WEBVTT 00:00:00.000 --> 00:00:02.395 align:middle line:90% [SQUEAKING] 00:00:02.395 --> 00:00:04.311 align:middle line:90% [RUSTLING] 00:00:04.311 --> 00:00:05.748 align:middle line:90% [CLICKING] 00:00:05.748 --> 00:00:11.017 align:middle line:90% 00:00:11.017 --> 00:00:14.070 align:middle line:84% ZACHARY ABEL: Good morning, good morning. 00:00:14.070 --> 00:00:16.430 align:middle line:90% Afternoon, I guess, technically. 00:00:16.430 --> 00:00:19.460 align:middle line:84% Welcome to lecture 15 of 6.1200-- 00:00:19.460 --> 00:00:20.310 align:middle line:90% 6.120. 00:00:20.310 --> 00:00:22.730 align:middle line:84% Again, I still don't know how to call it. 00:00:22.730 --> 00:00:26.960 align:middle line:84% Today we are talking about relations and counting. 00:00:26.960 --> 00:00:29.070 align:middle line:84% And there's a bunch of scary stuff on the board. 00:00:29.070 --> 00:00:31.340 align:middle line:90% Ignore that. 00:00:31.340 --> 00:00:32.689 align:middle line:90% Wonderful. 00:00:32.689 --> 00:00:36.110 align:middle line:84% So first thing we're talking about is relations. 00:00:36.110 --> 00:00:40.940 align:middle line:84% And relations are a very useful tool, very useful term, 00:00:40.940 --> 00:00:44.300 align:middle line:84% generalizing the idea of a function. 00:00:44.300 --> 00:00:47.150 align:middle line:84% So let's see what we mean by that. 00:00:47.150 --> 00:00:47.940 align:middle line:90% Let's see. 00:00:47.940 --> 00:00:52.010 align:middle line:90% So-- yes. 00:00:52.010 --> 00:01:01.470 align:middle line:84% So a relation R is a subset of A cross B. A and B are sets. 00:01:01.470 --> 00:01:03.070 align:middle line:90% They can be any sets you want. 00:01:03.070 --> 00:01:06.863 align:middle line:84% R is going to be a subset of pairs, where 00:01:06.863 --> 00:01:08.280 align:middle line:84% the first element is taken from A, 00:01:08.280 --> 00:01:10.510 align:middle line:84% the second element is taken from B. 00:01:10.510 --> 00:01:13.080 align:middle line:84% It might be all the pairs, none of the pairs, some subset 00:01:13.080 --> 00:01:14.760 align:middle line:90% of the pairs. 00:01:14.760 --> 00:01:18.300 align:middle line:84% All we know is that we need all three of these pieces of data. 00:01:18.300 --> 00:01:28.780 align:middle line:84% A relation consists of, first of all, a domain A, 00:01:28.780 --> 00:01:43.980 align:middle line:84% which is any set, a codomain B, which is a set, 00:01:43.980 --> 00:01:50.250 align:middle line:84% and a subset R, subset of A cross B. 00:01:50.250 --> 00:01:53.920 align:middle line:84% So you need to know what is your domain, what is your codomain, 00:01:53.920 --> 00:01:56.513 align:middle line:84% and which pairs are part of your relation. 00:01:56.513 --> 00:01:59.130 align:middle line:90% 00:01:59.130 --> 00:02:01.332 align:middle line:84% This generalizes the idea of a function 00:02:01.332 --> 00:02:02.790 align:middle line:84% because you can think of a function 00:02:02.790 --> 00:02:05.710 align:middle line:84% as you have your inputs, you have your outputs, 00:02:05.710 --> 00:02:08.100 align:middle line:84% and your function specifies which 00:02:08.100 --> 00:02:11.100 align:middle line:84% inputs map to which outputs by just encoding them 00:02:11.100 --> 00:02:13.260 align:middle line:90% in a pair like this. 00:02:13.260 --> 00:02:16.440 align:middle line:84% Let me give a more concrete example. 00:02:16.440 --> 00:02:17.910 align:middle line:90% Here we go. 00:02:17.910 --> 00:02:20.770 align:middle line:84% So imagine we have these three sets. 00:02:20.770 --> 00:02:22.690 align:middle line:84% We have the set of students that are Luke, 00:02:22.690 --> 00:02:25.230 align:middle line:90% Geralt, Quentin, and Willow. 00:02:25.230 --> 00:02:28.290 align:middle line:84% And we have some classes that these magic students might take 00:02:28.290 --> 00:02:30.240 align:middle line:90% in a magic school, I guess-- 00:02:30.240 --> 00:02:34.170 align:middle line:84% chemistry, sports, literature, and 6.1200. 00:02:34.170 --> 00:02:36.120 align:middle line:90% Totally makes sense. 00:02:36.120 --> 00:02:39.730 align:middle line:84% And we have some magic profs-- we have Galadriel, Zelda, 00:02:39.730 --> 00:02:43.050 align:middle line:90% Eda, and Doctor Strange. 00:02:43.050 --> 00:02:44.500 align:middle line:90% So we have some sets. 00:02:44.500 --> 00:02:47.400 align:middle line:84% We might specify a relation, let's call it 00:02:47.400 --> 00:02:49.470 align:middle line:84% the learning relation, L, which is 00:02:49.470 --> 00:02:53.380 align:middle line:84% going to be a subset of students across classes. 00:02:53.380 --> 00:02:55.530 align:middle line:84% So it's going to be all pairs where you have 00:02:55.530 --> 00:02:58.060 align:middle line:90% a student who's taking a class. 00:02:58.060 --> 00:03:02.140 align:middle line:84% And in this case, it looks like we have these students taking 00:03:02.140 --> 00:03:03.290 align:middle line:90% these classes. 00:03:03.290 --> 00:03:05.570 align:middle line:84% It looks like Luke is taking chemistry. 00:03:05.570 --> 00:03:07.700 align:middle line:90% Luke is also taking literature. 00:03:07.700 --> 00:03:09.910 align:middle line:84% Geralt is taking chemistry and sports 00:03:09.910 --> 00:03:12.370 align:middle line:90% and literature, et cetera. 00:03:12.370 --> 00:03:14.860 align:middle line:84% So we need all three of these pieces of information-- 00:03:14.860 --> 00:03:18.310 align:middle line:84% the domain, the codomain, and the set of pairs in order 00:03:18.310 --> 00:03:20.350 align:middle line:90% to specify this relation. 00:03:20.350 --> 00:03:22.510 align:middle line:84% Likewise, here's another relation-- 00:03:22.510 --> 00:03:25.640 align:middle line:84% which classes are being taught by which professors. 00:03:25.640 --> 00:03:30.220 align:middle line:84% So this is a subset of classes times professors. 00:03:30.220 --> 00:03:34.210 align:middle line:84% And it's saying that this class chemistry is being 00:03:34.210 --> 00:03:35.900 align:middle line:90% taught by this professor, Eda. 00:03:35.900 --> 00:03:39.050 align:middle line:84% Sports is being taught by Strange, lit by Galadriel, 00:03:39.050 --> 00:03:41.110 align:middle line:84% and 6.1200 is being taught by Zach-- 00:03:41.110 --> 00:03:42.260 align:middle line:90% I mean Zelda. 00:03:42.260 --> 00:03:43.330 align:middle line:90% [SOFT LAUGHTER] 00:03:43.330 --> 00:03:44.160 align:middle line:90% Thank you. 00:03:44.160 --> 00:03:46.720 align:middle line:90% 00:03:46.720 --> 00:03:49.250 align:middle line:90% This might look pretty familiar. 00:03:49.250 --> 00:03:54.010 align:middle line:84% This is almost how we define directed graphs, right? 00:03:54.010 --> 00:03:57.920 align:middle line:84% You have some set of vertices or whatnot, 00:03:57.920 --> 00:03:59.570 align:middle line:84% and you have some edges between them. 00:03:59.570 --> 00:04:02.470 align:middle line:84% This is a very closely related concept. 00:04:02.470 --> 00:04:08.380 align:middle line:84% And just like with graphs, that representation 00:04:08.380 --> 00:04:11.390 align:middle line:84% isn't the most useful way to represent it. 00:04:11.390 --> 00:04:14.560 align:middle line:84% It's kind of hard to see who's taking what 00:04:14.560 --> 00:04:16.579 align:middle line:84% and visualize and think about it. 00:04:16.579 --> 00:04:18.470 align:middle line:90% So let's draw it instead. 00:04:18.470 --> 00:04:23.170 align:middle line:84% So we already said here we have the learning relation, L. We 00:04:23.170 --> 00:04:24.880 align:middle line:90% said Luke is taking chemistry. 00:04:24.880 --> 00:04:27.520 align:middle line:84% And I'm going to draw it with a directed arrow. 00:04:27.520 --> 00:04:29.900 align:middle line:90% Luke is also taking literature. 00:04:29.900 --> 00:04:34.900 align:middle line:84% Geralt is taking chemistry and sports. 00:04:34.900 --> 00:04:38.590 align:middle line:84% Geralt is taking also literature. 00:04:38.590 --> 00:04:41.540 align:middle line:84% Then Quentin is taking sports and lit. 00:04:41.540 --> 00:04:44.800 align:middle line:84% Quentin is taking sports and lit. 00:04:44.800 --> 00:04:47.060 align:middle line:90% And Willow is taking lit. 00:04:47.060 --> 00:04:51.340 align:middle line:84% And apparently no one is taking 6.1200 at this magic school-- 00:04:51.340 --> 00:04:52.880 align:middle line:90% shame for them. 00:04:52.880 --> 00:04:56.640 align:middle line:84% But this is a very clear representation in my mind. 00:04:56.640 --> 00:04:59.250 align:middle line:84% But again, it's representing the same information. 00:04:59.250 --> 00:05:01.700 align:middle line:84% And we can see that, in fact, this is exactly 00:05:01.700 --> 00:05:08.030 align:middle line:84% a directed graph where the vertices are, 00:05:08.030 --> 00:05:10.530 align:middle line:84% what is it, students union classes. 00:05:10.530 --> 00:05:12.330 align:middle line:90% Those are allowed vertices. 00:05:12.330 --> 00:05:14.240 align:middle line:84% And it just has the special form where 00:05:14.240 --> 00:05:17.870 align:middle line:84% each one of our directed arrows always goes from a student 00:05:17.870 --> 00:05:19.050 align:middle line:90% to a class. 00:05:19.050 --> 00:05:22.130 align:middle line:84% But otherwise, it's just a directed graph. 00:05:22.130 --> 00:05:23.070 align:middle line:90% Does that make sense? 00:05:23.070 --> 00:05:25.430 align:middle line:84% Let's quickly draw the other one. 00:05:25.430 --> 00:05:28.700 align:middle line:84% So we have chemistry being taught by Eda. 00:05:28.700 --> 00:05:34.760 align:middle line:84% We have sports being taught by Strange, lit by Galadriel, 00:05:34.760 --> 00:05:37.610 align:middle line:90% and 6.1200 by Zelda. 00:05:37.610 --> 00:05:43.430 align:middle line:84% So this is the teaching relation. 00:05:43.430 --> 00:05:47.780 align:middle line:84% Are we clear on what these examples are? 00:05:47.780 --> 00:05:50.360 align:middle line:90% Wonderful. 00:05:50.360 --> 00:05:55.370 align:middle line:84% So for notation, there are lots of different notations 00:05:55.370 --> 00:05:57.950 align:middle line:84% that are used for relations because relations are 00:05:57.950 --> 00:06:00.170 align:middle line:90% used in lots of different ways. 00:06:00.170 --> 00:06:02.206 align:middle line:84% But here are some common notations. 00:06:02.206 --> 00:06:04.910 align:middle line:90% 00:06:04.910 --> 00:06:09.950 align:middle line:84% So if R is a relation from A to B, 00:06:09.950 --> 00:06:19.550 align:middle line:84% then we might write a relates to b to mean that the pair a comma 00:06:19.550 --> 00:06:22.680 align:middle line:84% b is one of the pairs in our relation, 00:06:22.680 --> 00:06:27.180 align:middle line:84% one of the directed edges in our graph, because, like I said, 00:06:27.180 --> 00:06:30.860 align:middle line:84% the relation is really about specifying which pairs relate 00:06:30.860 --> 00:06:32.130 align:middle line:90% and which pairs don't. 00:06:32.130 --> 00:06:33.177 align:middle line:90% It's a yes/no question. 00:06:33.177 --> 00:06:34.260 align:middle line:90% Some of the pairs are yes. 00:06:34.260 --> 00:06:35.550 align:middle line:90% Some of the pairs are no. 00:06:35.550 --> 00:06:39.980 align:middle line:84% And so we might write a relates to b with this infix notation 00:06:39.980 --> 00:06:42.350 align:middle line:90% to say that a relates to b. 00:06:42.350 --> 00:06:45.260 align:middle line:84% We're saying the same thing, just with different notation. 00:06:45.260 --> 00:06:53.040 align:middle line:84% And this might remind you of things like 3 is less than 7. 00:06:53.040 --> 00:06:56.730 align:middle line:84% A set S is a subset of a set T. This 00:06:56.730 --> 00:06:59.290 align:middle line:84% less-than relation, this subset equal relation, 00:06:59.290 --> 00:07:00.850 align:middle line:90% these are relations. 00:07:00.850 --> 00:07:02.880 align:middle line:84% And this general notion of relation 00:07:02.880 --> 00:07:05.740 align:middle line:84% is a way to generalize concepts like this. 00:07:05.740 --> 00:07:08.670 align:middle line:84% And so this is why we sometimes have our infix notation 00:07:08.670 --> 00:07:12.270 align:middle line:84% like that, so we can think about whatever other relation we're 00:07:12.270 --> 00:07:14.370 align:middle line:84% talking about similar to how we think 00:07:14.370 --> 00:07:16.770 align:middle line:90% about these more familiar ones. 00:07:16.770 --> 00:07:17.495 align:middle line:90% All right. 00:07:17.495 --> 00:07:20.670 align:middle line:90% 00:07:20.670 --> 00:07:23.530 align:middle line:84% We can also think of R as a predicate. 00:07:23.530 --> 00:07:28.770 align:middle line:90% We can say R of a comma b. 00:07:28.770 --> 00:07:32.070 align:middle line:84% So if R is a predicate that takes two arguments and spits 00:07:32.070 --> 00:07:34.290 align:middle line:84% out true or false, this is another way 00:07:34.290 --> 00:07:36.453 align:middle line:84% you can think of specifying a subset of pairs. 00:07:36.453 --> 00:07:39.000 align:middle line:90% 00:07:39.000 --> 00:07:44.620 align:middle line:84% So R a, b is true when a, b is part of our relation, 00:07:44.620 --> 00:07:45.845 align:middle line:90% and it's false when it's not. 00:07:45.845 --> 00:07:50.530 align:middle line:90% 00:07:50.530 --> 00:08:00.640 align:middle line:84% And phrase-wise, we might say that a relates to b. 00:08:00.640 --> 00:08:04.360 align:middle line:90% Again, means the same thing-- 00:08:04.360 --> 00:08:08.620 align:middle line:90% means that a, b is in R. 00:08:08.620 --> 00:08:12.340 align:middle line:84% At its core, we are just trying to give one bit of information 00:08:12.340 --> 00:08:13.220 align:middle line:90% for each pair. 00:08:13.220 --> 00:08:15.830 align:middle line:84% Either this pair relates, or this pair doesn't. 00:08:15.830 --> 00:08:17.810 align:middle line:84% This pair is in our set or it's not. 00:08:17.810 --> 00:08:20.210 align:middle line:84% This pair satisfies our predicate or it doesn't. 00:08:20.210 --> 00:08:23.680 align:middle line:84% All different ways of saying the same thing. 00:08:23.680 --> 00:08:25.070 align:middle line:90% Does that make sense? 00:08:25.070 --> 00:08:26.500 align:middle line:90% Wonderful. 00:08:26.500 --> 00:08:29.410 align:middle line:84% And by the way, when we say "relation" here, 00:08:29.410 --> 00:08:33.909 align:middle line:84% specifically we're talking about binary relation. 00:08:33.909 --> 00:08:36.133 align:middle line:84% Yeah, that should say a binary relation. 00:08:36.133 --> 00:08:40.062 align:middle line:90% 00:08:40.062 --> 00:08:41.770 align:middle line:84% But all the relations we're going to talk 00:08:41.770 --> 00:08:43.280 align:middle line:90% about today are binary. 00:08:43.280 --> 00:08:47.890 align:middle line:84% But it's the same relation as in relational database. 00:08:47.890 --> 00:08:50.410 align:middle line:84% You can imagine that this kind of thing 00:08:50.410 --> 00:08:53.450 align:middle line:84% is exactly how a database stores its information. 