1 00:00:00,560 --> 00:00:02,990 The following content is provided under a Creative 2 00:00:02,990 --> 00:00:04,510 Commons license. 3 00:00:04,510 --> 00:00:06,850 Your support will help MIT OpenCourseWare 4 00:00:06,850 --> 00:00:11,220 continue to offer high quality educational resources for free. 5 00:00:11,220 --> 00:00:13,820 To make a donation, or view additional materials 6 00:00:13,820 --> 00:00:17,726 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,726 --> 00:00:18,351 at ocw.mit.edu. 8 00:00:23,730 --> 00:00:26,310 PROFESSOR: Last week we talked about the key components 9 00:00:26,310 --> 00:00:31,380 of a proof, propositions, axioms, and logical deductions, 10 00:00:31,380 --> 00:00:34,580 and as you probably talked about during recitation, 11 00:00:34,580 --> 00:00:36,180 we're not going to worry too much 12 00:00:36,180 --> 00:00:40,360 about what axioms or logical deductions that you use. 13 00:00:40,360 --> 00:00:43,490 Anything that is reasonable, is fine by us. 14 00:00:43,490 --> 00:00:46,240 We're not going to ask you to know what modus ponens is, 15 00:00:46,240 --> 00:00:49,420 or to label some law when you make a logical deduction. 16 00:00:49,420 --> 00:00:51,360 Just, you know, be reasonable. 17 00:00:51,360 --> 00:00:54,860 Any facts you knew coming into this course about mathematics, 18 00:00:54,860 --> 00:00:57,470 probably close enough to use as an axiom. 19 00:00:57,470 --> 00:01:01,440 Want to make sure your axioms are consistent, but that's OK. 20 00:01:01,440 --> 00:01:05,160 Now the exception to this would be, is say we're on an exam, 21 00:01:05,160 --> 00:01:09,080 and we ask you to prove some proposition, p. 22 00:01:09,080 --> 00:01:13,730 Well you can't say, I already knew p, it's an axiom, check. 23 00:01:13,730 --> 00:01:14,550 That's not so good. 24 00:01:14,550 --> 00:01:16,710 We're asking you to prove it from some more 25 00:01:16,710 --> 00:01:20,060 elementary facts. 26 00:01:20,060 --> 00:01:22,980 OK it's also don't want me making wild leaps of faith, 27 00:01:22,980 --> 00:01:26,130 or saying it's obvious that, unless it really is obvious. 28 00:01:26,130 --> 00:01:27,920 That kind of stuff can get you in trouble. 29 00:01:27,920 --> 00:01:32,470 Much better to sort of explain the reasoning in the proof. 30 00:01:32,470 --> 00:01:35,480 Now the proofs that we covered last week in recitation, 31 00:01:35,480 --> 00:01:37,840 the problems set, were all examples 32 00:01:37,840 --> 00:01:40,140 of what are called direct proofs. 33 00:01:40,140 --> 00:01:42,820 You start with some axioms, you have some theorems 34 00:01:42,820 --> 00:01:45,100 you knew before or you proved along the way, 35 00:01:45,100 --> 00:01:46,960 and you make logical deductions until you 36 00:01:46,960 --> 00:01:49,960 get to where you want to go, the theorem. 37 00:01:49,960 --> 00:01:53,980 We're going to start today with an indirect proof. 38 00:01:53,980 --> 00:01:56,360 For example, a proof by contradiction. 39 00:01:56,360 --> 00:01:58,320 And this is a little bit different. 40 00:01:58,320 --> 00:02:02,740 In a proof by contradiction, you assume the opposite 41 00:02:02,740 --> 00:02:05,350 of what you're trying to prove. 42 00:02:05,350 --> 00:02:07,570 Then you just take steps for logical deductions 43 00:02:07,570 --> 00:02:10,610 forward until you arrive at a contradiction, 44 00:02:10,610 --> 00:02:13,920 something where you prove false equals true. 45 00:02:13,920 --> 00:02:15,420 Now if you can ever get to the point 46 00:02:15,420 --> 00:02:18,830 we approve something is false and true, 47 00:02:18,830 --> 00:02:21,750 that means what you assumed at the start had to be wrong. 48 00:02:21,750 --> 00:02:24,680 Namely, what you're trying to prove has to be true. 49 00:02:24,680 --> 00:02:26,180 So let's write that down, because it 50 00:02:26,180 --> 00:02:28,800 can be a little confusing the first time. 51 00:02:40,200 --> 00:02:49,000 So to prove a proposition p is true, 52 00:02:49,000 --> 00:02:54,060 you will assume that it's false. 53 00:02:59,430 --> 00:03:01,380 In other words, that not p is true. 54 00:03:10,210 --> 00:03:19,630 And then you use that hypothesis, namely the p 55 00:03:19,630 --> 00:03:27,040 is false, to derive a falsehood. 56 00:03:27,040 --> 00:03:29,590 In other words, you prove a falsehood is true. 57 00:03:29,590 --> 00:03:32,290 And this is called deriving a contradiction. 58 00:03:39,350 --> 00:03:41,720 And so it must be that, in fact, p is not false, 59 00:03:41,720 --> 00:03:43,980 namely that it's true. 60 00:03:43,980 --> 00:03:48,410 Now this works because if you can prove, 61 00:03:48,410 --> 00:03:59,660 if not p implies false is true, well from last time, 62 00:03:59,660 --> 00:04:05,530 the only way this is a true statement is if this is false, 63 00:04:05,530 --> 00:04:08,740 namely p is true. 64 00:04:08,740 --> 00:04:09,240 All right? 65 00:04:09,240 --> 00:04:11,520 So we can conclude that p is true, 66 00:04:11,520 --> 00:04:16,850 if we can show that not P implies a falsehood. 67 00:04:16,850 --> 00:04:18,779 Any questions about that? 68 00:04:18,779 --> 00:04:21,860 It's sort of a lot of sort of notation, 69 00:04:21,860 --> 00:04:25,300 and until you've seen, it can be confusing. 70 00:04:25,300 --> 00:04:28,970 So maybe we should do an example. 71 00:04:28,970 --> 00:04:33,005 Let's prove that square root of 2 is irrational. 72 00:04:39,967 --> 00:04:40,550 Is irrational. 73 00:04:45,540 --> 00:04:49,290 OK, everybody knows what an irrational number is? 74 00:04:49,290 --> 00:04:51,090 That's something that can't be expressed 75 00:04:51,090 --> 00:04:53,321 as the ratio of integers. 76 00:04:53,321 --> 00:04:54,820 OK, and probably most people already 77 00:04:54,820 --> 00:04:57,300 know that-- how many people have not 78 00:04:57,300 --> 00:05:01,780 seen a proof that square root of 2 is a rational before? 