A that aren't in B. 00:58:29.110 --> 00:58:31.090 align:middle line:84% So it's A, but take away everything 00:58:31.090 --> 00:58:33.880 align:middle line:84% in B. So in this case, it's going 00:58:33.880 --> 00:58:40.330 align:middle line:84% to equal 6 and 1 and not the two, but we still keep the 0. 00:58:40.330 --> 00:58:41.740 align:middle line:90% Yes? 00:58:41.740 --> 00:58:45.010 align:middle line:90% STUDENT: So that set is-- 00:58:45.010 --> 00:58:49.540 align:middle line:84% we basically [INAUDIBLE] the subset of everything, right? 00:58:49.540 --> 00:58:50.440 align:middle line:90% Zachary Abel: Yep. 00:58:50.440 --> 00:58:55.973 align:middle line:84% STUDENT: So [INAUDIBLE] also both in A and B and [INAUDIBLE]? 00:58:55.973 --> 00:58:57.140 align:middle line:90% Zachary Abel: Good question. 00:58:57.140 --> 00:58:58.900 align:middle line:84% So the question was, we've already 00:58:58.900 --> 00:59:02.650 align:middle line:84% said that the empty set is a subset of everything. 00:59:02.650 --> 00:59:05.890 align:middle line:84% So shouldn't it be in both A and B 00:59:05.890 --> 00:59:09.400 align:middle line:84% and therefore, in their intersection? 00:59:09.400 --> 00:59:11.770 align:middle line:90% Really good question. 00:59:11.770 --> 00:59:15.490 align:middle line:84% And the difference is that we have two different meanings 00:59:15.490 --> 00:59:17.290 align:middle line:90% of "in." 00:59:17.290 --> 00:59:20.820 align:middle line:84% We said that the empty set is a subset of everything, 00:59:20.820 --> 00:59:23.270 align:middle line:84% but not an element of everything. 00:59:23.270 --> 00:59:25.730 align:middle line:84% So the empty set is an element of B 00:59:25.730 --> 00:59:28.160 align:middle line:84% because it was written as one of the elements. 00:59:28.160 --> 00:59:30.740 align:middle line:84% But it's not an element of A, because these 00:59:30.740 --> 00:59:33.410 align:middle line:84% are all four elements of A. There's nothing else in there. 00:59:33.410 --> 00:59:35.040 align:middle line:90% So A is a subset-- 00:59:35.040 --> 00:59:39.630 align:middle line:84% sorry, the empty set is a subset of A and a subset of B. 00:59:39.630 --> 00:59:43.400 align:middle line:90% But it's not an element of both. 00:59:43.400 --> 00:59:44.340 align:middle line:90% Great question. 00:59:44.340 --> 00:59:45.330 align:middle line:90% Thank you for asking. 00:59:45.330 --> 00:59:45.920 align:middle line:90% Yes? 00:59:45.920 --> 00:59:49.880 align:middle line:84% STUDENT: Why does A have two zeros if there's no repeats? 00:59:49.880 --> 00:59:53.160 align:middle line:84% Zachary Abel: Why can A have two zeros if there's no repeats? 00:59:53.160 --> 00:59:54.830 align:middle line:90% Good question. 00:59:54.830 --> 00:59:58.520 align:middle line:84% It's because it doesn't actually have two zeros. 00:59:58.520 --> 01:00:00.750 align:middle line:90% I just wrote the two zeros. 01:00:00.750 --> 01:00:03.210 align:middle line:84% Each of them says, 0 is an element of this set. 01:00:03.210 --> 01:00:04.430 align:middle line:90% By the way, did you forget? 01:00:04.430 --> 01:00:07.070 align:middle line:90% 0 is an element of the set. 01:00:07.070 --> 01:00:08.280 align:middle line:90% So it doesn't have two. 01:00:08.280 --> 01:00:11.720 align:middle line:84% There's no way for it to remember it has two. 01:00:11.720 --> 01:00:14.400 align:middle line:84% But sometimes, in some weird cases, 01:00:14.400 --> 01:00:17.460 align:middle line:84% it might make sense to write a set where the same element is 01:00:17.460 --> 01:00:19.920 align:middle line:90% written twice. 01:00:19.920 --> 01:00:21.935 align:middle line:90% And so there's that example. 01:00:21.935 --> 01:00:27.540 align:middle line:90% 01:00:27.540 --> 01:00:28.840 align:middle line:90% Yes, question? 01:00:28.840 --> 01:00:30.656 align:middle line:84% STUDENT: Can you have A [INAUDIBLE]? 