00:08:53.450 --> 00:08:55.630 align:middle line:84% In this case, we're only talking about databases 00:08:55.630 --> 00:08:58.600 align:middle line:90% with two columns. 00:08:58.600 --> 00:09:03.590 align:middle line:84% So learning has a column for student and a column for class. 00:09:03.590 --> 00:09:06.440 align:middle line:84% And there's a row every time a student is taking a class. 00:09:06.440 --> 00:09:07.430 align:middle line:90% That's all this is. 00:09:07.430 --> 00:09:08.226 align:middle line:90% It's a database. 00:09:08.226 --> 00:09:13.990 align:middle line:90% 00:09:13.990 --> 00:09:16.420 align:middle line:90% Yes. 00:09:16.420 --> 00:09:19.450 align:middle line:90% So I said earlier that-- 00:09:19.450 --> 00:09:22.060 align:middle line:90% here, let's go back up here-- 00:09:22.060 --> 00:09:25.370 align:middle line:84% that relations are a generalization of functions. 00:09:25.370 --> 00:09:26.955 align:middle line:84% And let's see what we mean by that. 00:09:26.955 --> 00:09:30.010 align:middle line:90% 00:09:30.010 --> 00:09:41.470 align:middle line:84% So definition-- relation R from A to B 00:09:41.470 --> 00:09:53.750 align:middle line:84% is a function precisely when every element on the left side 00:09:53.750 --> 00:10:04.430 align:middle line:84% relates to, at most, one element on the right side. 00:10:04.430 --> 00:10:07.910 align:middle line:84% So if everything in the left set appears 00:10:07.910 --> 00:10:10.400 align:middle line:84% on the left side of at most one pair, 00:10:10.400 --> 00:10:12.900 align:middle line:84% it's the source of at most one arrow. 00:10:12.900 --> 00:10:20.450 align:middle line:84% So another way to say that is every a in A 00:10:20.450 --> 00:10:24.655 align:middle line:90% has at most one arrow out. 00:10:24.655 --> 00:10:27.680 align:middle line:90% 00:10:27.680 --> 00:10:31.190 align:middle line:84% All right, so these two examples, 00:10:31.190 --> 00:10:33.790 align:middle line:84% is this learning relation a function? 00:10:33.790 --> 00:10:36.530 align:middle line:90% 00:10:36.530 --> 00:10:39.432 align:middle line:90% Who can tell me why? 00:10:39.432 --> 00:10:41.932 align:middle line:84% AUDIENCE: One of the students has multiple arrows going out. 00:10:41.932 --> 00:10:43.610 align:middle line:84% ZACHARY ABEL: Yeah, we have a student 00:10:43.610 --> 00:10:45.300 align:middle line:90% with multiple arrows going out. 00:10:45.300 --> 00:10:47.430 align:middle line:84% That's precisely what we need to avoid. 00:10:47.430 --> 00:10:50.360 align:middle line:84% Its function says each node on the left 00:10:50.360 --> 00:10:53.000 align:middle line:84% is supposed to have at most one arrow going out, 00:10:53.000 --> 00:10:54.390 align:middle line:90% and this fails that. 00:10:54.390 --> 00:10:57.350 align:middle line:90% Is this a function? 00:10:57.350 --> 00:10:59.330 align:middle line:90% Yeah. 00:10:59.330 --> 00:11:03.020 align:middle line:84% Every class has at most one arrow out, so it's a function. 00:11:03.020 --> 00:11:08.930 align:middle line:84% If I deleted this arrow, so now we only have these three arrows 00:11:08.930 --> 00:11:14.330 align:middle line:84% and no one's teaching 6.1200, is this still a function? 00:11:14.330 --> 00:11:18.600 align:middle line:84% Yeah, it's fine-- at most one arrow, not exactly one arrow. 00:11:18.600 --> 00:11:21.450 align:middle line:84% So that's still fine even if that arrow is missing. 00:11:21.450 --> 00:11:25.880 align:middle line:84% But I'm going to put it back because that's our example. 00:11:25.880 --> 00:11:28.400 align:middle line:90% Let's see. 00:11:28.400 --> 00:11:37.490 align:middle line:84% When R is a function, we'll write R of a equals b to mean-- 00:11:37.490 --> 00:11:40.125 align:middle line:90% 00:11:40.125 --> 00:11:43.270 align:middle line:84% means that a, b is in the relation. 00:11:43.270 --> 00:11:48.640 align:middle line:84% And so R of a is the element that a points to if it exists. 00:11:48.640 --> 00:11:52.440 align:middle line:84% R of a might be undefined if the function doesn't have 00:11:52.440 --> 00:11:54.190 align:middle line:90% any arrow out at this point. 00:11:54.190 --> 00:11:56.980 align:middle line:84% But if there's only one, we can just call it R of a. 00:11:56.980 --> 00:11:59.300 align:middle line:84% This is the usual function notation that we're used to. 00:11:59.300 --> 00:12:08.100 align:middle line:90% 00:12:08.100 --> 00:12:11.400 align:middle line:84% Let's make that even more precise. 00:12:11.400 --> 00:12:17.105 align:middle line:84% Let's look at the function f from R to R. 00:12:17.105 --> 00:12:18.480 align:middle line:84% By the way, when it's a function, 00:12:18.480 --> 00:12:21.510 align:middle line:84% we can use this more specific notation, 00:12:21.510 --> 00:12:25.380 align:middle line:84% f is a function from R to R. We use the arrow instead 00:12:25.380 --> 00:12:29.670 align:middle line:90% of f is a subset of R cross R. 00:12:29.670 --> 00:12:32.790 align:middle line:84% So let's define the function where f of x 00:12:32.790 --> 00:12:37.140 align:middle line:84% is 1 over x squared, the kind of function you 00:12:37.140 --> 00:12:40.710 align:middle line:84% might see in a precalculus class or something. 00:12:40.710 --> 00:12:44.070 align:middle line:84% Everyone agrees this is a function? 00:12:44.070 --> 00:12:45.430 align:middle line:90% Well, I mean, kind of. 00:12:45.430 --> 00:12:49.020 align:middle line:84% How is it a function by our definition of relation? 00:12:49.020 --> 00:12:51.330 align:middle line:84% We're supposed to specify a domain-- 00:12:51.330 --> 00:12:53.530 align:middle line:90% check-- a codomain-- check-- 00:12:53.530 --> 00:12:55.120 align:middle line:90% and a set of pairs. 00:12:55.120 --> 00:12:56.860 align:middle line:90% So what are the set of pairs? 00:12:56.860 --> 00:13:00.450 align:middle line:84% In this case, the set of pairs is 00:13:00.450 --> 00:13:07.680 align:middle line:84% going to be x comma y in R cross R such that y equals 1 00:13:07.680 --> 00:13:09.810 align:middle line:90% over x squared. 00:13:09.810 --> 00:13:14.790 align:middle line:84% It's just the set of pairs that satisfy our relation. 00:13:14.790 --> 00:13:16.500 align:middle line:90% Another way to write this-- 00:13:16.500 --> 00:13:25.290 align:middle line:84% x comma 1 over x squared, where x is a real number without 0 00:13:25.290 --> 00:13:28.710 align:middle line:84% because we're not allowed to divide by 0. 00:13:28.710 --> 00:13:30.633 align:middle line:90% All right, so f is a function. 00:13:30.633 --> 00:13:32.550 align:middle line:84% Here's a way to write it in terms of the pairs 00:13:32.550 --> 00:13:33.305 align:middle line:90% like we've said. 00:13:33.305 --> 00:13:37.150 align:middle line:90% 00:13:37.150 --> 00:13:39.010 align:middle line:84% And it's totally fine that f of 0 00:13:39.010 --> 00:13:41.710 align:middle line:84% isn't defined because, again, you can't divide by 0. 00:13:41.710 --> 00:13:44.750 align:middle line:84% f says each input has at most one output. 00:13:44.750 --> 00:13:46.840 align:middle line:84% And it just so happens that the input 0 00:13:46.840 --> 00:13:48.940 align:middle line:90% has no outputs, which is fine. 00:13:48.940 --> 00:13:51.400 align:middle line:84% Sometimes you do want everything to have-- 00:13:51.400 --> 00:13:54.250 align:middle line:90% every input to have an output. 00:13:54.250 --> 00:13:56.660 align:middle line:84% So we have a different term for that. 00:13:56.660 --> 00:14:06.310 align:middle line:84% We'll say that a relation is total if 00:14:06.310 --> 00:14:15.923 align:middle line:84% and only if every a in A has at least one arrow out. 00:14:15.923 --> 00:14:21.430 align:middle line:90% 00:14:21.430 --> 00:14:24.910 align:middle line:84% In other words, for every a on the left side, 00:14:24.910 --> 00:14:28.340 align:middle line:84% there's at least one b on the right side that it relates to. 00:14:28.340 --> 00:14:30.920 align:middle line:84% Everything on the left has some arrow going out, 00:14:30.920 --> 00:14:32.380 align:middle line:90% possibly multiple. 00:14:32.380 --> 00:14:33.675 align:middle line:90% That's what we call total. 00:14:33.675 --> 00:14:38.090 align:middle line:90% 00:14:38.090 --> 00:14:41.180 align:middle line:90% Are our examples here total? 00:14:41.180 --> 00:14:44.630 align:middle line:90% Is this one total. 00:14:44.630 --> 00:14:47.900 align:middle line:84% Yeah, everything on the left has one arrow out-- 00:14:47.900 --> 00:14:50.210 align:middle line:90% has at least one arrow out. 00:14:50.210 --> 00:14:53.730 align:middle line:84% Here, everything on the left has at least one arrow out. 00:14:53.730 --> 00:14:54.462 align:middle line:90% Excellent. 00:14:54.462 --> 00:14:56.990 align:middle line:90% 00:14:56.990 --> 00:14:58.405 align:middle line:90% Is this function total? 00:14:58.405 --> 00:15:04.143 align:middle line:90% 00:15:04.143 --> 00:15:05.072 align:middle line:90% AUDIENCE: No. 00:15:05.072 --> 00:15:05.780 align:middle line:90% ZACHARY ABEL: No? 00:15:05.780 --> 00:15:07.652 align:middle line:90% Why not? 00:15:07.652 --> 00:15:09.110 align:middle line:90% AUDIENCE: 0 has no arrow out. 00:15:09.110 --> 00:15:12.030 align:middle line:84% ZACHARY ABEL: 0 has no arrow out. f of 0 doesn't exist. 00:15:12.030 --> 00:15:14.750 align:middle line:84% There is nothing that 0 relates to on the right. 00:15:14.750 --> 00:15:19.820 align:middle line:84% OK, let's give a different example. 00:15:19.820 --> 00:15:28.520 align:middle line:84% Let's define g from the set of R without 0 to the set of R 00:15:28.520 --> 00:15:30.710 align:middle line:90% without 0-- 00:15:30.710 --> 00:15:33.260 align:middle line:90% this is supposed to be a set-- 00:15:33.260 --> 00:15:38.840 align:middle line:84% where we're going to define g of x is 1 over x squared. 00:15:38.840 --> 00:15:43.070 align:middle line:90% All right, same function, right? 00:15:43.070 --> 00:15:45.890 align:middle line:90% Same set of pairs. 00:15:45.890 --> 00:15:48.373 align:middle line:90% Is this function g total? 00:15:48.373 --> 00:15:53.960 align:middle line:90% 00:15:53.960 --> 00:15:59.000 align:middle line:84% I saw exactly two nods and a bunch of blank stares. 00:15:59.000 --> 00:16:02.030 align:middle line:90% That is all right. 00:16:02.030 --> 00:16:04.490 align:middle line:84% Someone who's-- oh, a couple of more nods. 00:16:04.490 --> 00:16:07.815 align:middle line:84% Someone who's nodding, can you tell me why g is total? 00:16:07.815 --> 00:16:08.315 align:middle line:90% Yes. 00:16:08.315 --> 00:16:11.850 align:middle line:84% AUDIENCE: It's total because, like in the previous example, 00:16:11.850 --> 00:16:15.440 align:middle line:84% 0 [INAUDIBLE] arrow out because 0 isn't an option. 00:16:15.440 --> 00:16:16.250 align:middle line:90% [INAUDIBLE] 00:16:16.250 --> 00:16:19.730 align:middle line:90% 00:16:19.730 --> 00:16:20.820 align:middle line:90% ZACHARY ABEL: Right. 00:16:20.820 --> 00:16:23.630 align:middle line:84% So they were saying, since 0 is no longer 00:16:23.630 --> 00:16:27.410 align:middle line:84% an option as an input, all of the inputs 00:16:27.410 --> 00:16:30.050 align:middle line:84% that we're allowed to choose do have a valid output assigned 00:16:30.050 --> 00:16:31.250 align:middle line:90% to them. 00:16:31.250 --> 00:16:35.850 align:middle line:90% So this function g is total. 00:16:35.850 --> 00:16:43.110 align:middle line:84% This function f is not total, even though it's exactly 00:16:43.110 --> 00:16:47.280 align:middle line:84% the same set of pairs for both f and g. 00:16:47.280 --> 00:16:50.160 align:middle line:84% It's exactly the same set of real comma real, 00:16:50.160 --> 00:16:54.450 align:middle line:84% or x comma 1 over x squared, for all the nonzero values of x. 00:16:54.450 --> 00:16:58.120 align:middle line:84% All we've done is change the domain and codomain. 00:16:58.120 --> 00:17:00.990 align:middle line:84% And this goes to show that you need all three parts 00:17:00.990 --> 00:17:02.470 align:middle line:90% of that definition of relation. 00:17:02.470 --> 00:17:04.990 align:middle line:84% You need to know what set you're mapping from, 00:17:04.990 --> 00:17:09.150 align:middle line:84% what set you're mapping to, and what all of your related pairs 00:17:09.150 --> 00:17:09.670 align:middle line:90% are. 00:17:09.670 --> 00:17:12.240 align:middle line:84% All three of those pieces of information 00:17:12.240 --> 00:17:14.460 align:middle line:84% are important to determine things 00:17:14.460 --> 00:17:17.430 align:middle line:84% like whether it's a function, whether it's total. 00:17:17.430 --> 00:17:19.950 align:middle line:90% Does that make sense? 00:17:19.950 --> 00:17:21.790 align:middle line:90% Awesome. 00:17:21.790 --> 00:17:26.880 align:middle line:90% Now there's-- oh, by the way-- 00:17:26.880 --> 00:17:29.250 align:middle line:84% oh, also, these often go together. 00:17:29.250 --> 00:17:37.360 align:middle line:84% So a total function means it has-- 00:17:37.360 --> 00:17:40.840 align:middle line:84% so total means everything on the left has at least one arrow out. 00:17:40.840 --> 00:17:44.830 align:middle line:84% Function means everything on the left has at most one arrow out. 00:17:44.830 --> 00:17:48.870 align:middle line:84% So if you put those together, total function 00:17:48.870 --> 00:18:02.850 align:middle line:84% has exactly one arrow out for each a in A. 00:18:02.850 --> 00:18:05.920 align:middle line:84% And secretly, when mathematicians say "function," 00:18:05.920 --> 00:18:09.180 align:middle line:84% they usually mean total function. 00:18:09.180 --> 00:18:13.390 align:middle line:84% Usually, the fact that some inputs don't have outputs, 00:18:13.390 --> 00:18:16.000 align:middle line:84% that's supposed to be a special case. 00:18:16.000 --> 00:18:18.600 align:middle line:84% And the default when we say function 00:18:18.600 --> 00:18:20.850 align:middle line:84% is that everything on the left has 00:18:20.850 --> 00:18:23.190 align:middle line:90% a definite output on the right. 00:18:23.190 --> 00:18:26.110 align:middle line:84% So when mathematicians say "function," 00:18:26.110 --> 00:18:29.050 align:middle line:90% we usually mean total function. 00:18:29.050 --> 00:18:32.240 align:middle line:84% When we're not assuming that a function is total, 00:18:32.240 --> 00:18:35.360 align:middle line:84% mathematicians will often say "partial function." 00:18:35.360 --> 00:18:54.760 align:middle line:90% 00:18:54.760 --> 00:18:56.560 align:middle line:84% So partial function means a function 00:18:56.560 --> 00:18:58.750 align:middle line:90% that isn't necessarily total. 00:18:58.750 --> 00:19:00.560 align:middle line:90% It might be total. 00:19:00.560 --> 00:19:03.100 align:middle line:84% Confusingly, a total function is an example 00:19:03.100 --> 00:19:04.700 align:middle line:90% of a partial function. 00:19:04.700 --> 00:19:06.920 align:middle line:90% Partial does not mean not total. 00:19:06.920 --> 00:19:10.000 align:middle line:84% Partial does not mean that it's definitely missing inputs. 