79 00:05:01,780 --> 00:05:03,820 So most of you have seen a proof of that, good. 80 00:05:03,820 --> 00:05:05,945 You know, if you try to do a direct proof for this, 81 00:05:05,945 --> 00:05:06,840 it's pretty hard. 82 00:05:06,840 --> 00:05:08,900 How do you show there's no way to represent 83 00:05:08,900 --> 00:05:12,950 the square root of 2 is as integers a over b? 84 00:05:12,950 --> 00:05:17,040 But it's very easy, if we do a proof by contradiction. 85 00:05:17,040 --> 00:05:19,010 Now when you're doing a proof by contradiction, 86 00:05:19,010 --> 00:05:23,540 always start off by saying, by contradiction. 87 00:05:23,540 --> 00:05:25,910 Write that down. 88 00:05:25,910 --> 00:05:28,140 And then what you do next is you say, 89 00:05:28,140 --> 00:05:41,690 I'm going to assume, for the purpose of contradiction, 90 00:05:41,690 --> 00:05:47,350 that p is false and when not p is true. 91 00:05:47,350 --> 00:05:50,365 In this case, that would be square root of 2 is rational. 92 00:05:56,170 --> 00:05:59,520 So in this case, here's what we're trying to prove. 93 00:05:59,520 --> 00:06:01,730 That's p. 94 00:06:01,730 --> 00:06:04,740 I'm going to assume not p, namely that square root of 2 95 00:06:04,740 --> 00:06:06,570 is irrational number. 96 00:06:06,570 --> 00:06:08,830 Then I'm going to get a contradiction or falsehood, 97 00:06:08,830 --> 00:06:14,540 and then I'm going to know that p was true after all. 98 00:06:14,540 --> 00:06:18,250 All right, so let's see where this leads us. 99 00:06:18,250 --> 00:06:19,980 Well, if square root of 2 is rational 100 00:06:19,980 --> 00:06:23,850 then we can express it as a over b, where a over b 101 00:06:23,850 --> 00:06:25,325 is a fraction in lowest terms. 102 00:06:30,520 --> 00:06:32,415 That means a and b have no common divisors. 103 00:06:36,140 --> 00:06:39,720 And then I can square both sides, 104 00:06:39,720 --> 00:06:41,850 and I get two is a squared over b squared. 105 00:06:45,370 --> 00:06:48,100 Then I multiply by b squared, and I get 106 00:06:48,100 --> 00:06:51,900 2b squared equals a squared. 107 00:06:51,900 --> 00:06:55,340 And what does that imply about a? 108 00:06:55,340 --> 00:06:57,710 What can you tell me about a if it equals-- 109 00:06:57,710 --> 00:06:59,080 if a squared is 2b squared? 110 00:07:04,840 --> 00:07:05,910 Anything special about a? 111 00:07:05,910 --> 00:07:07,460 Could a be anything? 112 00:07:09,990 --> 00:07:12,180 A squared is even, all right? 113 00:07:12,180 --> 00:07:15,360 Because 2b squared is an even number, so a squared is even, 114 00:07:15,360 --> 00:07:18,605 and what does that mean about a? 115 00:07:18,605 --> 00:07:20,140 A is even. 116 00:07:20,140 --> 00:07:23,570 B squared is even, then a is even. 117 00:07:23,570 --> 00:07:26,860 So a is even. 118 00:07:26,860 --> 00:07:30,080 So we could write that as two divides a. 119 00:07:30,080 --> 00:07:31,250 That's the divide symbol. 120 00:07:31,250 --> 00:07:34,700 You'll see a lot of that next week. 121 00:07:34,700 --> 00:07:37,940 All right if a is even, what do I know about a squared? 122 00:07:37,940 --> 00:07:39,340 I know more than just, it's even. 123 00:07:44,010 --> 00:07:44,510 What is it? 124 00:07:44,510 --> 00:07:47,160 It's multiple of 4, yeah. 125 00:07:47,160 --> 00:07:53,030 So that means that four divides a squared. 126 00:07:53,030 --> 00:07:55,150 A squared is 2b squared, so that means 127 00:07:55,150 --> 00:07:57,970 that four divides 2b squared. 128 00:07:57,970 --> 00:08:02,780 Divide each side by 2 means 2 divides b squared. 129 00:08:02,780 --> 00:08:04,418 What does that imply about b? 130 00:08:07,764 --> 00:08:10,870 B is even. 131 00:08:10,870 --> 00:08:13,750 All right, b is even, good. 132 00:08:13,750 --> 00:08:19,620 Well, I've got a is even and b is even. 133 00:08:19,620 --> 00:08:21,225 I got a contradiction here, somewhere? 134 00:08:24,080 --> 00:08:28,260 Yeah a over b was not a fraction in lowest terms, 135 00:08:28,260 --> 00:08:31,410 because both and b are even. 136 00:08:31,410 --> 00:08:41,565 All right, so that implies a over b is not in lowest terms. 137 00:08:46,060 --> 00:08:50,500 And that is a contradiction. 138 00:08:50,500 --> 00:08:52,660 Now you'll see that written lots of ways. 139 00:08:52,660 --> 00:08:54,410 One is, you can say a contradiction, 140 00:08:54,410 --> 00:08:57,586 sometimes you'll see just this sharp symbol written, 141 00:08:57,586 --> 00:08:59,460 and that means you've got to a contradiction. 142 00:08:59,460 --> 00:09:03,510 Because here, we had it being in lowest terms, 143 00:09:03,510 --> 00:09:05,260 and here we have it not in lowest terms. 144 00:09:05,260 --> 00:09:06,801 You can't have both at the same time, 145 00:09:06,801 --> 00:09:08,910 so you got a contradiction. 146 00:09:08,910 --> 00:09:14,080 And that means we've now proved that this assumption was wrong. 147 00:09:14,080 --> 00:09:16,750 Square root of 2 is not rational, 148 00:09:16,750 --> 00:09:17,840 so it must be irrational. 149 00:09:24,420 --> 00:09:26,540 and then we put a little box here at the end, 150 00:09:26,540 --> 00:09:28,620 or sometimes you'll see a check, sometimes you'll 151 00:09:28,620 --> 00:09:32,910 see QED, that says the proof's done now. 152 00:09:32,910 --> 00:09:34,640 It's over. 153 00:09:34,640 --> 00:09:38,790 Any questions about that proof? 154 00:09:38,790 --> 00:09:41,740 We're going to do a lot of proofs by contradiction. 155 00:09:41,740 --> 00:09:45,630 Actually, there's an interesting story behind this proof. 156 00:09:45,630 --> 00:09:49,600 As far as we know, it was first discovered by the Pythagoreans, 157 00:09:49,600 --> 00:09:52,270 way back when in ancient Greece. 158 00:09:52,270 --> 00:09:55,330 And the Pythagoreans were a religious society 159 00:09:55,330 --> 00:09:59,350 started by Pythagoras, of Pythagoras theorem frame. 160 00:09:59,350 --> 00:10:03,850 Now it sounds weird today, but back then, in ancient Greece, 161 00:10:03,850 --> 00:10:06,845 math was a religion, all right? 