01:00:30.656 --> 01:00:36.887 align:middle line:90% 01:00:36.887 --> 01:00:38.470 align:middle line:84% Zachary Abel: That's a great question. 01:00:38.470 --> 01:00:43.110 align:middle line:84% So if A also had a 3 and 4 in it-- 01:00:43.110 --> 01:00:46.450 align:middle line:84% so let's pretend this also has a 3 and 4 in it. 01:00:46.450 --> 01:00:50.430 align:middle line:84% And then we ask about A minus B. We take away 01:00:50.430 --> 01:00:52.740 align:middle line:84% from A everything that's in B, everything 01:00:52.740 --> 01:00:54.840 align:middle line:90% that's an element of B. 01:00:54.840 --> 01:00:58.810 align:middle line:84% So would we be removing the 3 and 4? 01:00:58.810 --> 01:01:03.900 align:middle line:84% So if we have a 6 and 1 and a 2 and a 0 and a 3 and 4, 01:01:03.900 --> 01:01:07.500 align:middle line:84% and we subtract the set-- what was it? 01:01:07.500 --> 01:01:12.930 align:middle line:90% 2 and 3, 4, and empty set. 01:01:12.930 --> 01:01:16.650 align:middle line:84% And the answer is no, the 3 and 4 don't go away. 01:01:16.650 --> 01:01:20.460 align:middle line:84% Because we want to take all the elements of the first set 01:01:20.460 --> 01:01:23.610 align:middle line:84% that are not elements of the second set. 01:01:23.610 --> 01:01:26.230 align:middle line:84% And 3 and 4 are not elements of the second set. 01:01:26.230 --> 01:01:29.370 align:middle line:90% They're nested too deep. 01:01:29.370 --> 01:01:31.630 align:middle line:90% 3 Is not an element of this set. 01:01:31.630 --> 01:01:35.910 align:middle line:84% This whole set is an element of this set. 01:01:35.910 --> 01:01:36.820 align:middle line:90% Make sense? 01:01:36.820 --> 01:01:37.360 align:middle line:90% Cool. 01:01:37.360 --> 01:01:42.120 align:middle line:84% So this difference would keep the 6 and the 1, not the 2, 01:01:42.120 --> 01:01:46.830 align:middle line:84% would keep the 0 and keep the 3 and keep the 4. 01:01:46.830 --> 01:01:49.030 align:middle line:90% Awesome. 01:01:49.030 --> 01:01:52.290 align:middle line:90% Similar to sets, we have-- 01:01:52.290 --> 01:01:56.760 align:middle line:84% oh, sorry, before that, one of the reasons I'm 01:01:56.760 --> 01:01:59.910 align:middle line:84% talking about sets right now, beyond the fact that we need 01:01:59.910 --> 01:02:03.870 align:middle line:84% to talk about sets, is the fact that sets 01:02:03.870 --> 01:02:08.500 align:middle line:84% and propositions and predicates are very closely related. 01:02:08.500 --> 01:02:10.535 align:middle line:84% And they can both be useful for the other. 01:02:10.535 --> 01:02:14.040 align:middle line:90% 01:02:14.040 --> 01:02:17.473 align:middle line:84% Specifically, I want to talk about set builder notation. 01:02:17.473 --> 01:02:29.490 align:middle line:90% 01:02:29.490 --> 01:02:32.170 align:middle line:84% And that is where you take a set, 01:02:32.170 --> 01:02:38.400 align:middle line:84% and you describe a subset a set by using a predicate. 01:02:38.400 --> 01:02:43.110 align:middle line:84% So for example, we can look at the set of all n 01:02:43.110 --> 01:02:47.465 align:middle line:84% in the natural numbers such that n is prime. 01:02:47.465 --> 01:02:54.840 align:middle line:90% 01:02:54.840 --> 01:02:58.410 align:middle line:84% So we have this predicate of n over here on the right. 01:02:58.410 --> 01:03:00.660 align:middle line:84% And we're pulling out all of the elements 01:03:00.660 --> 01:03:02.980 align:middle line:84% of this set, the natural numbers, 01:03:02.980 --> 01:03:06.420 align:middle line:84% but only the ones that satisfy our predicate. 01:03:06.420 --> 01:03:08.380 align:middle line:90% So this is set builder notation. 01:03:08.380 --> 01:03:11.970 align:middle line:84% You have elements a set such that this predicate 01:03:11.970 --> 01:03:13.100 align:middle line:90% is satisfied. 