00:19:10.000 --> 00:19:12.970 align:middle line:84% Partial just means that you're not assuming 00:19:12.970 --> 00:19:15.550 align:middle line:90% there aren't missing inputs. 00:19:15.550 --> 00:19:18.440 align:middle line:84% So by our definitions that we've set out today, 00:19:18.440 --> 00:19:21.190 align:middle line:84% partial function just means function. 00:19:21.190 --> 00:19:22.960 align:middle line:84% And total function means function 00:19:22.960 --> 00:19:25.095 align:middle line:84% that has nothing missing, no inputs missing. 00:19:25.095 --> 00:19:27.970 align:middle line:90% 00:19:27.970 --> 00:19:31.030 align:middle line:84% From now on in this class, whenever we use the word 00:19:31.030 --> 00:19:35.440 align:middle line:84% "function," we're going to try to specify whether we mean 00:19:35.440 --> 00:19:39.070 align:middle line:84% a total function or whether we're allowed to have some 00:19:39.070 --> 00:19:42.320 align:middle line:84% missing inputs because usually when we say "function," 00:19:42.320 --> 00:19:44.740 align:middle line:84% we're going to mean total function. 00:19:44.740 --> 00:19:47.300 align:middle line:84% And so let's decide from now on in this class, 00:19:47.300 --> 00:19:49.810 align:middle line:84% at least when we give problem statements to you, 00:19:49.810 --> 00:19:53.890 align:middle line:84% that the word "function" on its own is hereby ambiguous. 00:19:53.890 --> 00:19:56.590 align:middle line:84% Whenever we say "function," we should really be telling you, 00:19:56.590 --> 00:19:58.060 align:middle line:90% do we mean total function? 00:19:58.060 --> 00:20:01.110 align:middle line:84% Or do we mean function that may or may not be total? 00:20:01.110 --> 00:20:04.990 align:middle line:90% 00:20:04.990 --> 00:20:07.210 align:middle line:84% So if you see it on a problem set or test 00:20:07.210 --> 00:20:09.800 align:middle line:84% and it's not qualified like that, please ask. 00:20:09.800 --> 00:20:11.510 align:middle line:90% We want to clarify that for you. 00:20:11.510 --> 00:20:14.740 align:middle line:84% We're not trying to be tricky with edge 00:20:14.740 --> 00:20:16.040 align:middle line:90% cases of this definition. 00:20:16.040 --> 00:20:19.250 align:middle line:84% If you see "function" on its own without qualifiers, please ask. 00:20:19.250 --> 00:20:19.870 align:middle line:90% Yes? 00:20:19.870 --> 00:20:23.320 align:middle line:84% AUDIENCE: On top, [INAUDIBLE] like-- 00:20:23.320 --> 00:20:27.860 align:middle line:84% it says that the relationship is total if every a in A 00:20:27.860 --> 00:20:29.074 align:middle line:90% has at least one. 00:20:29.074 --> 00:20:31.915 align:middle line:84% On the bottom, it's total if it has exactly one arrow. 00:20:31.915 --> 00:20:35.840 align:middle line:84% ZACHARY ABEL: So a relation is total if everything on the left 00:20:35.840 --> 00:20:37.650 align:middle line:90% has at least one arrow out. 00:20:37.650 --> 00:20:41.108 align:middle line:90% A relation is a total function-- 00:20:41.108 --> 00:20:42.820 align:middle line:90% AUDIENCE: Oh. 00:20:42.820 --> 00:20:45.530 align:middle line:84% ZACHARY ABEL: --if its total and also a function, which 00:20:45.530 --> 00:20:48.030 align:middle line:84% means that everything on the left has exactly one arrow out. 00:20:48.030 --> 00:20:51.140 align:middle line:90% Great question. 00:20:51.140 --> 00:20:51.850 align:middle line:90% All right. 00:20:51.850 --> 00:20:57.050 align:middle line:90% 00:20:57.050 --> 00:20:59.078 align:middle line:84% Yeah, sorry that partial function does not 00:20:59.078 --> 00:21:01.370 align:middle line:84% mean function that isn't total, even though that's what 00:21:01.370 --> 00:21:02.490 align:middle line:90% it sounds like it should be. 00:21:02.490 --> 00:21:03.990 align:middle line:84% Partial function just means function 00:21:03.990 --> 00:21:05.640 align:middle line:90% that may or may not be total. 00:21:05.640 --> 00:21:07.015 align:middle line:90% Naming things is hard. 00:21:07.015 --> 00:21:09.920 align:middle line:90% 00:21:09.920 --> 00:21:13.100 align:middle line:84% Or rather, using bad names that someone else has already 00:21:13.100 --> 00:21:15.050 align:middle line:90% defined for us is hard. 00:21:15.050 --> 00:21:17.540 align:middle line:90% I would change it if I could. 00:21:17.540 --> 00:21:21.830 align:middle line:84% Anyway-- (SING-SONGY) do, do, do, do. 00:21:21.830 --> 00:21:23.450 align:middle line:90% Great. 00:21:23.450 --> 00:21:28.440 align:middle line:84% We have similar terminology for arrows in instead of arrows out. 00:21:28.440 --> 00:21:30.560 align:middle line:84% So far, we've looked at total and function, which 00:21:30.560 --> 00:21:32.630 align:middle line:84% are talking about how many arrows are coming out 00:21:32.630 --> 00:21:34.670 align:middle line:90% of the left side. 00:21:34.670 --> 00:21:39.020 align:middle line:84% We can also ask how many arrows are coming into the right side. 00:21:39.020 --> 00:21:42.840 align:middle line:84% So we'll say that R, which is a relation between A and B, 00:21:42.840 --> 00:21:56.930 align:middle line:84% is injective if and only if every element on the right side 00:21:56.930 --> 00:22:03.410 align:middle line:90% has at most one arrow in. 00:22:03.410 --> 00:22:15.650 align:middle line:84% And we'll call it surjective if and only if every b in B 00:22:15.650 --> 00:22:22.170 align:middle line:90% has at least one arrow in. 00:22:22.170 --> 00:22:26.670 align:middle line:84% So injective means you never hit the same thing twice. 00:22:26.670 --> 00:22:30.000 align:middle line:84% Surjective means you hit everything possibly 00:22:30.000 --> 00:22:32.550 align:middle line:90% more than once. 00:22:32.550 --> 00:22:37.400 align:middle line:84% Back to these examples, is L injective? 00:22:37.400 --> 00:22:40.678 align:middle line:90% 00:22:40.678 --> 00:22:42.720 align:middle line:84% In other words, does everything on the right side 00:22:42.720 --> 00:22:45.720 align:middle line:90% have at most one arrow in? 00:22:45.720 --> 00:22:46.710 align:middle line:90% No. 00:22:46.710 --> 00:22:47.620 align:middle line:90% Thumbs down. 00:22:47.620 --> 00:22:49.560 align:middle line:90% Thank you. 00:22:49.560 --> 00:22:54.070 align:middle line:84% For example, chemistry has multiple arrows in. 00:22:54.070 --> 00:22:57.010 align:middle line:84% That alone is enough to show that it's not injective. 00:22:57.010 --> 00:22:59.370 align:middle line:90% Is it surjective? 00:22:59.370 --> 00:23:00.370 align:middle line:90% Also no. 00:23:00.370 --> 00:23:01.270 align:middle line:90% Another thumbs down. 00:23:01.270 --> 00:23:02.700 align:middle line:90% Thank you. 00:23:02.700 --> 00:23:07.080 align:middle line:84% It's not surjective because 6.1200 doesn't have any arrows 00:23:07.080 --> 00:23:07.620 align:middle line:90% in. 00:23:07.620 --> 00:23:10.330 align:middle line:84% Surjective means you're supposed to hit everything on the right, 00:23:10.330 --> 00:23:11.410 align:middle line:90% and we don't. 00:23:11.410 --> 00:23:14.740 align:middle line:84% So this is neither surjective nor injective, 00:23:14.740 --> 00:23:18.270 align:middle line:90% whereas this is both. 00:23:18.270 --> 00:23:21.370 align:middle line:84% Everything on the right has at least one arrow 00:23:21.370 --> 00:23:24.560 align:middle line:84% in-- so it's subjective-- and at most one arrow in. 00:23:24.560 --> 00:23:25.516 align:middle line:90% So it's injective. 00:23:25.516 --> 00:23:34.330 align:middle line:90% 00:23:34.330 --> 00:23:39.040 align:middle line:84% Now, one of the reasons we're defining all of these similar 00:23:39.040 --> 00:23:41.120 align:middle line:90% confusing-sounding terms-- 00:23:41.120 --> 00:23:44.680 align:middle line:84% apologies for that-- is that we have 00:23:44.680 --> 00:23:48.910 align:middle line:84% some theorems that these terms are enough to help us 00:23:48.910 --> 00:23:50.960 align:middle line:84% relate the sizes of various sets. 00:23:50.960 --> 00:23:55.390 align:middle line:84% So we have our first theorem of the day. 00:23:55.390 --> 00:24:05.500 align:middle line:84% If A and B are finite sets and R, a relation from A to B, 00:24:05.500 --> 00:24:08.846 align:middle line:90% is a total injection-- 00:24:08.846 --> 00:24:15.670 align:middle line:90% 00:24:15.670 --> 00:24:18.430 align:middle line:84% so if we have finite sets and a total injection 00:24:18.430 --> 00:24:20.060 align:middle line:90% from the first to the second-- 00:24:20.060 --> 00:24:28.810 align:middle line:84% then the size of A is at most the size of B. 00:24:28.810 --> 00:24:32.330 align:middle line:84% If we can find a total injection from one set to another set, 00:24:32.330 --> 00:24:35.020 align:middle line:84% then the set on the left is smaller or at most 00:24:35.020 --> 00:24:35.930 align:middle line:90% the same size. 00:24:35.930 --> 00:24:38.190 align:middle line:84% And let's draw a picture to see why. 00:24:38.190 --> 00:24:42.340 align:middle line:90% 00:24:42.340 --> 00:24:44.980 align:middle line:84% So total injection means that everything on the left 00:24:44.980 --> 00:24:46.820 align:middle line:90% has at least one arrow out. 00:24:46.820 --> 00:24:48.370 align:middle line:90% Total means that. 00:24:48.370 --> 00:24:51.235 align:middle line:84% And injection means we never hit the same thing on the right side 00:24:51.235 --> 00:24:51.735 align:middle line:90% twice. 00:24:51.735 --> 00:24:55.690 align:middle line:90% 00:24:55.690 --> 00:24:58.460 align:middle line:84% So there's always an arrow out on the left side. 00:24:58.460 --> 00:25:02.530 align:middle line:84% And there's at most one arrow out on the right side. 00:25:02.530 --> 00:25:06.130 align:middle line:84% And so all this is saying is that the number of elements 00:25:06.130 --> 00:25:09.430 align:middle line:84% on the left is at most the number of arrows 00:25:09.430 --> 00:25:12.640 align:middle line:84% because everything on the left has at least one arrow out. 00:25:12.640 --> 00:25:15.370 align:middle line:84% And the number of elements on the right side 00:25:15.370 --> 00:25:18.950 align:middle line:84% is at least the number of arrows because everything on the right 00:25:18.950 --> 00:25:20.900 align:middle line:90% has at most one arrow out. 00:25:20.900 --> 00:25:22.740 align:middle line:90% So that's why this is true. 00:25:22.740 --> 00:25:25.430 align:middle line:84% Size of A is less or equal to the number of arrows, which 00:25:25.430 --> 00:25:28.850 align:middle line:84% is less or equal to the size of B. So often, 00:25:28.850 --> 00:25:31.710 align:middle line:84% when you want to show that one set is smaller than another set, 00:25:31.710 --> 00:25:33.830 align:middle line:84% you just define a total injection between them, 00:25:33.830 --> 00:25:35.690 align:middle line:90% and that's your proof. 00:25:35.690 --> 00:25:43.880 align:middle line:84% Similarly, on the other side of things, theorem 2-- 00:25:43.880 --> 00:25:51.650 align:middle line:84% let's see-- if, again, A and B are finite sets 00:25:51.650 --> 00:26:09.110 align:middle line:84% and R is a surjective function, then the size of A 00:26:09.110 --> 00:26:12.470 align:middle line:84% is greater or equal to the size of B. 00:26:12.470 --> 00:26:15.630 align:middle line:84% And I claim that's just exactly this picture but 00:26:15.630 --> 00:26:18.310 align:middle line:90% in the opposite direction. 00:26:18.310 --> 00:26:20.400 align:middle line:84% Instead of total meaning everything 00:26:20.400 --> 00:26:23.340 align:middle line:84% on the left has an arrow out, we have 00:26:23.340 --> 00:26:25.590 align:middle line:84% surjective, which means everything on the right 00:26:25.590 --> 00:26:27.150 align:middle line:90% has an arrow in. 00:26:27.150 --> 00:26:30.150 align:middle line:84% And instead of injection, which means 00:26:30.150 --> 00:26:32.950 align:middle line:84% everything on the right, has at most one out-- 00:26:32.950 --> 00:26:36.870 align:middle line:84% or sorry, at most one in, we have function, which says 00:26:36.870 --> 00:26:41.500 align:middle line:84% everything on the left has at most one arrow out. 00:26:41.500 --> 00:26:44.470 align:middle line:84% So we've just swapped the roles of in and out. 00:26:44.470 --> 00:26:46.620 align:middle line:84% That's really all that happened to translate 00:26:46.620 --> 00:26:47.915 align:middle line:90% this theorem to that theorem. 00:26:47.915 --> 00:26:50.940 align:middle line:90% 00:26:50.940 --> 00:26:55.656 align:middle line:84% And by our powers combined, definition-- 00:26:55.656 --> 00:26:58.470 align:middle line:90% 00:26:58.470 --> 00:27:11.550 align:middle line:84% if R is injective, surjective, total, and a function, 00:27:11.550 --> 00:27:15.960 align:middle line:84% if it has all four of these properties, 00:27:15.960 --> 00:27:18.186 align:middle line:84% then we say that R is a bijection. 00:27:18.186 --> 00:27:24.090 align:middle line:90% 00:27:24.090 --> 00:27:26.140 align:middle line:84% So that's the definition of bijection. 00:27:26.140 --> 00:27:29.760 align:middle line:84% Bijection means you have all four of these nice properties. 00:27:29.760 --> 00:27:33.190 align:middle line:84% To summarize, everything on the left has exactly one arrow out. 00:27:33.190 --> 00:27:36.690 align:middle line:84% Everything on the right has exactly one arrow in. 00:27:36.690 --> 00:27:48.150 align:middle line:84% And theorem-- if R from A to B is a bijection, 00:27:48.150 --> 00:27:52.860 align:middle line:84% then the size of A equals the size of B. 00:27:52.860 --> 00:27:55.140 align:middle line:84% All we're doing is applying both of these theorems, 00:27:55.140 --> 00:27:56.530 align:middle line:90% and we get both inequalities. 00:27:56.530 --> 00:27:59.910 align:middle line:90% So the sets have the same size. 00:27:59.910 --> 00:28:02.610 align:middle line:84% All right, so bijection is the main tool 00:28:02.610 --> 00:28:06.270 align:middle line:84% that we're going to be using in the second half of this lecture 00:28:06.270 --> 00:28:09.930 align:middle line:90% to compare the sizes of sets. 00:28:09.930 --> 00:28:11.090 align:middle line:90% Questions about that? 00:28:11.090 --> 00:28:13.680 align:middle line:90% 00:28:13.680 --> 00:28:16.500 align:middle line:90% Wonderful. 00:28:16.500 --> 00:28:20.850 align:middle line:84% So next, I want to talk about a couple special kinds 00:28:20.850 --> 00:28:22.270 align:middle line:90% of relations. 00:28:22.270 --> 00:28:26.460 align:middle line:84% So first of all, I never said that the sets A and B 00:28:26.460 --> 00:28:30.060 align:middle line:84% have to be disjoint or even distinct. 00:28:30.060 --> 00:28:35.830 align:middle line:84% So let's look at the special case 00:28:35.830 --> 00:28:43.110 align:middle line:84% where A equals B. So R is a subset of A cross A. We call 00:28:43.110 --> 00:28:54.450 align:middle line:90% this a binary relation on A. 00:28:54.450 --> 00:28:57.120 align:middle line:84% And this, I claim, is basically exactly how 00:28:57.120 --> 00:28:59.460 align:middle line:90% we define directed graphs. 