162 00:10:06,845 --> 00:10:08,970 Every once in while you'll see somebody around MIT, 163 00:10:08,970 --> 00:10:11,920 and you'll think he must think math is a religion, 164 00:10:11,920 --> 00:10:14,860 but back then it really was, and it was ruled by God, 165 00:10:14,860 --> 00:10:16,620 because this is ancient Greece. 166 00:10:16,620 --> 00:10:20,430 And there were two key gods in this religion, Apeiron 167 00:10:20,430 --> 00:10:21,790 and Peros. 168 00:10:21,790 --> 00:10:26,730 Now Apeiron was the bad god, and he was the god of infinity, 169 00:10:26,730 --> 00:10:29,645 because infinity was considered all that was bad. 170 00:10:29,645 --> 00:10:32,020 And I don't think we'll do a lot with infinity this term, 171 00:10:32,020 --> 00:10:35,270 but if we do you'll appreciate why that's the case. 172 00:10:35,270 --> 00:10:37,540 And Peros was a good god. 173 00:10:37,540 --> 00:10:39,580 He was the god of the finite world, 174 00:10:39,580 --> 00:10:43,570 and represented everything that was good to the ancient Greeks. 175 00:10:43,570 --> 00:10:46,930 Now one of their main axioms, or beliefs, 176 00:10:46,930 --> 00:10:50,290 was that there were no irrational numbers. 177 00:10:50,290 --> 00:10:53,070 They just didn't exist. 178 00:10:53,070 --> 00:10:55,570 Now the reason is, they didn't like irrational numbers. 179 00:10:55,570 --> 00:10:56,690 They were bad. 180 00:10:56,690 --> 00:10:58,420 Because, well they're infinite. 181 00:10:58,420 --> 00:11:01,390 You can't write them as a decimal 182 00:11:01,390 --> 00:11:05,130 without repeating forever, and you know, just an infinite sort 183 00:11:05,130 --> 00:11:07,160 of decimal representation. 184 00:11:07,160 --> 00:11:11,530 They liked rational numbers, you know, like one seventh? 185 00:11:11,530 --> 00:11:13,540 That's a good number, because you can write it 186 00:11:13,540 --> 00:11:19,330 as 0.142857 repeating. 187 00:11:19,330 --> 00:11:21,370 So rational numbers are finite, in that 188 00:11:21,370 --> 00:11:24,100 you can always find a repeating pattern of finite length. 189 00:11:24,100 --> 00:11:25,540 Irrational numbers are not. 190 00:11:25,540 --> 00:11:28,924 They're infinite in that sense of the ancient Greeks. 191 00:11:28,924 --> 00:11:30,840 So they said there were no irrational numbers. 192 00:11:30,840 --> 00:11:33,380 That was an early axiom. 193 00:11:33,380 --> 00:11:36,810 Now they also had an axiom that said 194 00:11:36,810 --> 00:11:43,717 that every length of a line was finite, therefore rational. 195 00:11:43,717 --> 00:11:45,300 All right, you know the ancient Greeks 196 00:11:45,300 --> 00:11:48,130 were good with a compass, and drawing lines 197 00:11:48,130 --> 00:11:49,570 with straight edges and stuff. 198 00:11:49,570 --> 00:11:52,090 So they said any line that you can construct 199 00:11:52,090 --> 00:11:54,270 has a finite length to it, so therefore it 200 00:11:54,270 --> 00:11:55,390 has a rational length. 201 00:11:55,390 --> 00:11:58,740 That was axiom number two. 202 00:11:58,740 --> 00:12:01,910 Now they were good geometers, so they knew, of course, 203 00:12:01,910 --> 00:12:04,070 they developed the Pythagorean theorem. 204 00:12:04,070 --> 00:12:07,520 But if you took a triangle, and you have side lengths one, 205 00:12:07,520 --> 00:12:11,177 the length of the hypotenuse was square root 2. 206 00:12:11,177 --> 00:12:12,760 Therefore they had a theorem that said 207 00:12:12,760 --> 00:12:14,946 square root 2 was rational. 208 00:12:14,946 --> 00:12:18,430 Because you've got the 2 axioms there, right? 209 00:12:18,430 --> 00:12:23,480 Now eventually, they discovered a proof like that, 210 00:12:23,480 --> 00:12:24,740 that it wasn't rational. 211 00:12:24,740 --> 00:12:26,380 It was irrational. 212 00:12:26,380 --> 00:12:28,010 This caused quite a stir. 213 00:12:28,010 --> 00:12:31,180 First it meant that their axioms were inconsistent. 214 00:12:31,180 --> 00:12:34,590 Every theorem they proved was now suspect, 215 00:12:34,590 --> 00:12:36,600 once you have inconsistent axioms. 216 00:12:36,600 --> 00:12:41,800 Even worse, the devil is in their midst. 217 00:12:41,800 --> 00:12:44,300 Square root 2 is infinite is the bad god, 218 00:12:44,300 --> 00:12:48,690 and this is the most basic length they had, besides one. 219 00:12:48,690 --> 00:12:49,849 So that's a very bad thing. 220 00:12:49,849 --> 00:12:51,390 Sort of like, you know, today we were 221 00:12:51,390 --> 00:12:54,310 to wake up and discover that there were only nine 222 00:12:54,310 --> 00:12:57,950 commandments, and the 10th was planted there by the devil. 223 00:12:57,950 --> 00:12:59,710 And you're not sure which one, maybe. 224 00:12:59,710 --> 00:13:03,270 That would be a big mess, sacrilege. 225 00:13:03,270 --> 00:13:04,520 Well, so what were they to do? 226 00:13:04,520 --> 00:13:07,370 This is a disaster of major proportions. 227 00:13:07,370 --> 00:13:11,490 So they covered it up, and they denied the result. 228 00:13:11,490 --> 00:13:13,570 So they didn't want to publish that proof. 229 00:13:13,570 --> 00:13:16,760 So they kept on saying square root 2 was rational. 230 00:13:16,760 --> 00:13:20,130 But then, according to legend there was a Deep Throat. 231 00:13:20,130 --> 00:13:23,339 Somebody who let out the word and the proof the square root 2 232 00:13:23,339 --> 00:13:25,630 is irrational, because that would be very destabilizing 233 00:13:25,630 --> 00:13:29,810 for the society, and so they killed them. 234 00:13:29,810 --> 00:13:31,042 This is the legend. 235 00:13:31,042 --> 00:13:32,500 Now, hard to imagine getting killed 236 00:13:32,500 --> 00:13:35,505 over the irrationality of 2. 237 00:13:35,505 --> 00:13:37,130 All right, we're going to do a lot more 238 00:13:37,130 --> 00:13:39,080 of these kind of proofs in homework 239 00:13:39,080 --> 00:13:40,800 and throughout the term. 240 00:13:40,800 --> 00:13:42,750 The next proof I want to show you 241 00:13:42,750 --> 00:13:45,410 is one of my favorite proof techniques. 242 00:13:45,410 --> 00:13:46,950 One of my favorite proofs. 243 00:13:46,950 --> 00:13:48,912 And that is a false proof. 