01:03:13.100 --> 01:03:15.430 align:middle line:90% This is the notation we'll use. 01:03:15.430 --> 01:03:20.350 align:middle line:84% And this set, of course, will return the set of primes-- 01:03:20.350 --> 01:03:27.300 align:middle line:84% 2, 3, 5, 7, not 9, globe, 11, and so on. 01:03:27.300 --> 01:03:30.910 align:middle line:90% 01:03:30.910 --> 01:03:34.700 align:middle line:84% Very useful, we're going to use that all the time. 01:03:34.700 --> 01:03:36.810 align:middle line:90% Similar to sets, we have tuples. 01:03:36.810 --> 01:03:39.970 align:middle line:90% 01:03:39.970 --> 01:03:42.543 align:middle line:84% This is an ordered list of elements. 01:03:42.543 --> 01:03:49.480 align:middle line:90% 01:03:49.480 --> 01:03:50.955 align:middle line:90% And repeats are OK. 01:03:50.955 --> 01:03:54.610 align:middle line:90% 01:03:54.610 --> 01:03:56.440 align:middle line:84% And it's written with parentheses 01:03:56.440 --> 01:03:57.453 align:middle line:90% instead of curly braces. 01:03:57.453 --> 01:04:07.270 align:middle line:90% 01:04:07.270 --> 01:04:13.880 align:middle line:84% For example, the tuple 6, 1, 2, 0 01:04:13.880 --> 01:04:17.210 align:middle line:84% is a sequence with four elements. 01:04:17.210 --> 01:04:23.120 align:middle line:84% This is not the same as 6, 1, 2, 0, 0, which 01:04:23.120 --> 01:04:26.330 align:middle line:90% is a tuple with five elements. 01:04:26.330 --> 01:04:27.780 align:middle line:90% As sets, they're the same. 01:04:27.780 --> 01:04:29.220 align:middle line:90% They have the same elements. 01:04:29.220 --> 01:04:31.880 align:middle line:90% But this one has repeats. 01:04:31.880 --> 01:04:33.230 align:middle line:90% So they're different. 01:04:33.230 --> 01:04:39.553 align:middle line:84% And likewise, neither of these is equivalent to 2, 1, 6, 0. 01:04:39.553 --> 01:04:41.720 align:middle line:84% Because the order is different, even if the elements 01:04:41.720 --> 01:04:42.750 align:middle line:90% are the same. 01:04:42.750 --> 01:04:46.550 align:middle line:84% So a tuple cares about the order. 01:04:46.550 --> 01:04:51.500 align:middle line:84% By the way, last I checked 2.160 is identification, estimation, 01:04:51.500 --> 01:04:55.230 align:middle line:84% and learning in MechE, not the same, 01:04:55.230 --> 01:04:58.558 align:middle line:84% even though they have the same digits. 01:04:58.558 --> 01:05:00.650 align:middle line:84% It doesn't really make sense to ask 01:05:00.650 --> 01:05:02.300 align:middle line:84% for the intersection or the union 01:05:02.300 --> 01:05:05.390 align:middle line:90% or the difference of tuples. 01:05:05.390 --> 01:05:08.060 align:middle line:90% So we're not even going to try. 01:05:08.060 --> 01:05:12.030 align:middle line:84% Any questions about sets or tuples? 01:05:12.030 --> 01:05:14.740 align:middle line:90% Nice. 01:05:14.740 --> 01:05:19.070 align:middle line:84% Let's go back to the definition I probably deleted already. 01:05:19.070 --> 01:05:21.750 align:middle line:90% 01:05:21.750 --> 01:05:24.720 align:middle line:84% A proof is a verification of a proposition 01:05:24.720 --> 01:05:28.170 align:middle line:84% by a sequence of logical deductions from a base 01:05:28.170 --> 01:05:29.770 align:middle line:90% set of axioms. 01:05:29.770 --> 01:05:31.240 align:middle line:90% I hope I got that right. 01:05:31.240 --> 01:05:33.990 align:middle line:84% We've already talked at length about propositions, 01:05:33.990 --> 01:05:36.210 align:middle line:90% one of those three terms. 01:05:36.210 --> 01:05:40.260 align:middle line:84% Let's turn to the next one, axioms. 01:05:40.260 --> 01:05:42.845 align:middle line:84% We've used both of those boards, on to this board. 01:05:42.845 --> 01:05:53.410 align:middle line:90% 01:05:53.410 --> 01:05:53.910 align:middle line:90% Nice. 01:05:53.910 --> 01:05:57.720 align:middle line:90% 01:05:57.720 --> 01:06:14.940 align:middle line:84% And axiom is a proposition we assume is true. 01:06:14.940 --> 01:06:18.580 align:middle line:90% 01:06:18.580 --> 01:06:21.257 align:middle line:84% You might have heard, when you're 01:06:21.257 --> 01:06:23.840 align:middle line:84% doing mathematical proofs, you shouldn't be assuming anything. 