00:28:59.460 --> 00:29:03.450 align:middle line:84% You have a set of nodes, which in this case is A 00:29:03.450 --> 00:29:06.030 align:middle line:84% and serves as both our domain and codomain. 00:29:06.030 --> 00:29:11.800 align:middle line:84% And you have a set of directed edges between them, 00:29:11.800 --> 00:29:12.650 align:middle line:90% and that's it. 00:29:12.650 --> 00:29:14.540 align:middle line:84% That's how we define directed graph. 00:29:14.540 --> 00:29:19.480 align:middle line:84% So a relation on a is basically identical to a directed graph 00:29:19.480 --> 00:29:22.720 align:middle line:84% whose vertices are A, which means 00:29:22.720 --> 00:29:25.930 align:middle line:84% that all of the directed graph examples we've already talked 00:29:25.930 --> 00:29:30.250 align:middle line:84% about, like who follows whom on Twitter, like reachability 00:29:30.250 --> 00:29:36.190 align:middle line:84% in Google Maps, all of those digraphs are also relations. 00:29:36.190 --> 00:29:41.920 align:middle line:84% They're relations on the set of Twitter users or locations 00:29:41.920 --> 00:29:44.950 align:middle line:90% in Google Maps. 00:29:44.950 --> 00:29:46.810 align:middle line:84% Another good example that I really 00:29:46.810 --> 00:29:51.370 align:middle line:84% like of a relation on a set is the relation 00:29:51.370 --> 00:29:54.710 align:middle line:84% of who likes whom, not like romantically or anything, 00:29:54.710 --> 00:29:57.050 align:middle line:84% just as a person, who likes whom. 00:29:57.050 --> 00:29:58.523 align:middle line:90% Fix a moment in time. 00:29:58.523 --> 00:30:00.940 align:middle line:84% Everyone measures who they like among all the other people 00:30:00.940 --> 00:30:01.700 align:middle line:90% in the world. 00:30:01.700 --> 00:30:03.730 align:middle line:90% Bam, you have your relation. 00:30:03.730 --> 00:30:05.800 align:middle line:84% And the reason I like it is that it has tons 00:30:05.800 --> 00:30:08.290 align:middle line:90% and tons of chaotic structure. 00:30:08.290 --> 00:30:11.420 align:middle line:84% For example, if I like you, that doesn't 00:30:11.420 --> 00:30:13.620 align:middle line:84% mean that you like me back necessarily, 00:30:13.620 --> 00:30:15.930 align:middle line:84% especially this week when the problem set's a bit long. 00:30:15.930 --> 00:30:17.780 align:middle line:90% Apologies. 00:30:17.780 --> 00:30:20.510 align:middle line:84% If I like you and you like them, that 00:30:20.510 --> 00:30:23.060 align:middle line:90% doesn't mean that I like them. 00:30:23.060 --> 00:30:24.510 align:middle line:90% Not always, right? 00:30:24.510 --> 00:30:26.790 align:middle line:84% That first example, I like you means you like me, 00:30:26.790 --> 00:30:29.810 align:middle line:84% that's what we're going to call symmetry in a little bit. 00:30:29.810 --> 00:30:32.100 align:middle line:84% That second example, "if I like you, you like them, 00:30:32.100 --> 00:30:34.550 align:middle line:84% that means I like them," we call that transitivity. 00:30:34.550 --> 00:30:37.590 align:middle line:84% So this graph is neither symmetric nor transitive. 00:30:37.590 --> 00:30:40.220 align:middle line:90% It's also not reflexive. 00:30:40.220 --> 00:30:42.360 align:middle line:84% There are days when I don't like myself, 00:30:42.360 --> 00:30:44.960 align:middle line:84% so I wouldn't have a self-loop in this graph. 00:30:44.960 --> 00:30:47.460 align:middle line:84% Now, I don't mean that in like a "cry for help" kind of way. 00:30:47.460 --> 00:30:50.430 align:middle line:84% But you know, some days you're a little down on yourself. 00:30:50.430 --> 00:30:51.030 align:middle line:90% It can happen. 00:30:51.030 --> 00:30:51.650 align:middle line:90% It's OK. 00:30:51.650 --> 00:30:55.140 align:middle line:84% If it happens too much, you have support. 00:30:55.140 --> 00:30:56.185 align:middle line:90% Please seek it out. 00:30:56.185 --> 00:30:58.250 align:middle line:90% [SOFT LAUGHTER] 00:30:58.250 --> 00:30:59.520 align:middle line:90% So funny story. 00:30:59.520 --> 00:31:02.600 align:middle line:84% I gave this same example a couple years ago. 00:31:02.600 --> 00:31:06.230 align:middle line:84% I got the same half-hearted chuckle that we got just now. 00:31:06.230 --> 00:31:08.521 align:middle line:84% And then someone went home and wrote about it on "MIT 00:31:08.521 --> 00:31:09.063 align:middle line:90% Confessions." 00:31:09.063 --> 00:31:10.520 align:middle line:90% [LAUGHTER] 00:31:10.520 --> 00:31:12.390 align:middle line:84% I don't remember exactly what they said, 00:31:12.390 --> 00:31:14.543 align:middle line:84% but they maybe left off some of the context. 00:31:14.543 --> 00:31:17.330 align:middle line:90% 00:31:17.330 --> 00:31:19.410 align:middle line:84% And then, for the next couple days, 00:31:19.410 --> 00:31:23.070 align:middle line:84% I was getting multiple Facebook messages from multiple friends, 00:31:23.070 --> 00:31:25.340 align:middle line:90% 'Zach, are you OK?' 00:31:25.340 --> 00:31:28.400 align:middle line:84% 'I know we haven't caught up in a while, but I'm here to chat. 00:31:28.400 --> 00:31:31.410 align:middle line:84% And I'd love to talk to if you're available. 00:31:31.410 --> 00:31:33.110 align:middle line:90% I hope you're doing well.' 00:31:33.110 --> 00:31:35.875 align:middle line:84% And I just want to say I felt loved. 00:31:35.875 --> 00:31:37.130 align:middle line:90% [LAUGHTER] 00:31:37.130 --> 00:31:39.300 align:middle line:84% So thank you to whoever did that. 00:31:39.300 --> 00:31:41.185 align:middle line:84% But maybe let's not have a repeat. 00:31:41.185 --> 00:31:43.130 align:middle line:90% [LAUGHTER] 00:31:43.130 --> 00:31:44.560 align:middle line:90% Anyway, back to graphs. 00:31:44.560 --> 00:31:47.450 align:middle line:90% 00:31:47.450 --> 00:31:49.520 align:middle line:84% So one of the main reasons we want 00:31:49.520 --> 00:31:53.330 align:middle line:84% to look at relations on a set is because we already 00:31:53.330 --> 00:31:56.420 align:middle line:84% have lots of examples, and especially lots of examples 00:31:56.420 --> 00:32:00.890 align:middle line:84% that we don't necessarily think of as graphs. 00:32:00.890 --> 00:32:04.580 align:middle line:84% For example, A equals B is a relation 00:32:04.580 --> 00:32:07.200 align:middle line:90% that we use all the time. 00:32:07.200 --> 00:32:11.400 align:middle line:84% A is congruent to B mod 10 is a relation 00:32:11.400 --> 00:32:14.100 align:middle line:84% on the set of natural numbers, of integers 00:32:14.100 --> 00:32:16.890 align:middle line:84% that we use all the time and don't necessarily 00:32:16.890 --> 00:32:18.010 align:middle line:90% think of as a graph. 00:32:18.010 --> 00:32:20.410 align:middle line:90% But it's definitely a relation. 00:32:20.410 --> 00:32:22.870 align:middle line:84% It's also a graph because that's secretly the same thing. 00:32:22.870 --> 00:32:26.040 align:middle line:84% But the way you think about it can change if you want to. 00:32:26.040 --> 00:32:34.140 align:middle line:84% Likewise, A less equal B. We have set A is a subset of set B. 00:32:34.140 --> 00:32:39.660 align:middle line:84% We have A divides B. Lots of examples of relations 00:32:39.660 --> 00:32:43.110 align:middle line:84% that we want to be able to talk about. 00:32:43.110 --> 00:32:46.420 align:middle line:84% Another good example is reachability. 00:32:46.420 --> 00:32:47.693 align:middle line:90% So if G is a digraph. 00:32:47.693 --> 00:32:51.990 align:middle line:90% 00:32:51.990 --> 00:32:57.000 align:middle line:84% And if V and W are nodes of our graph, 00:32:57.000 --> 00:32:59.023 align:middle line:84% then we can define the walk relation. 00:32:59.023 --> 00:33:05.790 align:middle line:90% 00:33:05.790 --> 00:33:08.170 align:middle line:90% Sorry, that was out of order. 00:33:08.170 --> 00:33:10.890 align:middle line:84% If G is a digraph, we can define the walk relation 00:33:10.890 --> 00:33:15.240 align:middle line:84% on the set of vertices of our graph, where we define 00:33:15.240 --> 00:33:21.113 align:middle line:84% that V relates to W under this walk relation 00:33:21.113 --> 00:33:22.530 align:middle line:84% that we're going to call G prime-- 00:33:22.530 --> 00:33:34.350 align:middle line:84% or sorry, G star if and only if there exists a walk from V to W. 00:33:34.350 --> 00:33:37.980 align:middle line:84% That's a relation we can define on a digraph, 00:33:37.980 --> 00:33:39.530 align:middle line:90% and it's a really useful one. 00:33:39.530 --> 00:33:44.670 align:middle line:90% 00:33:44.670 --> 00:33:47.440 align:middle line:84% One thing I want to warn you about, we started with a graph. 00:33:47.440 --> 00:33:50.280 align:middle line:90% We defined a relation, G star. 00:33:50.280 --> 00:33:53.760 align:middle line:84% We know that relations on this set of vertices 00:33:53.760 --> 00:33:56.940 align:middle line:84% can be thought of as a graph on that set of vertices. 00:33:56.940 --> 00:34:01.990 align:middle line:84% We're not necessarily talking about the same graph. 00:34:01.990 --> 00:34:03.700 align:middle line:84% This relation, G star, is something 00:34:03.700 --> 00:34:07.990 align:middle line:84% we constructed from this digraph G. But the graph we constructed 00:34:07.990 --> 00:34:09.940 align:middle line:84% isn't necessarily the same as the graph G 00:34:09.940 --> 00:34:11.330 align:middle line:90% that we started with. 00:34:11.330 --> 00:34:14.780 align:middle line:84% So it can be a little confusing thinking of, oh, that's a graph, 00:34:14.780 --> 00:34:19.480 align:middle line:84% and that's also a graph that was constructed from this other. 00:34:19.480 --> 00:34:21.130 align:middle line:90% It's fun to think about. 00:34:21.130 --> 00:34:23.072 align:middle line:84% But just a warning, even though we're 00:34:23.072 --> 00:34:24.489 align:middle line:84% talking about two different graphs 00:34:24.489 --> 00:34:26.139 align:middle line:84% on the same set of vertices, they 00:34:26.139 --> 00:34:28.169 align:middle line:84% might actually be two different graphs. 00:34:28.169 --> 00:34:30.880 align:middle line:90% 00:34:30.880 --> 00:34:31.420 align:middle line:90% Cool. 00:34:31.420 --> 00:34:34.480 align:middle line:84% So there are two special kinds of relations 00:34:34.480 --> 00:34:36.730 align:middle line:84% on a set that are especially useful because we 00:34:36.730 --> 00:34:38.060 align:middle line:90% have lots of examples. 00:34:38.060 --> 00:34:41.199 align:middle line:84% And it's useful to be able to generalize those examples. 00:34:41.199 --> 00:34:45.710 align:middle line:84% The first one I want to talk about 00:34:45.710 --> 00:34:47.733 align:middle line:84% is called an equivalence relation. 00:34:47.733 --> 00:34:51.489 align:middle line:90% 00:34:51.489 --> 00:34:54.610 align:middle line:84% And some of the examples I wrote down, 00:34:54.610 --> 00:35:01.360 align:middle line:84% for example a equals b or a is congruent to b mod 10, 00:35:01.360 --> 00:35:04.556 align:middle line:84% these are somehow capturing this idea of sameness. 00:35:04.556 --> 00:35:11.800 align:middle line:84% a and b are the same. a and b have the same remainder, mod 10. 00:35:11.800 --> 00:35:13.960 align:middle line:84% If we're talking about connectivity 00:35:13.960 --> 00:35:18.070 align:middle line:84% in an undirected graph, we say that two nodes 00:35:18.070 --> 00:35:20.140 align:middle line:84% are connected with each other, meaning 00:35:20.140 --> 00:35:22.690 align:middle line:84% there's a walk from one to the other, which 00:35:22.690 --> 00:35:26.180 align:middle line:84% means that they belong to the same connected component. 00:35:26.180 --> 00:35:29.560 align:middle line:84% There's another idea of sameness there. 00:35:29.560 --> 00:35:33.160 align:middle line:84% So an equivalence relation is an idea 00:35:33.160 --> 00:35:37.180 align:middle line:84% that generalizes this idea of sameness. 00:35:37.180 --> 00:35:39.560 align:middle line:84% And so let's see if we can do that. 00:35:39.560 --> 00:35:43.150 align:middle line:84% What does it mean to somehow encapsulate 00:35:43.150 --> 00:35:46.340 align:middle line:84% that the two things are somehow equivalent to each other, 00:35:46.340 --> 00:35:49.240 align:middle line:90% even if they're not identical? 00:35:49.240 --> 00:35:53.800 align:middle line:84% And it turns out there are three easy-- 00:35:53.800 --> 00:35:58.240 align:middle line:84% by "easy," I mean possibly confusing, but I apologize-- 00:35:58.240 --> 00:36:02.670 align:middle line:84% three easily stated properties that a relation might have. 00:36:02.670 --> 00:36:04.880 align:middle line:84% And if it has all three, then I claim 00:36:04.880 --> 00:36:07.700 align:middle line:84% it captures everything it needs to capture 00:36:07.700 --> 00:36:09.870 align:middle line:90% about this idea of sameness. 00:36:09.870 --> 00:36:14.240 align:middle line:84% Everything I've just said is extremely vague. 00:36:14.240 --> 00:36:16.250 align:middle line:90% Let's make it precise. 00:36:16.250 --> 00:36:23.000 align:middle line:84% So let's say that R is a relation on A. We're 00:36:23.000 --> 00:36:27.040 align:middle line:90% going to say R is reflexive. 00:36:27.040 --> 00:36:32.000 align:middle line:90% 00:36:32.000 --> 00:36:41.870 align:middle line:84% This means that for all a in A, a relates to itself. 00:36:41.870 --> 00:36:45.240 align:middle line:84% Reflexive means everything relates to itself. 00:36:45.240 --> 00:36:47.250 align:middle line:84% Everything should be the same as itself 00:36:47.250 --> 00:36:48.750 align:middle line:90% if sameness means anything. 00:36:48.750 --> 00:36:52.490 align:middle line:84% So this is definitely something we would want to have. 00:36:52.490 --> 00:36:55.520 align:middle line:84% In terms of a digraph, it means that every vertex 00:36:55.520 --> 00:36:58.770 align:middle line:90% has a self-loop. 00:36:58.770 --> 00:37:10.320 align:middle line:84% All right, next one, R is symmetric if and only 00:37:10.320 --> 00:37:22.050 align:middle line:84% if for all a and b in our set, a relates to b if and only if b 00:37:22.050 --> 00:37:24.330 align:middle line:90% relates to a. 00:37:24.330 --> 00:37:27.370 align:middle line:84% So symmetry is saying that order doesn't matter. 00:37:27.370 --> 00:37:30.840 align:middle line:84% If I'm your friend, then you're my friend-- 00:37:30.840 --> 00:37:32.765 align:middle line:90% so Facebook, not Twitter. 00:37:32.765 --> 00:37:35.260 align:middle line:90% 00:37:35.260 --> 00:37:35.760 align:middle line:90% Cool. 00:37:35.760 --> 00:37:40.290 align:middle line:84% In terms of digraph, remember a relation is always a digraph. 00:37:40.290 --> 00:37:43.150 align:middle line:84% This means that every time we have an edge in one direction, 00:37:43.150 --> 00:37:46.590 align:middle line:84% we also have the edge in the other direction. 00:37:46.590 --> 00:37:49.770 align:middle line:84% All right, so symmetric means order doesn't matter. 