244 00:13:48,912 --> 00:13:51,370 And we're going to see a lot of these during the term, too. 245 00:13:51,370 --> 00:13:54,030 And if we could bring the screens down. 246 00:13:54,030 --> 00:13:55,420 Somebody back there to-- yeah. 247 00:13:55,420 --> 00:13:55,730 Great. 248 00:13:55,730 --> 00:13:57,313 Bring the screens down, I'm going to-- 249 00:13:57,313 --> 00:13:58,750 and then turn this on for me. 250 00:13:58,750 --> 00:14:05,160 So I'm going to prove to you that 90 is bigger than 92. 251 00:14:05,160 --> 00:14:05,660 All right? 252 00:14:05,660 --> 00:14:09,360 And that hopefully, is not really true. 253 00:14:09,360 --> 00:14:13,470 But, you know, watch this proof and see if there's 254 00:14:13,470 --> 00:14:14,450 any problems with it. 255 00:14:17,480 --> 00:14:19,016 All right, the proof, by PowerPoint. 256 00:14:19,016 --> 00:14:20,515 Right away you should be suspicious. 257 00:14:23,300 --> 00:14:27,004 I'm going to take two triangles with total area 90 258 00:14:27,004 --> 00:14:28,920 and put them together, and then divide them up 259 00:14:28,920 --> 00:14:34,110 into four triangles with area greater than 92. 260 00:14:34,110 --> 00:14:37,020 And by the conservation of area axiom, which 261 00:14:37,020 --> 00:14:39,230 you want to be maybe thinking about, 262 00:14:39,230 --> 00:14:43,110 this will imply an area of 90 is larger than an area of 92. 263 00:14:43,110 --> 00:14:47,460 And therefore that 90 is bigger than 92. 264 00:14:47,460 --> 00:14:48,900 All right, those are my triangles. 265 00:14:48,900 --> 00:14:53,150 They are right triangles, 9 by 10. 266 00:14:53,150 --> 00:14:58,510 If I put them together, I get a 9 by 10 rectangle. 267 00:14:58,510 --> 00:15:03,140 Of course it has area 90, 9 times 10. 268 00:15:03,140 --> 00:15:10,170 Now I'm going to slide these triangles across their diagonal 269 00:15:10,170 --> 00:15:12,620 so that I get, instead of having 10 on the side, 270 00:15:12,620 --> 00:15:15,030 I'm going to slide it so I get a 2 and an 8. 271 00:15:15,030 --> 00:15:17,560 OK I had 10 there before, now it's 2 and 8. 272 00:15:17,560 --> 00:15:19,060 And then you see I've got this-- I'm 273 00:15:19,060 --> 00:15:20,940 going to cut off along the dotted line, 274 00:15:20,940 --> 00:15:22,977 and as you can see that dotted line, yeah 275 00:15:22,977 --> 00:15:24,560 but won't compute it exactly, but it's 276 00:15:24,560 --> 00:15:26,371 a little bit bigger than 2. 277 00:15:26,371 --> 00:15:26,870 All right? 278 00:15:26,870 --> 00:15:28,828 It's a little bit longer than that 2 there, you 279 00:15:28,828 --> 00:15:31,090 could see that. 280 00:15:31,090 --> 00:15:33,390 All right, so I'll call it 2 plus, bigger than 2. 281 00:15:33,390 --> 00:15:36,970 Now I slice off along the dotted line, 282 00:15:36,970 --> 00:15:40,040 and now I'm going to put those two triangles together, 283 00:15:40,040 --> 00:15:42,560 and I create a rectangle. 284 00:15:42,560 --> 00:15:44,595 Two by two-- little more than two. 285 00:15:44,595 --> 00:15:46,095 Now the area of the little rectangle 286 00:15:46,095 --> 00:15:49,850 is bigger than 4, cause I got 2 times something bigger than 2, 287 00:15:49,850 --> 00:15:52,740 and you can see here I have the 8 left over, 288 00:15:52,740 --> 00:15:56,020 and I have 9 plus 2 plus, means it's a little bit bigger 289 00:15:56,020 --> 00:15:58,120 than 11 by 8. 290 00:15:58,120 --> 00:16:03,250 I got area bigger than 88, add them up, get area bigger 291 00:16:03,250 --> 00:16:04,820 than 92. 292 00:16:04,820 --> 00:16:07,655 So I started with 90 and I created more than 92. 293 00:16:10,220 --> 00:16:13,441 All right, what's the problem here? 294 00:16:13,441 --> 00:16:14,940 This would be good if I could do it. 295 00:16:14,940 --> 00:16:17,140 I'd get some gold, you know, bars of gold 296 00:16:17,140 --> 00:16:20,540 and cut them up, and do the game, and I make more gold. 297 00:16:20,540 --> 00:16:25,340 That would be pretty good if I could do that. 298 00:16:25,340 --> 00:16:28,300 Because of the conservation of area axiom? 299 00:16:28,300 --> 00:16:30,570 Did I assume something there that was too powerful 300 00:16:30,570 --> 00:16:34,382 that if I manipulate the area of rectangles like this, 301 00:16:34,382 --> 00:16:35,840 that the area needs to be the same? 302 00:16:39,050 --> 00:16:41,630 Yeah? 303 00:16:41,630 --> 00:16:44,170 My bigger ones aren't closed. 304 00:16:44,170 --> 00:16:46,320 Well, let's see, I don't know. 305 00:16:46,320 --> 00:16:50,230 Let's go back and see I made those bigger ones. 306 00:16:50,230 --> 00:16:52,700 All right, I got my triangles, right? 307 00:16:52,700 --> 00:16:56,287 I'm just chopping along the line there, and I got 2 plus and 9. 308 00:16:59,210 --> 00:17:01,457 Those are rectangles. 309 00:17:01,457 --> 00:17:02,540 They look like rectangles. 310 00:17:05,670 --> 00:17:06,541 Yeah. 311 00:17:06,541 --> 00:17:09,319 AUDIENCE: [INAUDIBLE] 312 00:17:09,319 --> 00:17:15,050 PROFESSOR: Well it looks bigger than 2, doesn't it? 313 00:17:15,050 --> 00:17:16,670 Yeah, 2 pluses. 314 00:17:16,670 --> 00:17:19,470 Well bigger than 2 is bigger than 2, but, yeah, maybe that 315 00:17:19,470 --> 00:17:21,329 should have been a 2 minus. 316 00:17:21,329 --> 00:17:24,190 Would that change anything? 317 00:17:24,190 --> 00:17:27,030 Yeah, if that had been a 2 minus, 318 00:17:27,030 --> 00:17:32,860 then we'd have area less than 4, area less than 88, and then 90 319 00:17:32,860 --> 00:17:34,420 would be less than 92. 320 00:17:34,420 --> 00:17:35,780 That would be OK. 321 00:17:35,780 --> 00:17:38,882 But it sure as heck looks bigger than-- the 2 plus looks 322 00:17:38,882 --> 00:17:40,090 bigger than the 2 doesn't it? 323 00:17:40,090 --> 00:17:43,240 I don't know, do they got distortion out there? 324 00:17:43,240 --> 00:17:45,590 I guarantee you, if I measured on my screen, 325 00:17:45,590 --> 00:17:48,864 that 2 plus is bigger than the length of the tube. 326 00:17:48,864 --> 00:17:49,530 I guarantee you. 327 00:17:49,530 --> 00:17:52,930 We can to take a ruler on my screen, 328 00:17:52,930 --> 00:17:56,510 and the 2 plus will be longer on the ruler than the 2. 