01:06:23.840 --> 01:06:25.540 align:middle line:90% You have to prove everything. 01:06:25.540 --> 01:06:27.170 align:middle line:90% And that's not quite true. 01:06:27.170 --> 01:06:28.550 align:middle line:90% We need somewhere to start. 01:06:28.550 --> 01:06:32.620 align:middle line:84% We need some foundation to build up from. 01:06:32.620 --> 01:06:35.780 align:middle line:84% The trick is to declare all of them in advance. 01:06:35.780 --> 01:06:38.050 align:middle line:90% These are the axioms I'm using. 01:06:38.050 --> 01:06:43.270 align:middle line:84% And if you assume these axioms, then these proofs 01:06:43.270 --> 01:06:46.420 align:middle line:84% tell you other things that also have to be true. 01:06:46.420 --> 01:06:49.130 align:middle line:84% So the axioms are the things you start with. 01:06:49.130 --> 01:06:51.080 align:middle line:90% We take them to be true. 01:06:51.080 --> 01:06:53.840 align:middle line:84% Sometimes, people say we take them to be self-evident. 01:06:53.840 --> 01:06:55.690 align:middle line:90% But I don't like that phrasing. 01:06:55.690 --> 01:07:03.380 align:middle line:84% Because they're not necessarily true in some holistic sense. 01:07:03.380 --> 01:07:05.380 align:middle line:84% We just assume them to be true for the sake 01:07:05.380 --> 01:07:07.520 align:middle line:90% of the math we're doing now. 01:07:07.520 --> 01:07:09.980 align:middle line:84% And in fact, you can often have axioms 01:07:09.980 --> 01:07:12.830 align:middle line:84% that contradict each other, as long as you're not 01:07:12.830 --> 01:07:15.230 align:middle line:90% using them at the same time. 01:07:15.230 --> 01:07:18.780 align:middle line:90% For example, let's see. 01:07:18.780 --> 01:07:32.090 align:middle line:84% For example, we have the famous Euclid's parallel postulate. 01:07:32.090 --> 01:07:35.670 align:middle line:84% So Euclid, in 200 BC, or whenever he was alive, 01:07:35.670 --> 01:07:39.050 align:middle line:84% maybe 2000-- someone can tell me-- 01:07:39.050 --> 01:07:42.390 align:middle line:84% wrote a book, The Elements, Euclid's Elements, 01:07:42.390 --> 01:07:45.150 align:middle line:84% that chronicled all of Euclidean geometry. 01:07:45.150 --> 01:07:47.130 align:middle line:84% That's why it's called Euclidean geometry. 01:07:47.130 --> 01:07:49.910 align:middle line:84% He literally wrote the book on it. 01:07:49.910 --> 01:07:53.460 align:middle line:84% And at the beginning of the book, he had five axioms. 01:07:53.460 --> 01:07:56.820 align:middle line:84% "Postulate" is another word for "axiom," a little outdated, 01:07:56.820 --> 01:07:59.360 align:middle line:90% but still there. 01:07:59.360 --> 01:08:03.260 align:middle line:84% And parallel postulate, the fifth of the five axioms, 01:08:03.260 --> 01:08:07.120 align:middle line:84% famously for centuries, millennia, people 01:08:07.120 --> 01:08:09.820 align:middle line:84% kept trying to prove that actually, the parallel postulate 01:08:09.820 --> 01:08:11.540 align:middle line:90% doesn't have to be an axiom. 01:08:11.540 --> 01:08:14.890 align:middle line:84% You can prove it's true if you only assume the first four 01:08:14.890 --> 01:08:16.479 align:middle line:90% axioms. 01:08:16.479 --> 01:08:18.140 align:middle line:84% So it doesn't have to be an axiom. 01:08:18.140 --> 01:08:19.750 align:middle line:90% It can be a theorem. 01:08:19.750 --> 01:08:22.279 align:middle line:84% And people kept trying and kept trying and kept trying. 01:08:22.279 --> 01:08:24.880 align:middle line:84% By the way, the statement of the parallel postulate 01:08:24.880 --> 01:08:33.880 align:middle line:84% is that for every point p and line 01:08:33.880 --> 01:08:41.439 align:middle line:84% l that doesn't contain p, p not in l, 01:08:41.439 --> 01:08:59.740 align:middle line:84% there exists a unique line l prime through p parallel to l. 01:08:59.740 --> 01:09:03.710 align:middle line:84% In picture, if you have a point here and some line that 01:09:03.710 --> 01:09:06.859 align:middle line:84% doesn't go through that point, then there's 01:09:06.859 --> 01:09:11.279 align:middle line:84% a unique line parallel and going through your point. 