00:37:49.770 --> 00:37:59.590 align:middle line:84% And finally, R is transitive if and only if for all a, 00:37:59.590 --> 00:38:06.210 align:middle line:84% b, and c in A, if a relates to b and b 00:38:06.210 --> 00:38:10.145 align:middle line:84% relates to c, then a relates to c. 00:38:10.145 --> 00:38:13.080 align:middle line:90% 00:38:13.080 --> 00:38:16.050 align:middle line:84% So if this is the same as that, and that's 00:38:16.050 --> 00:38:18.720 align:middle line:84% the same as this other thing, then this 00:38:18.720 --> 00:38:21.900 align:middle line:84% should be the same as this other thing-- 00:38:21.900 --> 00:38:24.420 align:middle line:84% another thing that we might want to be true 00:38:24.420 --> 00:38:27.750 align:middle line:84% if we're trying to capture what we mean by sameness. 00:38:27.750 --> 00:38:30.960 align:middle line:84% In terms of graphs, it means that any time we 00:38:30.960 --> 00:38:36.960 align:middle line:84% have a walk of length 2, then there was already 00:38:36.960 --> 00:38:40.620 align:middle line:84% a directed edge from the start to the end. 00:38:40.620 --> 00:38:42.637 align:middle line:84% If you can get from here to there and from there 00:38:42.637 --> 00:38:44.970 align:middle line:84% to this other place, then you can get straight from here 00:38:44.970 --> 00:38:48.430 align:middle line:90% to this other place. 00:38:48.430 --> 00:38:51.060 align:middle line:84% And both of these examples that I mentioned-- 00:38:51.060 --> 00:38:54.170 align:middle line:90% all three of these examples-- 00:38:54.170 --> 00:38:57.824 align:middle line:90% same connected component. 00:38:57.824 --> 00:39:01.270 align:middle line:90% 00:39:01.270 --> 00:39:04.660 align:middle line:84% So these circled relations, I claim, all have all three 00:39:04.660 --> 00:39:06.220 align:middle line:90% of these properties. 00:39:06.220 --> 00:39:10.100 align:middle line:84% Every number is congruent to itself mod 10. 00:39:10.100 --> 00:39:12.740 align:middle line:84% If two numbers are congruent to each other mod 10, 00:39:12.740 --> 00:39:15.370 align:middle line:84% then they're congruent the other way mod 10. 00:39:15.370 --> 00:39:18.110 align:middle line:84% And if a and b are congruent and b and c are congruent, 00:39:18.110 --> 00:39:21.880 align:middle line:84% then a and c are congruent mod 10. 00:39:21.880 --> 00:39:24.518 align:middle line:84% And you can check that all of those examples 00:39:24.518 --> 00:39:25.310 align:middle line:90% have this property. 00:39:25.310 --> 00:39:30.760 align:middle line:84% And in fact, I claim that these three properties are enough. 00:39:30.760 --> 00:39:36.250 align:middle line:84% Unfortunately, I have to put the definition very far away from-- 00:39:36.250 --> 00:39:41.110 align:middle line:84% or sorry, the theorem very far away from that definition. 00:39:41.110 --> 00:39:42.180 align:middle line:90% We'll make it work. 00:39:42.180 --> 00:39:48.820 align:middle line:90% 00:39:48.820 --> 00:39:51.430 align:middle line:90% Bye, eraser 00:39:51.430 --> 00:39:57.216 align:middle line:84% OK, definition-- R is an equivalence relation-- 00:39:57.216 --> 00:40:02.080 align:middle line:90% 00:40:02.080 --> 00:40:11.320 align:middle line:84% relation-- if and only if R is all three of those things-- 00:40:11.320 --> 00:40:13.900 align:middle line:90% symmetric-- 00:40:13.900 --> 00:40:20.316 align:middle line:84% sorry, reflexive, symmetric, and transitive. 00:40:20.316 --> 00:40:25.570 align:middle line:90% 00:40:25.570 --> 00:40:27.640 align:middle line:84% And we have a theorem that backs up 00:40:27.640 --> 00:40:31.090 align:middle line:84% my claim that says this is all we need for sameness. 00:40:31.090 --> 00:40:44.020 align:middle line:84% Theorem-- if R, which is a relation on A, is an equivalence 00:40:44.020 --> 00:40:44.806 align:middle line:90% relation-- 00:40:44.806 --> 00:40:47.740 align:middle line:90% 00:40:47.740 --> 00:40:52.730 align:middle line:84% if R is an equivalence relation, then R 00:40:52.730 --> 00:41:03.120 align:middle line:84% partitions A into some collection of subsets, A1 union 00:41:03.120 --> 00:41:06.750 align:middle line:90% A2 union A3 union, and so on. 00:41:06.750 --> 00:41:12.860 align:middle line:84% And this is going to be a disjoint union such 00:41:12.860 --> 00:41:26.660 align:middle line:84% that every a in A belongs to exactly one of our subsets. 00:41:26.660 --> 00:41:28.970 align:middle line:84% That's what it means to be a partition, 00:41:28.970 --> 00:41:31.340 align:middle line:84% that we're breaking A into smaller pieces, where 00:41:31.340 --> 00:41:36.410 align:middle line:84% every element of A is in exactly one of our pieces 00:41:36.410 --> 00:41:43.040 align:middle line:84% such that a relates to b if and only if a and b 00:41:43.040 --> 00:41:45.793 align:middle line:90% are in the same subset. 00:41:45.793 --> 00:41:49.550 align:middle line:90% 00:41:49.550 --> 00:41:51.770 align:middle line:84% So any time we have an equivalence relation 00:41:51.770 --> 00:41:55.700 align:middle line:84% that satisfies these three axioms, then that exactly 00:41:55.700 --> 00:41:59.850 align:middle line:84% forms a partition which tells us precisely what our relation is. 00:41:59.850 --> 00:42:02.340 align:middle line:84% Our relation just says you're related if and only 00:42:02.340 --> 00:42:04.940 align:middle line:90% if you're in the same subset. 00:42:04.940 --> 00:42:08.810 align:middle line:84% For example, if we're talking about congruence mod 10, 00:42:08.810 --> 00:42:13.370 align:middle line:84% the parts of our partition are, well, we have the set 00:42:13.370 --> 00:42:18.710 align:middle line:90% 0, 10, minus 10, 20, minus 20. 00:42:18.710 --> 00:42:21.060 align:middle line:90% This is one part of a partition. 00:42:21.060 --> 00:42:24.260 align:middle line:84% Everything in the set is congruent to everything else 00:42:24.260 --> 00:42:28.940 align:middle line:84% in the set and not congruent to anything outside the set. 00:42:28.940 --> 00:42:32.510 align:middle line:84% Then, we also have 1 and 11 and negative 9 00:42:32.510 --> 00:42:35.390 align:middle line:90% and 21 and negative 19. 00:42:35.390 --> 00:42:37.693 align:middle line:84% Here's another part of our partition. 00:42:37.693 --> 00:42:39.110 align:middle line:84% All of these numbers are congruent 00:42:39.110 --> 00:42:42.710 align:middle line:84% to each other and congruent to none of the other numbers. 00:42:42.710 --> 00:42:48.060 align:middle line:84% And in total, there are 10 of these parts. 00:42:48.060 --> 00:42:50.680 align:middle line:84% So anytime we have an equivalence relation, 00:42:50.680 --> 00:42:55.170 align:middle line:84% we can partition our set into subsets. 00:42:55.170 --> 00:42:57.825 align:middle line:84% That exactly tells us what our relation is. 00:42:57.825 --> 00:42:58.325 align:middle line:90% Yes? 00:42:58.325 --> 00:43:00.330 align:middle line:84% AUDIENCE: What's the union with the dot? 00:43:00.330 --> 00:43:02.470 align:middle line:84% ZACHARY ABEL: Yeah, so the union with the dot here, 00:43:02.470 --> 00:43:04.530 align:middle line:84% the dot is there to emphasize that we're 00:43:04.530 --> 00:43:07.800 align:middle line:84% taking the union of sets that are disjoint, 00:43:07.800 --> 00:43:11.500 align:middle line:84% that no element is in multiple of these sets. 00:43:11.500 --> 00:43:13.180 align:middle line:84% I know I wrote it out here in words. 00:43:13.180 --> 00:43:17.580 align:middle line:84% But I like to emphasize it in the formula as well. 00:43:17.580 --> 00:43:19.140 align:middle line:84% When we're talking about connectivity 00:43:19.140 --> 00:43:25.227 align:middle line:84% in an undirected graph, being connected to each other, 00:43:25.227 --> 00:43:27.060 align:middle line:84% a pair of vertices are connected if and only 00:43:27.060 --> 00:43:30.270 align:middle line:84% if they're in the same connected component. 00:43:30.270 --> 00:43:32.520 align:middle line:84% And so the connected components form 00:43:32.520 --> 00:43:36.330 align:middle line:90% the pieces of our partition. 00:43:36.330 --> 00:43:38.620 align:middle line:84% So we've seen multiple examples of this. 00:43:38.620 --> 00:43:41.110 align:middle line:84% This is just making this more general and more abstract. 00:43:41.110 --> 00:43:47.680 align:middle line:84% Any time we satisfy all three of those axioms, we get this. 00:43:47.680 --> 00:43:50.500 align:middle line:90% All right. 00:43:50.500 --> 00:43:53.440 align:middle line:84% so the other kind of subset I want to talk about-- 00:43:53.440 --> 00:43:57.730 align:middle line:90% 00:43:57.730 --> 00:44:01.070 align:middle line:84% actually, before that, we have a couple more examples. 00:44:01.070 --> 00:44:07.480 align:middle line:84% We had this example of a less equal b, a divides b, 00:44:07.480 --> 00:44:12.730 align:middle line:84% set A is a subset of set B. These are other common relations 00:44:12.730 --> 00:44:13.970 align:middle line:90% that we work with. 00:44:13.970 --> 00:44:17.990 align:middle line:84% And these are not equivalence relations. 00:44:17.990 --> 00:44:19.540 align:middle line:90% We know that 2 divides 10. 00:44:19.540 --> 00:44:21.410 align:middle line:90% But 10 doesn't divide 2. 00:44:21.410 --> 00:44:23.620 align:middle line:84% So it's not symmetric like we wanted 00:44:23.620 --> 00:44:25.480 align:middle line:90% with equivalence relations. 00:44:25.480 --> 00:44:31.390 align:middle line:84% But in many ways, a divides b behaves a lot like a less equal 00:44:31.390 --> 00:44:33.620 align:middle line:90% b or "a" subset of b. 00:44:33.620 --> 00:44:36.760 align:middle line:90% There's some commonality there. 00:44:36.760 --> 00:44:41.560 align:middle line:84% And that commonality we're going to define as what 00:44:41.560 --> 00:44:43.020 align:middle line:90% are called weak partial orders. 00:44:43.020 --> 00:45:03.460 align:middle line:90% 00:45:03.460 --> 00:45:04.790 align:middle line:90% So weak partial order. 00:45:04.790 --> 00:45:12.170 align:middle line:84% Our goal is going to be, behaves like a less or equal b, 00:45:12.170 --> 00:45:17.260 align:middle line:84% A is a subset of B, a divides b, et cetera. 00:45:17.260 --> 00:45:20.440 align:middle line:84% So let's see if we can capture what these all have in common 00:45:20.440 --> 00:45:23.020 align:middle line:90% and prove something about them. 00:45:23.020 --> 00:45:27.640 align:middle line:90% So again-- OK, so let's see. 00:45:27.640 --> 00:45:32.150 align:middle line:90% R is a weak partial order-- 00:45:32.150 --> 00:45:37.183 align:middle line:84% weak partial order-- if and only if R is-- 00:45:37.183 --> 00:45:40.540 align:middle line:90% 00:45:40.540 --> 00:45:42.890 align:middle line:90% are these all reflexive? 00:45:42.890 --> 00:45:45.450 align:middle line:84% Is a number always less or equal to itself? 00:45:45.450 --> 00:45:47.000 align:middle line:90% Does it divide itself? 00:45:47.000 --> 00:45:48.630 align:middle line:90% Is a set a subset of itself? 00:45:48.630 --> 00:45:49.190 align:middle line:90% Yes. 00:45:49.190 --> 00:45:53.150 align:middle line:90% So these are all reflexive. 00:45:53.150 --> 00:45:56.060 align:middle line:84% So let's add that to our desired definition 00:45:56.060 --> 00:45:57.540 align:middle line:90% for weak partial order. 00:45:57.540 --> 00:45:59.370 align:middle line:84% We already said it's not symmetric. 00:45:59.370 --> 00:46:02.030 align:middle line:84% So we're probably going to have to replace that somehow. 00:46:02.030 --> 00:46:04.340 align:middle line:90% Are these transitive? 00:46:04.340 --> 00:46:08.240 align:middle line:84% If a less equal b and b less equal c, is a less equal c? 00:46:08.240 --> 00:46:10.110 align:middle line:90% Yeah, same for subset. 00:46:10.110 --> 00:46:11.250 align:middle line:90% Same for divides. 00:46:11.250 --> 00:46:12.220 align:middle line:90% So let's put that. 00:46:12.220 --> 00:46:16.490 align:middle line:90% 00:46:16.490 --> 00:46:19.700 align:middle line:84% So this is looking very similar to the definition 00:46:19.700 --> 00:46:21.120 align:middle line:90% of equivalence relation. 00:46:21.120 --> 00:46:22.370 align:middle line:90% We're going to make one tweak. 00:46:22.370 --> 00:46:24.740 align:middle line:84% Instead of requiring it to be symmetric, 00:46:24.740 --> 00:46:31.280 align:middle line:84% we're going to require it to be antisymmetric, which 00:46:31.280 --> 00:46:33.090 align:middle line:90% does not mean not symmetric. 00:46:33.090 --> 00:46:34.410 align:middle line:90% It's a different concept. 00:46:34.410 --> 00:46:37.370 align:middle line:90% Let's define that. 00:46:37.370 --> 00:46:49.550 align:middle line:84% So R is antisymmetric means for all a 00:46:49.550 --> 00:47:03.590 align:middle line:84% and b, if a relates to b and b relates to a, then a equals b. 00:47:03.590 --> 00:47:06.410 align:middle line:84% So the only way that you relate in both directions 00:47:06.410 --> 00:47:10.040 align:middle line:84% is if you're comparing yourself to yourself. 00:47:10.040 --> 00:47:12.540 align:middle line:84% That is certainly true for less equal. 00:47:12.540 --> 00:47:15.510 align:middle line:84% If we know that a is less equal b and a is greater or equal b, 00:47:15.510 --> 00:47:17.390 align:middle line:90% then a equals b. 00:47:17.390 --> 00:47:19.530 align:middle line:84% That's true for a subset, as well. 00:47:19.530 --> 00:47:22.590 align:middle line:84% If we have two sets and each of them is a subset of the other, 00:47:22.590 --> 00:47:25.310 align:middle line:84% then they're exactly the same set. 00:47:25.310 --> 00:47:27.300 align:middle line:84% And so this is going to be our axiom. 00:47:27.300 --> 00:47:32.660 align:middle line:84% If r is reflexive, antisymmetric like that, and transitive, 00:47:32.660 --> 00:47:35.040 align:middle line:84% we're going to define that to be a weak partial order. 00:47:35.040 --> 00:47:38.300 align:middle line:84% And that's going to be our notion of behaving 00:47:38.300 --> 00:47:41.320 align:middle line:90% like these familiar examples. 00:47:41.320 --> 00:47:43.842 align:middle line:84% And so why is a weak partial order useful? 00:47:43.842 --> 00:47:45.050 align:middle line:90% What can we prove about them? 00:47:45.050 --> 00:47:47.320 align:middle line:90% What can we do with them? 00:47:47.320 --> 00:47:51.250 align:middle line:84% Well, just like equivalence relations had 00:47:51.250 --> 00:47:56.470 align:middle line:84% a different characterization in terms of partitions of a set, 00:47:56.470 --> 00:48:00.140 align:middle line:84% we can prove that equivalence relations-- sorry, 00:48:00.140 --> 00:48:05.240 align:middle line:84% prove that weak partial orders are related to DAGs. 00:48:05.240 --> 00:48:14.200 align:middle line:90% Claim-- yes. 00:48:14.200 --> 00:48:30.670 align:middle line:84% If G is a digraph, then the walk relation G star 00:48:30.670 --> 00:48:37.910 align:middle line:84% is a weak partial order if and only if G is a DAG. 00:48:37.910 --> 00:48:44.310 align:middle line:90% 00:48:44.310 --> 00:48:45.990 align:middle line:84% So the point of weak partial order 00:48:45.990 --> 00:48:49.830 align:middle line:84% is that we can put things in somewhat 00:48:49.830 --> 00:48:52.830 align:middle line:90% increasing order of size. 00:48:52.830 --> 00:48:56.220 align:middle line:84% If a is less than b-- less equal b, less equal c, less equal d, 00:48:56.220 --> 00:49:00.450 align:middle line:84% less equal e, what we want to avoid is coming back around 00:49:00.450 --> 00:49:06.810 align:middle line:84% and e is less equal a because, as it turns out, 00:49:06.810 --> 00:49:13.380 align:middle line:84% that's going to break some of these axioms. 