329 00:17:56,510 --> 00:18:01,090 That I guarantee you, but you're on the right track. 330 00:18:01,090 --> 00:18:04,700 Yeah 331 00:18:04,700 --> 00:18:08,200 Good, all right, now how did you-- 332 00:18:08,200 --> 00:18:10,871 but it is true, if I measured on my screen, it's bigger. 333 00:18:10,871 --> 00:18:11,370 Yeah. 334 00:18:13,887 --> 00:18:15,970 AUDIENCE: Are the triangles, like, drawn to scale? 335 00:18:15,970 --> 00:18:18,502 It's like, on the SAT they always say, not to scale, 336 00:18:18,502 --> 00:18:19,350 right? 337 00:18:19,350 --> 00:18:20,720 PROFESSOR: Yeah. 338 00:18:20,720 --> 00:18:22,780 Yeah, they're not drawn to scale. 339 00:18:22,780 --> 00:18:29,850 In fact, if I go back to the beginning, it says 9 by 10. 340 00:18:29,850 --> 00:18:34,080 But in fact, if I measure the nine, it's bigger than the 10. 341 00:18:34,080 --> 00:18:35,690 All right? 342 00:18:35,690 --> 00:18:40,706 So this is one of the big problems of proofs by picture. 343 00:18:40,706 --> 00:18:42,580 Because look at it, you're not thinking which 344 00:18:42,580 --> 00:18:44,030 is bigger, 9 or 10 up there? 345 00:18:44,030 --> 00:18:46,199 10's bigger even though it's not, it's shorter. 346 00:18:46,199 --> 00:18:47,740 But when you get down into the proof, 347 00:18:47,740 --> 00:18:49,860 well, clearly the 2 plus was bigger than the 2, 348 00:18:49,860 --> 00:18:50,943 because it looks that way. 349 00:18:50,943 --> 00:18:52,480 It is, on the paper. 350 00:18:52,480 --> 00:18:52,980 All right? 351 00:18:52,980 --> 00:18:55,271 In fact if you go on a computer, you're probably right, 352 00:18:55,271 --> 00:18:57,410 it's 1.8, not 2 plus. 353 00:18:57,410 --> 00:19:00,940 But that's how the error crept in and how it was drawn, 354 00:19:00,940 --> 00:19:02,665 and then you're going along with a proof, 355 00:19:02,665 --> 00:19:04,040 and everything else is just fine. 356 00:19:04,040 --> 00:19:05,840 So the mistake is right up front. 357 00:19:05,840 --> 00:19:07,680 I drew it wrong. 358 00:19:07,680 --> 00:19:09,320 OK. 359 00:19:09,320 --> 00:19:10,960 And this happens, you're doing graphs-- 360 00:19:10,960 --> 00:19:13,168 we'll talk about graphs here in a couple weeks-- over 361 00:19:13,168 --> 00:19:14,519 and over again, people do this. 362 00:19:14,519 --> 00:19:16,560 They draw it, it looks like this, you accept that 363 00:19:16,560 --> 00:19:17,620 and then you're dead. 364 00:19:17,620 --> 00:19:20,190 Everything else, there's no hope. 365 00:19:20,190 --> 00:19:23,080 Everybody clear what went wrong in this picture? 366 00:19:23,080 --> 00:19:25,630 How we got off track? 367 00:19:25,630 --> 00:19:29,100 This is sort of a nasty one. 368 00:19:29,100 --> 00:19:29,600 OK. 369 00:19:32,120 --> 00:19:34,586 Now one of the nice things about proofs 370 00:19:34,586 --> 00:19:36,710 is that when there is a bug, if you really write it 371 00:19:36,710 --> 00:19:38,775 out step by step, and there's a bug, 372 00:19:38,775 --> 00:19:40,620 you can go back and find it. 373 00:19:40,620 --> 00:19:42,240 And so this, I didn't sort of really 374 00:19:42,240 --> 00:19:43,770 write it out very well step by step, 375 00:19:43,770 --> 00:19:46,740 and it was harder to find. 376 00:19:46,740 --> 00:19:49,190 Now, all right, so we can pull the screens up. 377 00:19:52,543 --> 00:19:54,460 Thanks. 378 00:19:54,460 --> 00:19:57,915 OK, so for the rest of today and the rest of this week, 379 00:19:57,915 --> 00:20:00,040 we're going to talk about a different kind of proof 380 00:20:00,040 --> 00:20:03,600 technique, which is induction. 381 00:20:03,600 --> 00:20:07,510 Now, induction is by far the most powerful and commonly used 382 00:20:07,510 --> 00:20:10,350 proof technique in computer science. 383 00:20:10,350 --> 00:20:11,880 If there's one thing you should know 384 00:20:11,880 --> 00:20:14,640 by the time you're done with this class, 385 00:20:14,640 --> 00:20:16,415 it's how to do a proof by induction. 386 00:20:16,415 --> 00:20:17,790 In fact, if there's one thing you 387 00:20:17,790 --> 00:20:20,280 will know by the time we're done with this class, 388 00:20:20,280 --> 00:20:22,020 is how to do a proof by induction. 389 00:20:22,020 --> 00:20:25,240 And in some sense, when we get to grading the final, how 390 00:20:25,240 --> 00:20:27,020 we measure ourselves as instructors, 391 00:20:27,020 --> 00:20:29,770 the first test is, can you do the proof 392 00:20:29,770 --> 00:20:32,250 by induction question, and do a good proof there? 393 00:20:32,250 --> 00:20:35,090 You will see it on the midterm and the final, 394 00:20:35,090 --> 00:20:38,410 probably in multiple instances. 395 00:20:38,410 --> 00:20:40,960 Now just to make sure you become intimately familiar 396 00:20:40,960 --> 00:20:44,480 with induction, we're going to do over a dozen proofs 397 00:20:44,480 --> 00:20:46,720 with induction in class, and many more in homework 398 00:20:46,720 --> 00:20:49,610 over the next five to six weeks. 399 00:20:49,610 --> 00:20:51,670 You'll probably be dreaming about induction, 400 00:20:51,670 --> 00:20:53,940 if we're successful here. 401 00:20:53,940 --> 00:20:55,130 Soon. 402 00:20:55,130 --> 00:20:57,984 The good news is that induction is very easy. 403 00:20:57,984 --> 00:21:00,150 Once you get your mind around it, get some practice, 404 00:21:00,150 --> 00:21:02,910 it really is not a hard thing. 405 00:21:02,910 --> 00:21:05,600 In fact, induction is really just an axiom. 406 00:21:08,290 --> 00:21:09,350 So let me state it. 407 00:21:16,950 --> 00:21:19,720 Let P(n) be a predicate. 