01:09:11.279 --> 01:09:11.930 align:middle line:90% Here's p. 01:09:11.930 --> 01:09:13.520 align:middle line:90% Here's l. 01:09:13.520 --> 01:09:14.760 align:middle line:90% Intuitive, right? 01:09:14.760 --> 01:09:16.670 align:middle line:90% Makes sense. 01:09:16.670 --> 01:09:21.109 align:middle line:84% And Euclid took this as an axiom and went with it. 01:09:21.109 --> 01:09:25.229 align:middle line:84% But can we prove this, based on the other four axioms? 01:09:25.229 --> 01:09:27.140 align:middle line:84% And turns out, thousands of years 01:09:27.140 --> 01:09:29.569 align:middle line:84% later, we finally know the answer. 01:09:29.569 --> 01:09:31.160 align:middle line:90% No. 01:09:31.160 --> 01:09:32.270 align:middle line:90% No. 01:09:32.270 --> 01:09:33.569 align:middle line:90% This is not a theorem. 01:09:33.569 --> 01:09:36.229 align:middle line:84% It does not follow from the other four axioms. 01:09:36.229 --> 01:09:43.370 align:middle line:84% Because we know what happens if you change the axiom. 01:09:43.370 --> 01:09:46.370 align:middle line:84% Things do happen in a meaningful way. 01:09:46.370 --> 01:09:49.250 align:middle line:84% For example, instead of saying there 01:09:49.250 --> 01:09:57.080 align:middle line:84% is a unique line, a unique line, let's change that to no lines. 01:09:57.080 --> 01:10:01.110 align:middle line:84% There are no lines through your point parallel to your line. 01:10:01.110 --> 01:10:03.090 align:middle line:84% The concept of parallel lines doesn't 01:10:03.090 --> 01:10:08.640 align:middle line:84% exist if we use that axiom instead of Euclid's version 01:10:08.640 --> 01:10:13.470 align:middle line:84% and the other four axioms that Euclid did, 01:10:13.470 --> 01:10:14.873 align:middle line:90% you can get spherical geometry. 01:10:14.873 --> 01:10:19.650 align:middle line:90% 01:10:19.650 --> 01:10:25.890 align:middle line:84% In spherical geometry, your plane, your canvas, 01:10:25.890 --> 01:10:28.560 align:middle line:90% is a sphere instead of a plane. 01:10:28.560 --> 01:10:32.220 align:middle line:90% A line is a great circle. 01:10:32.220 --> 01:10:36.000 align:middle line:84% It's a circle whose center is the center of the sphere. 01:10:36.000 --> 01:10:39.450 align:middle line:84% And a point isn't just a point, but it's a point, 01:10:39.450 --> 01:10:44.460 align:middle line:84% and it's antipode, the one directly across from it 01:10:44.460 --> 01:10:45.780 align:middle line:90% on the sphere. 01:10:45.780 --> 01:10:49.200 align:middle line:84% And if that's how we define points and lines, 01:10:49.200 --> 01:10:50.800 align:middle line:84% then there are no parallel lines. 01:10:50.800 --> 01:10:54.570 align:middle line:84% All great circles intersect each other. 01:10:54.570 --> 01:10:57.900 align:middle line:84% And we get a meaningful geometry that we can prove stuff with, 01:10:57.900 --> 01:11:02.800 align:middle line:84% that we can prove different things with than Euclid could. 01:11:02.800 --> 01:11:04.820 align:middle line:84% Since we have a different set of axioms, 01:11:04.820 --> 01:11:07.040 align:middle line:84% we're going to have a different set of conclusions. 01:11:07.040 --> 01:11:09.910 align:middle line:90% But it's still meaningful. 01:11:09.910 --> 01:11:13.250 align:middle line:84% Likewise, if we replace this-- instead of no lines, 01:11:13.250 --> 01:11:19.285 align:middle line:84% we do infinitely many lines, well, 01:11:19.285 --> 01:11:21.410 align:middle line:84% there's another version of geometry that does that. 01:11:21.410 --> 01:11:32.570 align:middle line:84% It's hyperbolic geometry, lots of ways to draw it. 01:11:32.570 --> 01:11:34.820 align:middle line:84% But sometimes, it's drawn in a circle. 