00:49:13.380 --> 00:49:18.960 align:middle line:84% And so all we're saying is that satisfying these three axioms 00:49:18.960 --> 00:49:24.360 align:middle line:84% is equivalent to being the walk relation on a DAG. 00:49:24.360 --> 00:49:28.560 align:middle line:84% And DAGs were defined by not having directed cycles. 00:49:28.560 --> 00:49:32.253 align:middle line:84% All right, we're not going to prove this, but it's true. 00:49:32.253 --> 00:49:34.295 align:middle line:84% You can come ask me about it later if you'd like. 00:49:34.295 --> 00:49:36.820 align:middle line:90% 00:49:36.820 --> 00:49:39.890 align:middle line:90% Last definition for now-- 00:49:39.890 --> 00:49:41.175 align:middle line:90% once again, I apologize. 00:49:41.175 --> 00:49:42.675 align:middle line:84% There are lots of definitions today. 00:49:42.675 --> 00:49:47.860 align:middle line:90% 00:49:47.860 --> 00:49:51.220 align:middle line:84% But I want to bring our attention to this word 00:49:51.220 --> 00:49:51.980 align:middle line:90% "partial." 00:49:51.980 --> 00:49:56.650 align:middle line:84% A weak partial order, just like we had partial function that 00:49:56.650 --> 00:50:00.160 align:middle line:84% might be missing values, a weak partial order 00:50:00.160 --> 00:50:02.710 align:middle line:90% might be missing comparisons. 00:50:02.710 --> 00:50:10.135 align:middle line:84% For example, the set 2, 1 and the set 2, 00:50:10.135 --> 00:50:16.600 align:middle line:90% 7, is one a subset of the other? 00:50:16.600 --> 00:50:18.945 align:middle line:84% No, in neither direction is that true. 00:50:18.945 --> 00:50:23.650 align:middle line:90% 00:50:23.650 --> 00:50:26.680 align:middle line:84% So here are two sets, and neither 00:50:26.680 --> 00:50:31.840 align:middle line:84% is related to the other one under this subset relation. 00:50:31.840 --> 00:50:35.780 align:middle line:84% So this is a pair that cannot be compared. 00:50:35.780 --> 00:50:41.060 align:middle line:84% So if R is a weak partial order, we'll 00:50:41.060 --> 00:50:50.030 align:middle line:84% say that a and b are comparable if and only 00:50:50.030 --> 00:50:53.910 align:middle line:84% if the relation goes in at least one of the two directions, 00:50:53.910 --> 00:50:57.470 align:middle line:84% a relates to b or b relates to a. 00:50:57.470 --> 00:50:59.246 align:middle line:90% Otherwise, they're incomparable. 00:50:59.246 --> 00:51:08.000 align:middle line:90% 00:51:08.000 --> 00:51:10.610 align:middle line:84% So these two sets, 2, 1 and 2, 7, 00:51:10.610 --> 00:51:15.410 align:middle line:84% are incomparable under this subset equal relation. 00:51:15.410 --> 00:51:16.550 align:middle line:90% All right? 00:51:16.550 --> 00:51:31.040 align:middle line:84% And now, finally, a weak partial order 00:51:31.040 --> 00:51:38.810 align:middle line:84% is called a total order if and only 00:51:38.810 --> 00:51:45.046 align:middle line:90% if every pair is comparable. 00:51:45.046 --> 00:51:47.870 align:middle line:90% 00:51:47.870 --> 00:51:50.340 align:middle line:84% So whenever you have two elements, 00:51:50.340 --> 00:51:52.170 align:middle line:84% you can always put them in order. 00:51:52.170 --> 00:51:54.560 align:middle line:84% One of them is less or equal to the other one. 00:51:54.560 --> 00:51:58.010 align:middle line:84% So that's certainly true for the less equal relation. 00:51:58.010 --> 00:52:01.290 align:middle line:84% It's not true for the subset relation. 00:52:01.290 --> 00:52:03.470 align:middle line:84% And it's not true for the divides relation. 00:52:03.470 --> 00:52:07.430 align:middle line:84% 2 and 3 are incomparable under the divides relation. 00:52:07.430 --> 00:52:12.320 align:middle line:84% But if every pair can be compared, 00:52:12.320 --> 00:52:15.200 align:middle line:90% then it's called a total order. 00:52:15.200 --> 00:52:17.390 align:middle line:84% And if you have a total order, then really all you 00:52:17.390 --> 00:52:21.270 align:middle line:84% have is just an entire line of them organized by size. 00:52:21.270 --> 00:52:25.410 align:middle line:84% A total order is the most structure you can have. 00:52:25.410 --> 00:52:28.766 align:middle line:84% It's just everything is ordered, and there's no ambiguity. 00:52:28.766 --> 00:52:31.410 align:middle line:90% 00:52:31.410 --> 00:52:34.980 align:middle line:84% All right, so that is everything I 00:52:34.980 --> 00:52:38.790 align:middle line:84% wanted to say on the relations side of things today. 00:52:38.790 --> 00:52:42.210 align:middle line:84% Now let's move on to the other part of today's lecture, 00:52:42.210 --> 00:52:45.360 align:middle line:90% which is counting. 00:52:45.360 --> 00:52:48.670 align:middle line:84% And I don't mean counting like 1, 2, 3, 4, 5. 00:52:48.670 --> 00:52:51.240 align:middle line:84% I assume all of us are capable of counting 00:52:51.240 --> 00:52:55.050 align:middle line:90% at least to 300 or so. 00:52:55.050 --> 00:53:02.700 align:middle line:84% No, but by "counting," I mean counting the size a set. 00:53:02.700 --> 00:53:05.700 align:middle line:84% If you want to know-- like, here's 00:53:05.700 --> 00:53:08.590 align:middle line:84% a comically large deck of playing cards. 00:53:08.590 --> 00:53:10.750 align:middle line:84% How many different ways are there to shuffle them? 00:53:10.750 --> 00:53:12.480 align:middle line:84% How many different orderings are there 00:53:12.480 --> 00:53:14.770 align:middle line:90% for the 52 cards in the deck? 00:53:14.770 --> 00:53:18.060 align:middle line:90% Here's one of those orderings. 00:53:18.060 --> 00:53:19.743 align:middle line:84% Here's a different one, et cetera. 00:53:19.743 --> 00:53:21.910 align:middle line:84% How many orderings are there for this deck of cards? 00:53:21.910 --> 00:53:23.095 align:middle line:90% Does anyone know? 00:53:23.095 --> 00:53:23.595 align:middle line:90% Yes? 00:53:23.595 --> 00:53:25.086 align:middle line:90% AUDIENCE: 52 factorial. 00:53:25.086 --> 00:53:28.330 align:middle line:84% ZACHARY ABEL: 52 factorial, absolutely. 00:53:28.330 --> 00:53:38.890 align:middle line:84% So the number of orders for a deck of cards is 52 factorial. 00:53:38.890 --> 00:53:40.890 align:middle line:84% And we'll see why later in this course, 00:53:40.890 --> 00:53:43.410 align:middle line:84% or later in this lecture in fact. 00:53:43.410 --> 00:53:52.620 align:middle line:84% How many trees are there whose vertices are exactly 00:53:52.620 --> 00:53:58.830 align:middle line:90% the set 1, 2, 3, up to n? 00:53:58.830 --> 00:54:01.980 align:middle line:90% Does anyone know? 00:54:01.980 --> 00:54:03.370 align:middle line:90% I'd honestly be surprised. 00:54:03.370 --> 00:54:05.520 align:middle line:90% It's not an easy problem. 00:54:05.520 --> 00:54:07.620 align:middle line:84% But this is the kind of problem that we're 00:54:07.620 --> 00:54:09.910 align:middle line:84% going to try to be able to answer 00:54:09.910 --> 00:54:12.550 align:middle line:84% and we're going to look at techniques for trying to answer. 00:54:12.550 --> 00:54:15.660 align:middle line:84% In this case, the number of trees on that set of vertices 00:54:15.660 --> 00:54:19.480 align:middle line:84% is n to the n minus 2, which I love. 00:54:19.480 --> 00:54:20.723 align:middle line:90% I find that kind of magical. 00:54:20.723 --> 00:54:22.390 align:middle line:84% There are lots of cool ways to prove it. 00:54:22.390 --> 00:54:23.590 align:middle line:84% We're not going to prove it together, 00:54:23.590 --> 00:54:25.390 align:middle line:84% but come ask me if you're interested. 00:54:25.390 --> 00:54:26.997 align:middle line:84% But these are the kinds of questions 00:54:26.997 --> 00:54:28.080 align:middle line:90% we're going to try to ask. 00:54:28.080 --> 00:54:29.490 align:middle line:90% Here's a set of things. 00:54:29.490 --> 00:54:32.150 align:middle line:90% How many things are there? 00:54:32.150 --> 00:54:36.540 align:middle line:84% And to get us a little more in the mood for counting, 00:54:36.540 --> 00:54:40.700 align:middle line:84% I have a short story I want to tell you, the parable of the two 00:54:40.700 --> 00:54:41.790 align:middle line:90% shepherds. 00:54:41.790 --> 00:54:42.870 align:middle line:90% No one calls it that. 00:54:42.870 --> 00:54:45.050 align:middle line:90% That's just me. 00:54:45.050 --> 00:54:50.360 align:middle line:84% So this first shepherd-- this first shepherd, every morning, 00:54:50.360 --> 00:54:52.980 align:middle line:84% has to let the sheep out to graze every evening, 00:54:52.980 --> 00:54:55.650 align:middle line:84% bring the sheep back in to protect them from wolves, 00:54:55.650 --> 00:54:56.150 align:middle line:90% I guess. 00:54:56.150 --> 00:54:59.060 align:middle line:90% I don't know what shepherds do. 00:54:59.060 --> 00:55:01.430 align:middle line:84% But unfortunately, the shepherd does not 00:55:01.430 --> 00:55:05.360 align:middle line:84% know how to count grade-school style, 1, 2, 3. 00:55:05.360 --> 00:55:09.380 align:middle line:84% The shepherd doesn't have words for numbers above 3. 00:55:09.380 --> 00:55:13.550 align:middle line:84% So how do they make sure that all of their 50-something sheep 00:55:13.550 --> 00:55:16.580 align:middle line:84% go out in the morning and come back in the evening if they 00:55:16.580 --> 00:55:18.470 align:middle line:90% can't count the sheep? 00:55:18.470 --> 00:55:21.800 align:middle line:84% And a classic answer to this, which may or may not 00:55:21.800 --> 00:55:26.372 align:middle line:84% have historical merit, but all mathematicians 00:55:26.372 --> 00:55:28.330 align:middle line:84% who have studied counting have heard this story 00:55:28.330 --> 00:55:31.060 align:middle line:84% at least once, the classic example is 00:55:31.060 --> 00:55:33.260 align:middle line:84% every time you let a sheep out of the pen, 00:55:33.260 --> 00:55:36.130 align:middle line:90% you put a pebble in your pocket. 00:55:36.130 --> 00:55:37.580 align:middle line:90% Another sheep, another pebble. 00:55:37.580 --> 00:55:38.830 align:middle line:90% Another sheep, another pebble. 00:55:38.830 --> 00:55:41.247 align:middle line:84% At the end of the day, when you're bringing the sheep back 00:55:41.247 --> 00:55:43.030 align:middle line:84% in, every time a sheep comes in, you 00:55:43.030 --> 00:55:45.370 align:middle line:84% take a pebble out of your pocket. 00:55:45.370 --> 00:55:48.290 align:middle line:84% And when all the sheep have come in, 00:55:48.290 --> 00:55:50.890 align:middle line:84% that's the same moment that all the pebbles are out 00:55:50.890 --> 00:55:51.920 align:middle line:90% of your pocket. 00:55:51.920 --> 00:55:53.320 align:middle line:84% So when your pocket is empty, you 00:55:53.320 --> 00:55:54.760 align:middle line:90% know there are no more sheep. 00:55:54.760 --> 00:55:58.910 align:middle line:84% And vice versa, if there are pebbles left in your pocket, 00:55:58.910 --> 00:56:02.050 align:middle line:84% there are still sheep out there, and you have to go find them. 00:56:02.050 --> 00:56:04.640 align:middle line:84% If your pocket is empty and there are more sheep coming in, 00:56:04.640 --> 00:56:06.598 align:middle line:84% well, something weird happened and you probably 00:56:06.598 --> 00:56:08.020 align:middle line:90% stole from your neighbor. 00:56:08.020 --> 00:56:11.590 align:middle line:84% But the point is, for these next couple of lectures 00:56:11.590 --> 00:56:15.320 align:middle line:84% in our counting unit, we count with correspondences. 00:56:15.320 --> 00:56:17.680 align:middle line:84% We count with bijections, which we 00:56:17.680 --> 00:56:21.100 align:middle line:90% defined on one of these boards. 00:56:21.100 --> 00:56:23.180 align:middle line:84% If we can set up a one-to-one correspondence, 00:56:23.180 --> 00:56:25.820 align:middle line:84% everything in this set corresponds to everything 00:56:25.820 --> 00:56:28.080 align:middle line:84% in this set and everyone has a partner, 00:56:28.080 --> 00:56:31.160 align:middle line:84% no one has two partners, no one is missing a partner, 00:56:31.160 --> 00:56:34.820 align:middle line:84% then there are the same number of things on both sides. 00:56:34.820 --> 00:56:36.620 align:middle line:84% What that shepherd is doing is basically 00:56:36.620 --> 00:56:38.620 align:middle line:84% making sure that there's always a correspondence 00:56:38.620 --> 00:56:41.570 align:middle line:84% between sheep that are out there and pebbles in their pocket. 00:56:41.570 --> 00:56:45.120 align:middle line:84% They don't have to keep track of what that correspondence is. 00:56:45.120 --> 00:56:46.790 align:middle line:90% And in fact, it might change. 00:56:46.790 --> 00:56:52.010 align:middle line:84% If you initially put a pebble in your pocket for sheep Deborah. 00:56:52.010 --> 00:56:54.470 align:middle line:84% you're not keeping track of which pebble corresponds 00:56:54.470 --> 00:56:55.190 align:middle line:90% to sheep Deborah. 00:56:55.190 --> 00:56:56.607 align:middle line:84% So when Deborah comes back in, you 00:56:56.607 --> 00:56:59.490 align:middle line:84% might remove the pebble that was assigned to Sandy. 00:56:59.490 --> 00:57:03.950 align:middle line:84% I don't know why I'm picking these names. 00:57:03.950 --> 00:57:07.940 align:middle line:84% But you don't care what that correspondence is, as long 00:57:07.940 --> 00:57:10.580 align:middle line:84% as there is a correspondence between pebbles in your pocket 00:57:10.580 --> 00:57:13.130 align:middle line:84% and sheep that are still grazing. 00:57:13.130 --> 00:57:17.270 align:middle line:84% So the moral here is we count with correspondences. 00:57:17.270 --> 00:57:20.118 align:middle line:84% The second shepherd-- this second shepherd 00:57:20.118 --> 00:57:20.910 align:middle line:90% knows how to count. 00:57:20.910 --> 00:57:22.370 align:middle line:84% That's totally fine, not a problem. 00:57:22.370 --> 00:57:23.890 align:middle line:90% We have language. 00:57:23.890 --> 00:57:27.100 align:middle line:84% But the shepherd has an overeager apprentice. 00:57:27.100 --> 00:57:29.240 align:middle line:84% This apprentice runs in one evening and says, 00:57:29.240 --> 00:57:31.960 align:middle line:84% master, I brought all 40 sheep into the pen. 00:57:31.960 --> 00:57:34.010 align:middle line:90% And the shepherd looks confused. 00:57:34.010 --> 00:57:34.820 align:middle line:90% What do you mean? 00:57:34.820 --> 00:57:37.600 align:middle line:90% We only have 38 sheep. 