408 00:21:26,740 --> 00:21:44,940 If P(0) is true, and for all natural numbers n, p of n 409 00:21:44,940 --> 00:21:51,140 implies p of n plus 1 is true. 410 00:21:51,140 --> 00:21:53,850 So here I'm saying that this is true for all n. 411 00:21:59,525 --> 00:22:00,025 OK. 412 00:22:02,620 --> 00:22:11,860 Then for all n, natural numbers, P of n is true. 413 00:22:16,170 --> 00:22:18,220 I had another way of saying this without the 414 00:22:18,220 --> 00:22:25,090 for alls there, is that if P(0) is true, 415 00:22:25,090 --> 00:22:34,030 if P(0) implies P(1) is true, if P(1) implies P(2) is true, 416 00:22:34,030 --> 00:22:40,240 and so on, forever, then p of n is true for all that. 417 00:22:40,240 --> 00:22:48,990 So then, P(0), P(1), P(2), forever, are true. 418 00:22:57,760 --> 00:22:59,900 OK. 419 00:22:59,900 --> 00:23:04,600 Now you can sort of see why this is a reasonable axiom, 420 00:23:04,600 --> 00:23:10,440 because if P(0) is true, and P(0) implies P(1), then 421 00:23:10,440 --> 00:23:13,640 by that modus ponens thing or one of the logical deductions, 422 00:23:13,640 --> 00:23:16,000 we know P(1) is true. 423 00:23:16,000 --> 00:23:20,120 And if P(1) is true, and P(1) implies P(2) is true, 424 00:23:20,120 --> 00:23:21,770 then we know by the same reasoning, 425 00:23:21,770 --> 00:23:25,770 P(2) is true, and so on, forever. 426 00:23:25,770 --> 00:23:28,870 Now the reason it becomes an axiom is that and so on forever 427 00:23:28,870 --> 00:23:30,200 bit is part of it. 428 00:23:30,200 --> 00:23:33,370 That's why we need it as an axiom. 429 00:23:33,370 --> 00:23:37,652 You could sort of view this as a series of dominoes. 430 00:23:37,652 --> 00:23:57,300 You know, I got a domino for each n, 431 00:23:57,300 --> 00:24:01,710 and each domino knocks over the next in terms of truth, 432 00:24:01,710 --> 00:24:04,230 knocking over corresponds to the truth here. 433 00:24:04,230 --> 00:24:08,400 And if I know P(0) is true, that means I knock over P(0), 434 00:24:08,400 --> 00:24:11,210 P(0) implies P(1) means P(1) one goes down. 435 00:24:11,210 --> 00:24:15,070 P(1) implies P(2) means P(2) goes down, and so forth. 436 00:24:15,070 --> 00:24:16,230 Pretty basic. 437 00:24:19,035 --> 00:24:21,660 Raise your hand if you think you don't have a lot of experience 438 00:24:21,660 --> 00:24:23,745 with induction. 439 00:24:23,745 --> 00:24:26,200 All right, yeah, that's pretty typical. 440 00:24:26,200 --> 00:24:27,560 About a third to a half of you. 441 00:24:27,560 --> 00:24:30,640 So we're going to change that. 442 00:24:30,640 --> 00:24:32,390 Let's do a simple proof using induction. 443 00:24:36,580 --> 00:24:40,600 So let's prove that for all n bigger than or equal to 0, 444 00:24:40,600 --> 00:24:48,140 again natural numbers, that 1, plus 2, plus 3, plus n 445 00:24:48,140 --> 00:24:53,677 equals n times n plus 1 over 2. 446 00:24:53,677 --> 00:24:55,510 This is actually a useful thing to remember. 447 00:24:55,510 --> 00:24:58,820 We're going to use this all term, this identity. 448 00:24:58,820 --> 00:25:01,740 Now the first thing, before we prove it, 449 00:25:01,740 --> 00:25:04,080 I want to make sure we understand this dot, 450 00:25:04,080 --> 00:25:05,740 dot, dot, notation. 451 00:25:05,740 --> 00:25:08,690 Because it is the source of a lot of errors. 452 00:25:08,690 --> 00:25:14,180 What it means is that you need to fill in the pattern here, 453 00:25:14,180 --> 00:25:16,500 which is vague. 454 00:25:16,500 --> 00:25:19,380 What it means in this case, you fill in four, five, six all 455 00:25:19,380 --> 00:25:21,865 the way up to n minus 1, and then n. 456 00:25:21,865 --> 00:25:22,740 That's what it means. 457 00:25:22,740 --> 00:25:25,020 Figure out the pattern and fill it in. 458 00:25:25,020 --> 00:25:27,340 Pretty risky thing. 459 00:25:27,340 --> 00:25:29,490 Now in this case, because that's so vague, 460 00:25:29,490 --> 00:25:32,320 there's other terminology we use for this. 461 00:25:32,320 --> 00:25:36,200 For example, we would use a big sigma, 462 00:25:36,200 --> 00:25:40,250 capital sigma i equals 0 to n of i. 463 00:25:40,250 --> 00:25:44,940 That means the sum of i, where i is the integers from 0 464 00:25:44,940 --> 00:25:47,004 to n inclusive. 465 00:25:47,004 --> 00:25:48,170 Actually, let me put 1 here. 466 00:25:51,470 --> 00:25:54,830 Another way to write this-- these are all equivalent ways 467 00:25:54,830 --> 00:25:57,810 to write it-- is you could put 1 less than 468 00:25:57,810 --> 00:26:01,005 or equal to i, less than or equal to n of i. 469 00:26:01,005 --> 00:26:04,810 So you could put something i over the range, from one to n, 470 00:26:04,810 --> 00:26:07,000 or you can write it on the bottom if you want. 471 00:26:10,690 --> 00:26:12,730 So these are four different ways of writing 472 00:26:12,730 --> 00:26:16,106 the same thing, the sum of the natural numbers from 1 to n. 473 00:26:16,106 --> 00:26:17,730 And we'll use them all during the term. 474 00:26:20,810 --> 00:26:22,750 All right, now there's some special cases 475 00:26:22,750 --> 00:26:26,430 that make this a little more interesting. 476 00:26:26,430 --> 00:26:27,686 What if n equals 1? 477 00:26:30,520 --> 00:26:35,570 I've got 1 plus 2 plus dot, dot, dot, plus n. 478 00:26:38,490 --> 00:26:40,490 What do you suppose it equals if n equals 1? 479 00:26:43,290 --> 00:26:48,910 1, because we're summing the numbers from 1 to 1. 480 00:26:48,910 --> 00:26:51,390 That's just 1, the number 1. 481 00:26:51,390 --> 00:26:53,930 There is no 2. 482 00:26:53,930 --> 00:26:56,880 And there aren't two copies of 1. 483 00:26:56,880 --> 00:26:58,910 So this notation is very ambiguous. 484 00:26:58,910 --> 00:27:00,810 You'll see a 2 here in the sum. 485 00:27:00,810 --> 00:27:03,614 If n is 1, you'll see a 1 here and a 1 here. 486 00:27:03,614 --> 00:27:04,780 So you've got to be careful. 487 00:27:04,780 --> 00:27:07,080 I guarantee you, you'll make a mistake with this. 488 00:27:07,080 --> 00:27:09,371 In fact I'm going to show you another false proof later 489 00:27:09,371 --> 00:27:11,244 where this comes into play. 490 00:27:11,244 --> 00:27:13,160 What about if n is less than or equal to zero? 491 00:27:15,849 --> 00:27:16,515 What is it then? 492 00:27:20,950 --> 00:27:21,510 Any thoughts? 