01:11:34.820 --> 01:11:40.120 align:middle line:84% But now a line is a part of a circle that's 01:11:40.120 --> 01:11:42.370 align:middle line:90% perpendicular to the boundary. 01:11:42.370 --> 01:11:45.040 align:middle line:90% A point is just a point. 01:11:45.040 --> 01:11:48.500 align:middle line:84% But now if we have a line and a point not on that line, 01:11:48.500 --> 01:11:51.440 align:middle line:84% well, parallel just means they don't intersect. 01:11:51.440 --> 01:11:57.740 align:middle line:84% So, well, here is a line through p that doesn't intersect l. 01:11:57.740 --> 01:12:01.220 align:middle line:90% Well, here's another one. 01:12:01.220 --> 01:12:02.670 align:middle line:90% And here's another one. 01:12:02.670 --> 01:12:05.720 align:middle line:84% There are lots of them, infinitely many lines through p 01:12:05.720 --> 01:12:06.985 align:middle line:90% that don't touch l. 01:12:06.985 --> 01:12:09.950 align:middle line:90% 01:12:09.950 --> 01:12:12.290 align:middle line:84% And so this is a different kind of geometry 01:12:12.290 --> 01:12:16.640 align:middle line:84% that satisfies a different set of axioms. 01:12:16.640 --> 01:12:19.370 align:middle line:84% And so no, you can't prove the fifth axiom from the first four. 01:12:19.370 --> 01:12:21.950 align:middle line:84% Because you can change the fifth one and get new conclusions 01:12:21.950 --> 01:12:24.655 align:middle line:84% that make sense and aren't contradictory. 01:12:24.655 --> 01:12:26.030 align:middle line:84% Now, I've said a couple of times, 01:12:26.030 --> 01:12:31.730 align:middle line:84% we can change it and get a new system of geometry. 01:12:31.730 --> 01:12:34.610 align:middle line:84% Well, you can imagine you can always 01:12:34.610 --> 01:12:38.060 align:middle line:84% change your axioms to some other set of assumptions 01:12:38.060 --> 01:12:42.350 align:middle line:84% and make a mathematical theory out of it. 01:12:42.350 --> 01:12:45.530 align:middle line:84% Does any set of assumptions work? 01:12:45.530 --> 01:12:48.410 align:middle line:84% Are they all as good as the others? 01:12:48.410 --> 01:12:52.230 align:middle line:84% Well, there are a couple of things that can go wrong. 01:12:52.230 --> 01:12:54.890 align:middle line:84% There are a couple of properties you might want 01:12:54.890 --> 01:12:56.513 align:middle line:90% your set of axioms to satisfy. 01:12:56.513 --> 01:13:04.620 align:middle line:90% 01:13:04.620 --> 01:13:19.410 align:middle line:84% A set of axioms is consistent when you 01:13:19.410 --> 01:13:29.280 align:middle line:90% can't prove that false is true. 01:13:29.280 --> 01:13:32.220 align:middle line:90% And we need that, right? 01:13:32.220 --> 01:13:34.890 align:middle line:84% Let me see if that's how we defined it. 01:13:34.890 --> 01:13:38.730 align:middle line:84% Yeah, if no proposition can be proved and disproved, 01:13:38.730 --> 01:13:42.570 align:middle line:84% we need true and false to be different things. 01:13:42.570 --> 01:13:46.000 align:middle line:84% Because remember, false implies anything you want. 01:13:46.000 --> 01:13:48.190 align:middle line:84% That's sometimes called the principle of explosion, 01:13:48.190 --> 01:13:49.570 align:middle line:90% by the way, great name. 01:13:49.570 --> 01:13:51.160 align:middle line:90% False implies anything. 01:13:51.160 --> 01:13:55.520 align:middle line:84% So if false is true, and false implies anything, 01:13:55.520 --> 01:13:58.870 align:middle line:90% then everything is true. 01:13:58.870 --> 01:14:01.840 align:middle line:84% So we don't want to be able to prove 01:14:01.840 --> 01:14:05.710 align:middle line:84% that false is true, because then our theory is uninteresting. 01:14:05.710 --> 01:14:08.780 align:middle line:84% Because everything is true, and everything is false. 01:14:08.780 --> 01:14:13.420 align:middle line:84% So we definitely want a set of axioms that is consistent. 