00:57:37.600 --> 00:57:40.282 align:middle line:84% Don't worry, master, I rounded them up. 00:57:40.282 --> 00:57:41.728 align:middle line:90% [SOFT LAUGHTER] 00:57:41.728 --> 00:57:44.140 align:middle line:90% 00:57:44.140 --> 00:57:44.840 align:middle line:90% Thank you. 00:57:44.840 --> 00:57:45.350 align:middle line:90% No moral. 00:57:45.350 --> 00:57:47.694 align:middle line:90% I just like bad puns. 00:57:47.694 --> 00:57:49.622 align:middle line:90% [SOFT LAUGHTER] 00:57:49.622 --> 00:57:51.070 align:middle line:90% 00:57:51.070 --> 00:57:52.430 align:middle line:90% I heard one groan. 00:57:52.430 --> 00:57:53.420 align:middle line:90% Thank you very much. 00:57:53.420 --> 00:57:56.320 align:middle line:90% That warms my heart. 00:57:56.320 --> 00:57:59.840 align:middle line:84% And now that we're in a great mood, let's talk about counting. 00:57:59.840 --> 00:58:01.712 align:middle line:90% [LAUGHTER] 00:58:01.712 --> 00:58:02.650 align:middle line:90% 00:58:02.650 --> 00:58:07.720 align:middle line:84% All right, so let's actually talk about counting. 00:58:07.720 --> 00:58:09.340 align:middle line:90% Here. 00:58:09.340 --> 00:58:10.820 align:middle line:90% First of all, why do we care? 00:58:10.820 --> 00:58:16.460 align:middle line:84% Why are these tools that we need in a computer science class? 00:58:16.460 --> 00:58:18.620 align:middle line:84% Why do we care about counting things? 00:58:18.620 --> 00:58:22.660 align:middle line:84% Well, often when we're analyzing the runtime of an algorithm, 00:58:22.660 --> 00:58:24.988 align:middle line:84% we're doing some steps some number of times. 00:58:24.988 --> 00:58:27.030 align:middle line:84% And we need to be able to count how many times we 00:58:27.030 --> 00:58:30.368 align:middle line:84% do that step in order to bound the runtime of the algorithm 00:58:30.368 --> 00:58:31.535 align:middle line:90% and prove that it runs fast. 00:58:31.535 --> 00:58:35.770 align:middle line:90% 00:58:35.770 --> 00:58:38.730 align:middle line:84% It's also useful for things like probability. 00:58:38.730 --> 00:58:42.490 align:middle line:84% Probability is built very heavily on top of counting. 00:58:42.490 --> 00:58:43.950 align:middle line:84% And all of our counting techniques 00:58:43.950 --> 00:58:47.110 align:middle line:84% will transfer over to probability techniques, as well. 00:58:47.110 --> 00:58:50.460 align:middle line:84% And that'll be useful after quiz 2, which is coming up next week 00:58:50.460 --> 00:58:51.110 align:middle line:90% by the way. 00:58:51.110 --> 00:58:54.690 align:middle line:90% 00:58:54.690 --> 00:58:58.230 align:middle line:84% But let's learn-- where my chalk is. 00:58:58.230 --> 00:59:00.420 align:middle line:90% Found it. 00:59:00.420 --> 00:59:03.270 align:middle line:84% Let's learn some techniques for counting. 00:59:03.270 --> 00:59:10.350 align:middle line:84% The first one I want to mention is called the product rule, 00:59:10.350 --> 00:59:21.970 align:middle line:84% which says that for finite sets A and B, the size of A cross B 00:59:21.970 --> 00:59:25.480 align:middle line:84% equals the size of A times the size of B. 00:59:25.480 --> 00:59:29.500 align:middle line:84% Remember, A cross B is the set of all possible pairs where 00:59:29.500 --> 00:59:33.130 align:middle line:84% the first element is in A and the second element is in B. 00:59:33.130 --> 00:59:35.770 align:middle line:84% So the number of ways to choose an element from A 00:59:35.770 --> 00:59:37.780 align:middle line:84% and then choose an element from B 00:59:37.780 --> 00:59:41.860 align:middle line:84% is the size of A times the size of B. 00:59:41.860 --> 00:59:43.950 align:middle line:90% And this generalizes, as well. 00:59:43.950 --> 00:59:51.700 align:middle line:90% 00:59:51.700 --> 00:59:58.000 align:middle line:84% If we have n different sets and we're 00:59:58.000 --> 01:00:02.560 align:middle line:84% looking at ordered sequences where the first element comes 01:00:02.560 --> 01:00:04.090 align:middle line:84% from A1, the second element comes 01:00:04.090 --> 01:00:06.230 align:middle line:84% from A2, the third element from A3, 01:00:06.230 --> 01:00:09.880 align:middle line:84% and all the way down, the number of such sequences 01:00:09.880 --> 01:00:13.990 align:middle line:84% is just the size of A1 times the size of A2, 01:00:13.990 --> 01:00:18.250 align:middle line:90% all the way to the size of An. 01:00:18.250 --> 01:00:20.200 align:middle line:84% So if we have to choose an element of A1 01:00:20.200 --> 01:00:23.080 align:middle line:84% and then choose an element of A2 and then choose an element of A3 01:00:23.080 --> 01:00:25.720 align:middle line:84% and all the way down, the number of ways 01:00:25.720 --> 01:00:28.210 align:middle line:84% we can make that choice in its entirety 01:00:28.210 --> 01:00:32.170 align:middle line:84% is size of A1 times size of A2 times size 01:00:32.170 --> 01:00:33.730 align:middle line:90% of A3 all the way down. 01:00:33.730 --> 01:00:49.570 align:middle line:84% For example, the number of length n binary sequences-- 01:00:49.570 --> 01:00:51.700 align:middle line:84% so a binary sequence is a sequence 01:00:51.700 --> 01:00:54.520 align:middle line:84% of zeros and ones, just zeros and ones. 01:00:54.520 --> 01:00:56.680 align:middle line:84% And we're asking, how many length n binary 01:00:56.680 --> 01:00:58.510 align:middle line:90% sequences are there? 01:00:58.510 --> 01:01:01.210 align:middle line:84% Well, the set of length n binary sequences 01:01:01.210 --> 01:01:09.970 align:middle line:84% is 0, 1 times 0, 1 times 0, 1 n times. 01:01:09.970 --> 01:01:12.980 align:middle line:84% This is precisely the set of binary sequences. 01:01:12.980 --> 01:01:14.720 align:middle line:84% This is what we mean by binary sequence. 01:01:14.720 --> 01:01:16.500 align:middle line:90% First element is a 0 or a 1. 01:01:16.500 --> 01:01:19.620 align:middle line:84% Second element is a 0 or a 1, all the way down. 01:01:19.620 --> 01:01:21.170 align:middle line:84% And the generalized product rule-- 01:01:21.170 --> 01:01:26.060 align:middle line:84% or sorry, the general version of the product rule 01:01:26.060 --> 01:01:29.750 align:middle line:84% says that this set, the size of this set 01:01:29.750 --> 01:01:34.925 align:middle line:84% is just the size of 0, 1 times the size of 0, 01:01:34.925 --> 01:01:45.890 align:middle line:84% 1 all the way down, which is 2 times 2 times 2 times 2 n times. 01:01:45.890 --> 01:01:49.610 align:middle line:84% So there are 2 to the n binary sequences of length n. 01:01:49.610 --> 01:01:51.680 align:middle line:90% Does that make sense? 01:01:51.680 --> 01:01:53.570 align:middle line:90% Wonderful. 01:01:53.570 --> 01:01:56.725 align:middle line:84% Next, let's look at the bijection rule. 01:01:56.725 --> 01:02:11.390 align:middle line:90% 01:02:11.390 --> 01:02:19.890 align:middle line:84% If there exists a bijection from A to B, 01:02:19.890 --> 01:02:22.860 align:middle line:90% then A and B have the same size. 01:02:22.860 --> 01:02:24.655 align:middle line:84% This is what that first shepherd was doing. 01:02:24.655 --> 01:02:26.530 align:middle line:84% If there exists a bijection between two sets, 01:02:26.530 --> 01:02:28.447 align:middle line:84% the sets have the same size because we're just 01:02:28.447 --> 01:02:30.510 align:middle line:90% pairing them up one to one. 01:02:30.510 --> 01:02:31.540 align:middle line:90% No one is repeated. 01:02:31.540 --> 01:02:33.960 align:middle line:90% No one is excluded. 01:02:33.960 --> 01:02:45.960 align:middle line:84% And as an example of this, how many subsets of 1 through n 01:02:45.960 --> 01:02:46.655 align:middle line:90% are there? 01:02:46.655 --> 01:02:51.390 align:middle line:90% 01:02:51.390 --> 01:02:53.580 align:middle line:84% So if we're looking at all of the subsets of 1 01:02:53.580 --> 01:02:55.620 align:middle line:90% through n of any length-- 01:02:55.620 --> 01:02:57.970 align:middle line:84% could be length 0, where we take none of the elements; 01:02:57.970 --> 01:02:59.723 align:middle line:84% length n, where we take all the elements; 01:02:59.723 --> 01:03:01.140 align:middle line:84% or any length in between, where we 01:03:01.140 --> 01:03:03.190 align:middle line:84% take some of the elements and not others-- 01:03:03.190 --> 01:03:05.520 align:middle line:90% how many subsets are there? 01:03:05.520 --> 01:03:08.260 align:middle line:84% Well, let's use the bijection rule. 01:03:08.260 --> 01:03:17.820 align:middle line:84% I claim set of subsets of 1 through n 01:03:17.820 --> 01:03:27.450 align:middle line:90% is in bijection with Bn. 01:03:27.450 --> 01:03:31.638 align:middle line:90% And I'm going to define this. 01:03:31.638 --> 01:03:40.440 align:middle line:84% Bn is binary sequences of length n. 01:03:40.440 --> 01:03:42.790 align:middle line:84% So the thing we counted in the last example, 01:03:42.790 --> 01:03:44.850 align:middle line:84% I claim that the set of subsets of 1 through n 01:03:44.850 --> 01:03:50.310 align:middle line:84% is in bijection with the set of those length n binary sequences. 01:03:50.310 --> 01:03:54.240 align:middle line:84% And to prove that, we can just define our function 01:03:54.240 --> 01:03:57.000 align:middle line:84% and verify that it's a bijection. 01:03:57.000 --> 01:04:07.410 align:middle line:84% So given a binary sequence b1, b2, bn, where each of these 01:04:07.410 --> 01:04:13.560 align:middle line:84% is 0 or 1, I'm going to map that to a subset 01:04:13.560 --> 01:04:23.760 align:middle line:84% of 1 through n, i in 1 through n, such that bi equals 1. 01:04:23.760 --> 01:04:28.140 align:middle line:84% We send it to the subset of indices that have value 1. 01:04:28.140 --> 01:04:30.180 align:middle line:84% So you can think of this as a set-- 01:04:30.180 --> 01:04:32.530 align:middle line:84% as a sequence of binary decisions. 01:04:32.530 --> 01:04:35.410 align:middle line:84% Do we keep element 1 or throw it away? 01:04:35.410 --> 01:04:37.620 align:middle line:84% Do we keep element 2 or throw it away? 01:04:37.620 --> 01:04:38.550 align:middle line:90% All the way down. 01:04:38.550 --> 01:04:45.720 align:middle line:84% For example, 0, 1; 0; 0, 1; 1, if n 01:04:45.720 --> 01:04:52.916 align:middle line:84% is 6, then this maps to the subset 01:04:52.916 --> 01:05:00.120 align:middle line:84% 2, 5, and 6 because those are the indices that have ones. 01:05:00.120 --> 01:05:03.010 align:middle line:84% All right, and we're not going to do it now. 01:05:03.010 --> 01:05:04.500 align:middle line:84% But you can verify that this gives 01:05:04.500 --> 01:05:09.160 align:middle line:84% a bijection between the set Bn and the set of subsets of 1 01:05:09.160 --> 01:05:09.660 align:middle line:90% through n. 01:05:09.660 --> 01:05:13.180 align:middle line:90% 01:05:13.180 --> 01:05:16.386 align:middle line:90% Claim-- this is a bijection. 01:05:16.386 --> 01:05:19.120 align:middle line:90% 01:05:19.120 --> 01:05:22.570 align:middle line:84% You'll see more examples in your homework of what details 01:05:22.570 --> 01:05:25.240 align:middle line:84% we would need to check to verify more carefully that this really 01:05:25.240 --> 01:05:28.240 align:middle line:90% is a bijection. 01:05:28.240 --> 01:05:31.790 align:middle line:84% But intuitively, they're encoding the same information. 01:05:31.790 --> 01:05:35.630 align:middle line:84% Each number from 1 through n is either included or excluded. 01:05:35.630 --> 01:05:37.460 align:middle line:90% That's a binary decision. 01:05:37.460 --> 01:05:40.640 align:middle line:90% The i-th bit is either 1 or 0. 01:05:40.640 --> 01:05:42.080 align:middle line:90% That's a binary decision. 01:05:42.080 --> 01:05:45.310 align:middle line:84% So we're just corresponding those two things. 01:05:45.310 --> 01:05:46.335 align:middle line:90% All right, next. 01:05:46.335 --> 01:05:48.940 align:middle line:90% 01:05:48.940 --> 01:05:51.650 align:middle line:84% We have the product rule, the bijection rule. 01:05:51.650 --> 01:05:53.025 align:middle line:90% Now we have the sum rule. 01:05:53.025 --> 01:06:03.640 align:middle line:90% 01:06:03.640 --> 01:06:11.780 align:middle line:84% Sum rule says that if A and B are disjoint, 01:06:11.780 --> 01:06:15.620 align:middle line:84% so they have no elements in common, 01:06:15.620 --> 01:06:19.820 align:middle line:84% then the size of A union B is the size of A 01:06:19.820 --> 01:06:26.940 align:middle line:90% plus the size of B. 01:06:26.940 --> 01:06:31.970 align:middle line:84% So if I have five cards in my right hand 01:06:31.970 --> 01:06:34.760 align:middle line:84% and three cards in my left hand, and I 01:06:34.760 --> 01:06:37.260 align:middle line:84% don't have any cards in both hands, 01:06:37.260 --> 01:06:38.510 align:middle line:90% then I'm holding eight cards-- 01:06:38.510 --> 01:06:39.370 align:middle line:90% 5 plus 3. 01:06:39.370 --> 01:06:42.170 align:middle line:90% 01:06:42.170 --> 01:06:46.550 align:middle line:84% And more generally, if I'm taking 01:06:46.550 --> 01:06:52.310 align:middle line:84% the size of a disjoint union of n things, 01:06:52.310 --> 01:06:55.910 align:middle line:84% if all of these Ai's are disjoint with each other, 01:06:55.910 --> 01:07:01.880 align:middle line:84% then this is the size of A1 plus the size of A2 plus the size 01:07:01.880 --> 01:07:03.980 align:middle line:90% of An. 01:07:03.980 --> 01:07:05.695 align:middle line:90% And let's see that in action. 01:07:05.695 --> 01:07:13.370 align:middle line:90% 01:07:13.370 --> 01:07:19.790 align:middle line:84% OK, so all of us who have not been using password managers 01:07:19.790 --> 01:07:22.490 align:middle line:84% all our lives have frequently been told, 01:07:22.490 --> 01:07:26.480 align:middle line:84% choose another password that has these specific requirements that 01:07:26.480 --> 01:07:27.360 align:middle line:90% has this length. 01:07:27.360 --> 01:07:28.860 align:middle line:90% It must use a special character. 01:07:28.860 --> 01:07:31.220 align:middle line:84% It has to do all this other stuff, which just makes it 01:07:31.220 --> 01:07:33.950 align:middle line:90% harder for me to remember it. 01:07:33.950 --> 01:07:37.530 align:middle line:84% Anyway, I'm going to play that role now. 01:07:37.530 --> 01:07:55.130 align:middle line:84% How many passwords have length between 6 and 8 inclusive, 01:07:55.130 --> 01:08:04.660 align:middle line:84% start with a capital letter, and the rest 01:08:04.660 --> 01:08:14.850 align:middle line:84% are capital or lowercase letters and/or digits. 01:08:14.850 --> 01:08:17.560 align:middle line:90% 01:08:17.560 --> 01:08:19.700 align:middle line:84% The digits aren't capital or lowercase. 01:08:19.700 --> 01:08:20.840 align:middle line:90% They're just digits. 01:08:20.840 --> 01:08:23.510 align:middle line:84% So how many passwords have length between 6 and 8, 01:08:23.510 --> 01:08:28.090 align:middle line:84% start with a capital letter, and then finish with whatever 01:08:28.090 --> 01:08:31.630 align:middle line:90% letters or digits you want? 01:08:31.630 --> 01:08:34.810 align:middle line:84% And with this, it's useful to break it up 01:08:34.810 --> 01:08:38.710 align:middle line:84% into cases based on the length of your password. 