493 00:27:25,620 --> 00:27:26,680 0. 494 00:27:26,680 --> 00:27:32,889 There are no integers to sum, no 1, no 2, no n, because you 495 00:27:32,889 --> 00:27:34,930 never get started because-- sorry, n is less than 496 00:27:34,930 --> 00:27:37,550 or equal to 0-- because you're summing from 1 to 0, 497 00:27:37,550 --> 00:27:38,320 0 is below. 498 00:27:38,320 --> 00:27:40,710 You never-- it doesn't include anything. 499 00:27:40,710 --> 00:27:45,120 So these are the conventions to keep in mind with the edge 500 00:27:45,120 --> 00:27:45,710 cases here. 501 00:27:50,500 --> 00:27:53,300 All right, it's easy enough to check that the theorem is 502 00:27:53,300 --> 00:27:54,640 true for certain values of n. 503 00:27:54,640 --> 00:28:00,520 For example, if n equals 4, I've got 1 plus 2 plus 3 plus 4. 504 00:28:00,520 --> 00:28:02,220 That's 10. 505 00:28:02,220 --> 00:28:09,580 And 10 equals 4 times 5 over 2, plugging in the formula. 506 00:28:09,580 --> 00:28:13,030 So for any value of n you could check this formula is true. 507 00:28:13,030 --> 00:28:15,930 Proving is true for all n. 508 00:28:15,930 --> 00:28:19,820 It takes a little more effort unless you use induction. 509 00:28:19,820 --> 00:28:20,760 So let's do that. 510 00:28:23,430 --> 00:28:24,870 So we'll prove the theorem. 511 00:28:24,870 --> 00:28:27,200 Now, whenever you're using a proof by induction, 512 00:28:27,200 --> 00:28:30,480 first thing you do is you write down by induction 513 00:28:30,480 --> 00:28:33,920 so we know what you're going to do. 514 00:28:33,920 --> 00:28:35,980 And the next thing you need to do 515 00:28:35,980 --> 00:28:40,500 is figure out, what's your predicate. 516 00:28:40,500 --> 00:28:43,060 What's your inductive hypothesis? 517 00:28:43,060 --> 00:28:43,750 What's p? 518 00:28:46,310 --> 00:28:50,870 So usually p will be the thing you're 519 00:28:50,870 --> 00:28:53,820 trying to prove, namely that 1 plus 2 plus 3 up to n 520 00:28:53,820 --> 00:28:56,460 is n times n plus 1 over 2. 521 00:28:56,460 --> 00:28:57,390 And you state that. 522 00:28:57,390 --> 00:29:06,140 You say let p of n be the proposition, the predicate, 523 00:29:06,140 --> 00:29:13,880 that the sum i equals 1 to n of i equals n times n 524 00:29:13,880 --> 00:29:14,897 plus 1 over 2. 525 00:29:17,669 --> 00:29:19,210 And once you've got that established, 526 00:29:19,210 --> 00:29:24,300 now we're going to go verify that p 0 is true 527 00:29:24,300 --> 00:29:29,030 and that p n implies p n plus 1. 528 00:29:29,030 --> 00:29:31,510 So we always have to write this down. 529 00:29:31,510 --> 00:29:34,740 The next thing to do is to check what's called the base case, 530 00:29:34,740 --> 00:29:36,165 p zero. 531 00:29:36,165 --> 00:29:37,300 So let's do that. 532 00:29:52,510 --> 00:29:54,130 So we write down base case. 533 00:29:54,130 --> 00:29:55,615 Some people call it the basis step. 534 00:29:59,160 --> 00:30:03,060 And we have to check that p 0 is true. 535 00:30:05,860 --> 00:30:13,650 Well, what's the sum of i equals 1 to 0 of i? 536 00:30:13,650 --> 00:30:14,650 0. 537 00:30:14,650 --> 00:30:16,140 There are no terms in this sum. 538 00:30:18,750 --> 00:30:22,020 And if I look over there, n times n plus 1 over 2. 539 00:30:22,020 --> 00:30:24,995 If n is 0, it equals 0. 540 00:30:28,346 --> 00:30:29,720 So we're done with the base case. 541 00:30:29,720 --> 00:30:33,890 We've now proved that p0 is true. 542 00:30:33,890 --> 00:30:36,935 And the second part is called the inductive step. 543 00:30:42,460 --> 00:30:47,350 And here we have to show, for n greater than or equal to 0, 544 00:30:47,350 --> 00:30:53,020 we need to show that pn implies pn plus 1 is true. 545 00:31:01,620 --> 00:31:05,380 Now how do we show an implication is true? 546 00:31:05,380 --> 00:31:06,580 How do I show this is true? 547 00:31:06,580 --> 00:31:09,810 What am I going to do to show that's true in general? 548 00:31:09,810 --> 00:31:10,900 Yeah? 549 00:31:10,900 --> 00:31:11,775 AUDIENCE: [INAUDIBLE] 550 00:31:19,060 --> 00:31:19,780 PROFESSOR: Right. 551 00:31:19,780 --> 00:31:24,230 Because an implication is true in every case, except for true 552 00:31:24,230 --> 00:31:26,060 implies false. 553 00:31:26,060 --> 00:31:27,960 So if pn is false, we're done. 554 00:31:27,960 --> 00:31:29,410 This implication is true. 555 00:31:29,410 --> 00:31:32,530 The only case we have to worry about is p of n is true. 556 00:31:32,530 --> 00:31:34,600 So we assume p of n is true. 557 00:31:34,600 --> 00:31:37,500 And now we confirm that that means pn plus 1 is true. 558 00:31:40,920 --> 00:31:43,960 So we write that down. 559 00:31:43,960 --> 00:31:49,960 Assume pn is true. 560 00:31:49,960 --> 00:31:53,765 And you might also write down, "for purposes of induction" 561 00:31:53,765 --> 00:31:59,490 or "purposes of it verifying the inductive hypothesis." just 562 00:31:59,490 --> 00:32:01,152 to let us know why you're assuming it. 563 00:32:01,152 --> 00:32:02,610 In other words, you're not assuming 564 00:32:02,610 --> 00:32:05,615 pn is true for purposes of contradiction. 565 00:32:05,615 --> 00:32:07,490 You're assuming it for purposes of induction. 566 00:32:10,329 --> 00:32:11,370 All right, let's do that. 567 00:32:23,150 --> 00:32:27,400 That means, in this case, i.e. 568 00:32:27,400 --> 00:32:40,270 we assume 1 plus 2 plus up to n equals n times n plus 1 over 2. 569 00:32:40,270 --> 00:32:49,910 And we need to show that the n plus 1 case is true, 570 00:32:49,910 --> 00:32:58,350 and the 1 plus 2 plus n plus 1 equals n plus 1 times n 571 00:32:58,350 --> 00:33:00,960 plus 2 over 2. 572 00:33:04,710 --> 00:33:06,930 It's sort of weird, and this does 573 00:33:06,930 --> 00:33:10,830 confuse people that aren't familiar yet with induction. 574 00:33:10,830 --> 00:33:14,360 It looks like we just assumed what we're trying to prove. 