01:14:13.420 --> 01:14:17.770 align:middle line:84% Ideally, we would also have a set of axioms-- 01:14:17.770 --> 01:14:28.780 align:middle line:84% a set of axioms is complete when every true-- 01:14:28.780 --> 01:14:32.620 align:middle line:84% let's see, yeah, every true proposition 01:14:32.620 --> 01:14:40.980 align:middle line:90% can be proved from the axioms. 01:14:40.980 --> 01:14:46.030 align:middle line:90% 01:14:46.030 --> 01:14:48.770 align:middle line:84% We want things to be true or false. 01:14:48.770 --> 01:14:51.370 align:middle line:84% And we want proof to be the metric 01:14:51.370 --> 01:14:53.880 align:middle line:84% by which we decide whether something is true or false. 01:14:53.880 --> 01:14:56.840 align:middle line:84% If we can find a proof for it, it's true. 01:14:56.840 --> 01:14:58.490 align:middle line:84% Ideally, if it's not true, we'd be 01:14:58.490 --> 01:15:00.800 align:middle line:84% able to find a proof for the opposite, 01:15:00.800 --> 01:15:03.350 align:middle line:90% for the not of that proposition. 01:15:03.350 --> 01:15:06.500 align:middle line:84% So the baseline things, the two things 01:15:06.500 --> 01:15:08.870 align:middle line:84% you would want in a set of axioms 01:15:08.870 --> 01:15:11.820 align:middle line:84% that is nice to work with, you would want it to be consistent. 01:15:11.820 --> 01:15:14.600 align:middle line:84% You would want it to be complete. 01:15:14.600 --> 01:15:19.700 align:middle line:84% And then Kurt Godel comes along and says, nope, can't have both. 01:15:19.700 --> 01:15:23.450 align:middle line:84% There's what's called the Godel Incompleteness Theorem. 01:15:23.450 --> 01:15:29.540 align:middle line:84% That's G-O-umlaut-D-E-L. Godel was able to prove that you 01:15:29.540 --> 01:15:32.810 align:middle line:84% can't have a system of axioms that is both consistent 01:15:32.810 --> 01:15:38.420 align:middle line:84% and complete, as long as it's complicated enough to just do 01:15:38.420 --> 01:15:43.040 align:middle line:84% arithmetic, addition and multiplication. 01:15:43.040 --> 01:15:46.580 align:middle line:90% You can't have both. 01:15:46.580 --> 01:15:48.890 align:middle line:84% We certainly want to be able to add and multiply 01:15:48.890 --> 01:15:52.900 align:middle line:84% numbers, which means we have to pick consistency 01:15:52.900 --> 01:15:54.230 align:middle line:90% or completeness. 01:15:54.230 --> 01:15:56.330 align:middle line:84% And I know which one I'm picking. 01:15:56.330 --> 01:15:59.800 align:middle line:84% I'm going to pick consistent, so that true and false 01:15:59.800 --> 01:16:01.090 align:middle line:90% have meaning. 01:16:01.090 --> 01:16:05.770 align:middle line:84% What this means-- and this kind of messes 01:16:05.770 --> 01:16:07.840 align:middle line:90% with my head a little bit. 01:16:07.840 --> 01:16:09.820 align:middle line:84% What Gödel's completeness theorem-- 01:16:09.820 --> 01:16:13.750 align:middle line:84% sorry, Incompleteness Theorem tells us is that there are true 01:16:13.750 --> 01:16:17.350 align:middle line:84% statements in math, in whatever set of axioms we're working 01:16:17.350 --> 01:16:22.325 align:middle line:84% with, there are true statements that cannot be proved. 01:16:22.325 --> 01:16:24.200 align:middle line:84% You're never going to be able to disprove it. 01:16:24.200 --> 01:16:28.240 align:middle line:84% But if you want to prove it, you have to add more axioms. 01:16:28.240 --> 01:16:30.200 align:middle line:84% But then once you add more axioms, 01:16:30.200 --> 01:16:33.430 align:middle line:84% that bigger set of axioms now has other true statements 01:16:33.430 --> 01:16:37.300 align:middle line:84% that can't be proved with those axioms. 