01:08:38.710 --> 01:08:41.770 align:middle line:90% I claim that the set W-- 01:08:41.770 --> 01:08:47.740 align:middle line:84% I called it W in the notes for some reason, W for "password"-- 01:08:47.740 --> 01:08:56.170 align:middle line:84% I claim that W is the disjoint union of the set of passwords 01:08:56.170 --> 01:08:59.020 align:middle line:84% of length 6 that satisfy the other properties, 01:08:59.020 --> 01:09:00.850 align:middle line:84% set of passwords of length 7, and set 01:09:00.850 --> 01:09:02.380 align:middle line:90% of passwords of length 8. 01:09:02.380 --> 01:09:05.649 align:middle line:84% So if we can count each of these three subsets, 01:09:05.649 --> 01:09:08.640 align:middle line:84% then we add them up and get the size of W. 01:09:08.640 --> 01:09:10.770 align:middle line:90% And let's see if we can do that. 01:09:10.770 --> 01:09:13.680 align:middle line:90% So what is the size of W6? 01:09:13.680 --> 01:09:17.903 align:middle line:84% How many passwords satisfy this property? 01:09:17.903 --> 01:09:21.870 align:middle line:90% 01:09:21.870 --> 01:09:29.859 align:middle line:84% Well, I claim the set W6 itself as a set is, well, 01:09:29.859 --> 01:09:36.689 align:middle line:84% first of all, the set of capital letters for the first entry 01:09:36.689 --> 01:09:44.220 align:middle line:84% times the set of letters or digits 01:09:44.220 --> 01:09:47.310 align:middle line:84% to the fifth for the other five entries. 01:09:47.310 --> 01:09:49.420 align:middle line:84% This is exactly what this set is. 01:09:49.420 --> 01:09:51.340 align:middle line:84% The first entry has to be a capital letter. 01:09:51.340 --> 01:09:55.860 align:middle line:84% The other five characters can be anything from the set. 01:09:55.860 --> 01:10:00.880 align:middle line:84% So we have this product of six sets. 01:10:00.880 --> 01:10:04.480 align:middle line:84% And that brings us back to the regular product rule. 01:10:04.480 --> 01:10:07.690 align:middle line:84% So by the product rule, the size of W6 01:10:07.690 --> 01:10:09.890 align:middle line:84% is, well, how many capital letters are there? 01:10:09.890 --> 01:10:10.730 align:middle line:90% I think that's 26. 01:10:10.730 --> 01:10:13.240 align:middle line:90% 01:10:13.240 --> 01:10:16.010 align:middle line:84% And then how many letters and digits are there? 01:10:16.010 --> 01:10:18.767 align:middle line:90% 26, 26, and 10, so that's 50-- 01:10:18.767 --> 01:10:22.990 align:middle line:90% 62 to the fifth. 01:10:22.990 --> 01:10:25.755 align:middle line:84% We just use the regular product rule to get this. 01:10:25.755 --> 01:10:31.000 align:middle line:90% 01:10:31.000 --> 01:10:34.233 align:middle line:84% And we can do the same thing for W7 and W8. 01:10:34.233 --> 01:10:46.000 align:middle line:90% 01:10:46.000 --> 01:10:51.100 align:middle line:84% W7 is going to be 26 times 62, this time to the sixth 01:10:51.100 --> 01:10:53.500 align:middle line:84% because there are six extra digits after the-- 01:10:53.500 --> 01:10:56.420 align:middle line:84% extra characters after the first one. 01:10:56.420 --> 01:11:04.300 align:middle line:84% And the size of W8 is 26 times 62 to the seventh. 01:11:04.300 --> 01:11:10.840 align:middle line:84% So by the sum rule, since W is the disjoint union 01:11:10.840 --> 01:11:16.180 align:middle line:84% of these three different sets, the size of W is the size of W6 01:11:16.180 --> 01:11:24.130 align:middle line:84% plus the size of W7 plus the size of W8, which is 26 times 01:11:24.130 --> 01:11:29.380 align:middle line:84% 62 to the fifth, plus 26 times 62 to the sixth, plus 26 times 01:11:29.380 --> 01:11:32.740 align:middle line:90% 62 to the seventh. 01:11:32.740 --> 01:11:36.050 align:middle line:84% So we use the sum rule to break it up into three separate cases, 01:11:36.050 --> 01:11:38.720 align:middle line:84% and then use the product rule for each of those cases. 01:11:38.720 --> 01:11:41.140 align:middle line:90% Did that make sense? 01:11:41.140 --> 01:11:42.130 align:middle line:90% Wonderful. 01:11:42.130 --> 01:11:42.870 align:middle line:90% All right. 01:11:42.870 --> 01:11:46.180 align:middle line:90% 01:11:46.180 --> 01:11:49.780 align:middle line:84% So this is a common strategy here. 01:11:49.780 --> 01:11:56.320 align:middle line:84% Notice that we're using the sum rule when what we mean is "or." 01:11:56.320 --> 01:11:58.760 align:middle line:84% You can choose a password of length six 01:11:58.760 --> 01:12:03.930 align:middle line:84% or a password of length seven or a password of length eight. 01:12:03.930 --> 01:12:09.290 align:middle line:90% So "or" means sum. 01:12:09.290 --> 01:12:13.190 align:middle line:84% For product rule, you have to choose a capital letter 01:12:13.190 --> 01:12:16.140 align:middle line:84% and then choose one of these characters 01:12:16.140 --> 01:12:17.940 align:middle line:84% and then choose one of these characters 01:12:17.940 --> 01:12:19.898 align:middle line:84% and then choose one of these characters. 01:12:19.898 --> 01:12:21.440 align:middle line:84% I forget how many times I've said it, 01:12:21.440 --> 01:12:23.990 align:middle line:90% but five times for this part. 01:12:23.990 --> 01:12:25.990 align:middle line:90% We multiply when we mean "and." 01:12:25.990 --> 01:12:31.940 align:middle line:90% 01:12:31.940 --> 01:12:37.880 align:middle line:84% Said differently, if I have 10 shirts and 6 pants in my closet, 01:12:37.880 --> 01:12:41.490 align:middle line:84% if I want to choose an article of clothing, 01:12:41.490 --> 01:12:46.490 align:middle line:84% I can choose a shirt or a pair of pants. 01:12:46.490 --> 01:12:47.900 align:middle line:90% And there are 16 options-- 01:12:47.900 --> 01:12:50.630 align:middle line:90% 10 plus 6. 01:12:50.630 --> 01:12:51.660 align:middle line:90% "Or" means add. 01:12:51.660 --> 01:12:53.660 align:middle line:84% On the other hand, if I'm trying to put together 01:12:53.660 --> 01:12:55.440 align:middle line:84% an outfit for the day, I have to choose 01:12:55.440 --> 01:12:58.380 align:middle line:90% a shirt and a pair of pants. 01:12:58.380 --> 01:12:59.770 align:middle line:90% "And" means multiply. 01:12:59.770 --> 01:13:03.150 align:middle line:84% It's 10 times 6 options, so I have 60 outfits. 01:13:03.150 --> 01:13:04.870 align:middle line:84% Maybe not all of those look good. 01:13:04.870 --> 01:13:06.480 align:middle line:84% But I wear the same pants every day 01:13:06.480 --> 01:13:09.660 align:middle line:90% anyway, so it doesn't matter. 01:13:09.660 --> 01:13:11.500 align:middle line:90% Does this make sense? 01:13:11.500 --> 01:13:15.750 align:middle line:84% Generally speaking, "or" means add, "and" means times. 01:13:15.750 --> 01:13:17.130 align:middle line:90% OK. 01:13:17.130 --> 01:13:20.160 align:middle line:84% The last example I want to talk about today 01:13:20.160 --> 01:13:22.415 align:middle line:84% is called the generalized product rule. 01:13:22.415 --> 01:13:29.850 align:middle line:90% 01:13:29.850 --> 01:13:33.630 align:middle line:84% And example first-- in fact, an example we mentioned earlier. 01:13:33.630 --> 01:13:38.890 align:middle line:84% How many different permutations are there for a set of 52 cards? 01:13:38.890 --> 01:13:42.040 align:middle line:84% How many different orderings are there for a set of 52 cards? 01:13:42.040 --> 01:13:45.088 align:middle line:84% Let's talk through this together. 01:13:45.088 --> 01:13:45.880 align:middle line:90% Here are the cards. 01:13:45.880 --> 01:13:49.180 align:middle line:84% I can't hold them all at once with my notes and my chalk. 01:13:49.180 --> 01:13:51.780 align:middle line:90% So bye, bye, cards. 01:13:51.780 --> 01:14:00.960 align:middle line:84% So how many permutations of the 52 cards? 01:14:00.960 --> 01:14:02.590 align:middle line:90% Well, let's think about it. 01:14:02.590 --> 01:14:05.470 align:middle line:84% Let's choose the first card first. 01:14:05.470 --> 01:14:08.110 align:middle line:84% Which card is going to go at the beginning of our list? 01:14:08.110 --> 01:14:12.150 align:middle line:84% Well, how many options do we have? 01:14:12.150 --> 01:14:14.140 align:middle line:84% We have the whole deck to play with. 01:14:14.140 --> 01:14:18.400 align:middle line:84% OK, I think I do want to hold the cards. 01:14:18.400 --> 01:14:19.900 align:middle line:84% We have the whole deck to play with. 01:14:19.900 --> 01:14:23.310 align:middle line:84% We can choose any one of the cards as our first card. 01:14:23.310 --> 01:14:25.893 align:middle line:90% Yeah, maybe I choose this one. 01:14:25.893 --> 01:14:27.310 align:middle line:84% I'm doing a magic trick to myself. 01:14:27.310 --> 01:14:29.190 align:middle line:90% I don't know what card this is. 01:14:29.190 --> 01:14:30.960 align:middle line:84% So the 9 of spades is going to be 01:14:30.960 --> 01:14:33.930 align:middle line:90% the first card in our ordering. 01:14:33.930 --> 01:14:36.060 align:middle line:90% So this was the 9 of spades. 01:14:36.060 --> 01:14:39.300 align:middle line:84% And we had 52 options for it, yeah? 01:14:39.300 --> 01:14:39.880 align:middle line:90% Great. 01:14:39.880 --> 01:14:42.040 align:middle line:84% That goes at the beginning of our order. 01:14:42.040 --> 01:14:45.210 align:middle line:90% What's the next card we pick? 01:14:45.210 --> 01:14:49.470 align:middle line:84% Well how many options do we have left? 01:14:49.470 --> 01:14:50.555 align:middle line:90% Just shout it out. 01:14:50.555 --> 01:14:51.340 align:middle line:90% AUDIENCE: 51. 01:14:51.340 --> 01:14:52.550 align:middle line:90% ZACHARY ABEL: 51, yeah. 01:14:52.550 --> 01:14:54.830 align:middle line:90% We have 51 options. 01:14:54.830 --> 01:14:57.463 align:middle line:84% Maybe we choose this one, without dropping 01:14:57.463 --> 01:14:58.130 align:middle line:90% all the others-- 01:14:58.130 --> 01:14:59.210 align:middle line:90% 8 of clubs. 01:14:59.210 --> 01:15:04.270 align:middle line:84% Great, that's the second card in our order, 8 of clubs. 01:15:04.270 --> 01:15:06.920 align:middle line:84% Now let's all watch as I do this 50 more times. 01:15:06.920 --> 01:15:09.010 align:middle line:90% No. 01:15:09.010 --> 01:15:11.290 align:middle line:90% But we get the idea. 01:15:11.290 --> 01:15:13.550 align:middle line:84% The next card, the third card in my ordering-- 01:15:13.550 --> 01:15:15.320 align:middle line:84% well, I've used two cards already. 01:15:15.320 --> 01:15:19.180 align:middle line:84% I have to choose one of the remaining 50 cards, 01:15:19.180 --> 01:15:22.720 align:middle line:90% then 49, then 48-- 01:15:22.720 --> 01:15:24.320 align:middle line:90% then 2, then 1. 01:15:24.320 --> 01:15:28.180 align:middle line:84% This is the number of choices I have at each stage. 01:15:28.180 --> 01:15:30.740 align:middle line:84% So it feels like, by the product rule, 01:15:30.740 --> 01:15:35.770 align:middle line:84% this answer should be 52 times 51 times 50 times 01:15:35.770 --> 01:15:37.430 align:middle line:90% 49, and so on-- 01:15:37.430 --> 01:15:41.350 align:middle line:90% 52 factorial. 01:15:41.350 --> 01:15:45.820 align:middle line:84% The weird thing, though, is that this isn't the product rule. 01:15:45.820 --> 01:15:47.800 align:middle line:84% Product rule says you have some set 01:15:47.800 --> 01:15:50.180 align:middle line:84% of options at the beginning, which is true. 01:15:50.180 --> 01:15:53.080 align:middle line:84% We have the set of 52 cards at the beginning. 01:15:53.080 --> 01:15:55.480 align:middle line:84% But then we're supposed to have the same set of options 01:15:55.480 --> 01:15:57.940 align:middle line:90% for the next entry every time. 01:15:57.940 --> 01:16:01.300 align:middle line:84% But the set of options we have in the second round 01:16:01.300 --> 01:16:05.180 align:middle line:84% depends on which card we picked in the first round. 01:16:05.180 --> 01:16:08.140 align:middle line:84% There are always 51 options because we've always excluded 01:16:08.140 --> 01:16:10.070 align:middle line:90% one card from the first round. 01:16:10.070 --> 01:16:13.510 align:middle line:84% But the set of options changes depending on what 01:16:13.510 --> 01:16:15.940 align:middle line:90% we picked in the first round. 01:16:15.940 --> 01:16:17.930 align:middle line:84% So this isn't straight-up product rule. 01:16:17.930 --> 01:16:19.960 align:middle line:84% The set of permutations is not just 01:16:19.960 --> 01:16:24.560 align:middle line:84% a set of 52 cards times a set of 51 cards times set of 50 cards, 01:16:24.560 --> 01:16:27.790 align:middle line:84% even though the number of options at each stage 01:16:27.790 --> 01:16:29.030 align:middle line:90% is always the same. 01:16:29.030 --> 01:16:31.520 align:middle line:84% And that's what the generalized product rule does for us. 01:16:31.520 --> 01:16:34.930 align:middle line:84% It basically says, if the number of options 01:16:34.930 --> 01:16:39.640 align:middle line:84% at every step of the process is always consistent, 01:16:39.640 --> 01:16:43.340 align:middle line:84% even if this particular set of those options might change, 01:16:43.340 --> 01:16:46.000 align:middle line:84% then the number of ways to make your choices 01:16:46.000 --> 01:16:49.390 align:middle line:84% is just the product of the number of choices at each stage. 01:16:49.390 --> 01:16:54.230 align:middle line:90% 01:16:54.230 --> 01:17:00.580 align:middle line:84% So this is the generalized product rule. 01:17:00.580 --> 01:17:03.830 align:middle line:90% 01:17:03.830 --> 01:17:17.690 align:middle line:84% If A is a set of length k sequences 01:17:17.690 --> 01:17:29.030 align:middle line:84% where there are n1 choices for the first entry, for a1, 01:17:29.030 --> 01:17:45.800 align:middle line:84% and two choices for a2, no matter what a1 was chosen, 01:17:45.800 --> 01:17:50.180 align:middle line:90% et cetera, exactly-- 01:17:50.180 --> 01:17:56.360 align:middle line:84% sorry, exactly ni choices for ai, 01:17:56.360 --> 01:18:01.850 align:middle line:84% for the i-th entry in our sequence, 01:18:01.850 --> 01:18:11.120 align:middle line:84% no matter what a1 through ai minus 1 were chosen, 01:18:11.120 --> 01:18:15.350 align:middle line:84% and all the way down to nk at the end-- 01:18:15.350 --> 01:18:18.830 align:middle line:84% so if earlier choices don't change the number of options 01:18:18.830 --> 01:18:20.303 align:middle line:90% we have in later steps-- 01:18:20.303 --> 01:18:23.570 align:middle line:90% 01:18:23.570 --> 01:18:31.430 align:middle line:84% then the size of A is n1 times n2 up to nk. 01:18:31.430 --> 01:18:33.290 align:middle line:90% Sound good? 01:18:33.290 --> 01:18:34.100 align:middle line:90% Cool. 01:18:34.100 --> 01:18:37.550 align:middle line:84% We will get more practice with that next time. 01:18:37.550 --> 01:18:38.700 align:middle line:90% And I will see you then. 01:18:38.700 --> 01:18:40.450 align:middle line:90% Thank you. 01:18:40.450 --> 01:18:46.000 align:middle line:90%