01:16:37.300 --> 01:16:41.920 align:middle line:84% So conceivably, we might ask you to prove a problem in homework 01:16:41.920 --> 01:16:45.800 align:middle line:84% that can't be proved from the axioms, even though it's true. 01:16:45.800 --> 01:16:47.180 align:middle line:90% We're not going to do that. 01:16:47.180 --> 01:16:48.530 align:middle line:90% That's mean. 01:16:48.530 --> 01:16:51.050 align:middle line:84% Conceivably, Goldbach's Conjecture 01:16:51.050 --> 01:16:54.410 align:middle line:90% might be true, but unprovable. 01:16:54.410 --> 01:16:56.930 align:middle line:90% Maybe, I don't know. 01:16:56.930 --> 01:16:59.270 align:middle line:90% Question? 01:16:59.270 --> 01:17:00.410 align:middle line:90% STUDENT: [INAUDIBLE]. 01:17:00.410 --> 01:17:03.230 align:middle line:90% 01:17:03.230 --> 01:17:04.860 align:middle line:84% Zachary Abel: Yeah, excellent question. 01:17:04.860 --> 01:17:08.990 align:middle line:84% If something doesn't have a proof, how do we know it's true? 01:17:08.990 --> 01:17:10.680 align:middle line:90% That is a really good question. 01:17:10.680 --> 01:17:14.120 align:middle line:84% And that's also getting back into the set theory and logic 01:17:14.120 --> 01:17:16.280 align:middle line:84% and underpinnings of math that are too 01:17:16.280 --> 01:17:18.230 align:middle line:90% in the weeds for this course. 01:17:18.230 --> 01:17:22.070 align:middle line:84% But I'm happy to discuss it afterward if you'd like. 01:17:22.070 --> 01:17:24.950 align:middle line:84% Last thing I want to say about axioms. 01:17:24.950 --> 01:17:27.530 align:middle line:84% Every mathematician needs a set of axioms 01:17:27.530 --> 01:17:29.150 align:middle line:84% that they're working with so they 01:17:29.150 --> 01:17:30.890 align:middle line:84% know what they're allowed to assume 01:17:30.890 --> 01:17:34.790 align:middle line:90% as they're proving new things. 01:17:34.790 --> 01:17:38.930 align:middle line:84% What axioms are we going to use in this class? 01:17:38.930 --> 01:17:42.950 align:middle line:84% Roughly speaking, all of high school math. 01:17:42.950 --> 01:17:46.610 align:middle line:84% Let's say that any fact that you think 01:17:46.610 --> 01:17:50.940 align:middle line:84% is general enough and familiar enough and simple enough 01:17:50.940 --> 01:17:53.340 align:middle line:84% to be included in basically, every high school math 01:17:53.340 --> 01:17:54.460 align:middle line:90% curriculum. 01:17:54.460 --> 01:17:58.110 align:middle line:84% You're allowed to assume it, and go from there. 01:17:58.110 --> 01:18:00.850 align:middle line:84% Ideally, we ask you to identify it. 01:18:00.850 --> 01:18:03.700 align:middle line:90% Write down fact. 01:18:03.700 --> 01:18:05.085 align:middle line:84% Write the theorem you're stating, 01:18:05.085 --> 01:18:06.990 align:middle line:84% or the theorem you're assuming, like, 01:18:06.990 --> 01:18:10.710 align:middle line:84% the product of two even numbers is even. 01:18:10.710 --> 01:18:12.370 align:middle line:90% I know this from high school. 01:18:12.370 --> 01:18:14.220 align:middle line:90% And then go ahead and use it. 01:18:14.220 --> 01:18:17.040 align:middle line:84% Because ideally, at every step in a proof, 01:18:17.040 --> 01:18:20.680 align:middle line:90% we know why each thing is true. 01:18:20.680 --> 01:18:23.320 align:middle line:84% And if sometimes the reason is that's an axiom, 01:18:23.320 --> 01:18:25.385 align:middle line:84% I'm assuming it, we should say that. 01:18:25.385 --> 01:18:28.350 align:middle line:90% 01:18:28.350 --> 01:18:31.890 align:middle line:90% Are there any questions? 01:18:31.890 --> 01:18:33.490 align:middle line:90% All right, that's all I've got. 01:18:33.490 --> 01:18:35.760 align:middle line:84% Be sure to go to recitation tomorrow. 01:18:35.760 --> 01:18:37.720 align:middle line:84% You can check your time on Canvas. 01:18:37.720 --> 01:18:39.580 align:middle line:84% You can switch your time on Canvas. 01:18:39.580 --> 01:18:41.230 align:middle line:84% We look forward to seeing you then. 01:18:41.230 --> 01:18:43.250 align:middle line:90% Thank you so much. 01:18:43.250 --> 01:18:47.000 align:middle line:90%