575 00:33:14,360 --> 00:33:16,410 We're trying to prove this is true for all n. 576 00:33:16,410 --> 00:33:18,280 And we just assumed it. 577 00:33:18,280 --> 00:33:22,360 But we're assuming it in the context of establishing 578 00:33:22,360 --> 00:33:24,080 this implication is true. 579 00:33:24,080 --> 00:33:26,730 And then we apply the induction axiom to conclude pn 580 00:33:26,730 --> 00:33:27,480 is true for all n. 581 00:33:30,540 --> 00:33:39,040 All right, well, let's rewrite this as 1 plus 2 582 00:33:39,040 --> 00:33:42,540 plus n plus n plus 1. 583 00:33:46,140 --> 00:33:50,130 Because I've assumed pn, I can rewrite this 584 00:33:50,130 --> 00:33:54,960 as an n plus 1 over 2. 585 00:33:54,960 --> 00:33:59,600 And now plus n plus 1 out here. 586 00:33:59,600 --> 00:34:06,520 That equals, well, I got n squared plus n, here, 587 00:34:06,520 --> 00:34:10,544 over 2, plus 2 n plus 2. 588 00:34:15,290 --> 00:34:22,050 And that equals n plus one times n plus 2 over 2, 589 00:34:22,050 --> 00:34:25,830 which is what we're trying to show. 590 00:34:25,830 --> 00:34:28,350 So we've completed, now, the inductive step. 591 00:34:28,350 --> 00:34:33,976 We have shown that for all n, pn implies pn plus one for all n 592 00:34:33,976 --> 00:34:35,100 greater than or equal to 0. 593 00:34:41,710 --> 00:34:45,760 Any questions about that? 594 00:34:45,760 --> 00:34:49,370 So, the proof is done. 595 00:34:49,370 --> 00:34:51,830 We've done everything we need to imply induction. 596 00:34:51,830 --> 00:34:53,909 We've got p0 is true. 597 00:34:53,909 --> 00:34:56,820 And pn implies pn plus 1 for n bigger than or equal to zero. 598 00:34:59,650 --> 00:35:05,138 Now induction helped us prove the theorem. 599 00:35:05,138 --> 00:35:07,221 Did it help us understand why the theorem is true? 600 00:35:09,879 --> 00:35:11,670 Do you have any feel for why the theorem is 601 00:35:11,670 --> 00:35:12,836 true after seeing the proof? 602 00:35:15,410 --> 00:35:17,670 Not really. 603 00:35:17,670 --> 00:35:19,709 I don't think-- sometimes induction 604 00:35:19,709 --> 00:35:21,000 will give you an understanding. 605 00:35:21,000 --> 00:35:21,900 Sometimes it won't. 606 00:35:21,900 --> 00:35:24,950 Here you've got no understanding of why the theorem's true, 607 00:35:24,950 --> 00:35:27,160 which is sort of unfortunate. 608 00:35:27,160 --> 00:35:32,979 Did induction help you figure out the answer to the sum? 609 00:35:32,979 --> 00:35:34,520 Namely, say you were trying to derive 610 00:35:34,520 --> 00:35:35,810 this answer from this sum. 611 00:35:35,810 --> 00:35:39,730 Did induction give you the answer? 612 00:35:39,730 --> 00:35:40,450 No. 613 00:35:40,450 --> 00:35:44,720 You had to know the answer, namely this, in order 614 00:35:44,720 --> 00:35:47,320 to prove it was true. 615 00:35:47,320 --> 00:35:49,830 Now later we'll see examples where induction actually 616 00:35:49,830 --> 00:35:52,380 can give you the answer, but often it does not. 617 00:35:52,380 --> 00:35:55,280 Often, induction gives you no hints, no answer, 618 00:35:55,280 --> 00:35:59,090 just prove that it's right once you had the clever idea that, 619 00:35:59,090 --> 00:36:01,940 oh, maybe that's the answer. 620 00:36:01,940 --> 00:36:06,080 You'll see that with things like the beaver flu problem. 621 00:36:06,080 --> 00:36:10,040 Figuring out the inductive hypothesis or the answer, 622 00:36:10,040 --> 00:36:12,040 you know, the details of it is hard. 623 00:36:12,040 --> 00:36:14,840 But once you do it, then applying the induction, 624 00:36:14,840 --> 00:36:15,410 not so hard. 625 00:36:15,410 --> 00:36:16,660 It gives you a concrete proof. 626 00:36:19,309 --> 00:36:20,350 OK, let's do another one. 627 00:36:25,684 --> 00:36:27,850 In fact, we're just going to spend the rest of today 628 00:36:27,850 --> 00:36:29,110 doing induction proofs. 629 00:36:35,380 --> 00:36:48,780 So for all natural numbers n, 3 divides n cubed minus n. 630 00:36:48,780 --> 00:36:51,740 Means than n cubed minus n is a multiple of three. 631 00:36:51,740 --> 00:36:53,940 For example, n is 5. 632 00:36:58,220 --> 00:37:06,720 3 divides 125 minus 5, because that's 120. 633 00:37:06,720 --> 00:37:09,191 Let's prove that. 634 00:37:09,191 --> 00:37:10,565 And we're going to use induction. 635 00:37:17,620 --> 00:37:19,220 What do you suppose pn's going to be? 636 00:37:22,000 --> 00:37:26,810 What's my predicate or my inductive hypothesis here? 637 00:37:26,810 --> 00:37:27,480 Any thoughts? 638 00:37:30,390 --> 00:37:31,752 Yeah? 639 00:37:31,752 --> 00:37:32,684 PROFESSOR: [INAUDIBLE] 640 00:37:32,684 --> 00:37:33,850 PROFESSOR: Yeah, that's you. 641 00:37:33,850 --> 00:37:34,638 Go ahead. 642 00:37:34,638 --> 00:37:36,224 AUDIENCE: [INAUDIBLE] 643 00:37:36,224 --> 00:37:36,890 PROFESSOR: Yeah. 644 00:37:36,890 --> 00:37:38,710 First thing you always want to try 645 00:37:38,710 --> 00:37:42,460 is you assume that is your induction hypothesis. 646 00:37:42,460 --> 00:37:50,320 So we say, let pn be 3 divides n cubed minus n. 647 00:37:53,040 --> 00:37:56,230 What's the next thing we do in our proof? 648 00:37:56,230 --> 00:37:56,830 Base case. 649 00:37:59,540 --> 00:38:03,200 Always easy to forget, but not a good idea. 650 00:38:03,200 --> 00:38:05,550 Base case is n equals 0. 651 00:38:05,550 --> 00:38:09,850 And sure enough, 3 divides 0 minus 0. 652 00:38:09,850 --> 00:38:10,753 So that's done. 653 00:38:18,650 --> 00:38:20,210 What's the next step we do? 654 00:38:24,090 --> 00:38:24,590 What is it? 655 00:38:24,590 --> 00:38:25,270 Next step? 656 00:38:28,710 --> 00:38:29,810 Inductive step. 657 00:38:36,940 --> 00:38:41,960 And for that we're going to need to show for n bigger 658 00:38:41,960 --> 00:38:49,661 than or equal to 0, we want to show pn implies pn plus 1 659 00:38:49,661 --> 00:38:50,160 is true. 660 00:38:54,880 --> 00:38:57,130 So to do that we assume pn is true. 661 00:39:04,750 --> 00:39:11,590 In other words, we assume that 3 divides n cubed minus n. 662 00:39:11,590 --> 00:39:15,690 And we're trying to show that 3 divides n plus 1 cubed minus 663 00:39:15,690 --> 00:39:17,270 n plus 1. 664 00:39:17,270 --> 00:39:25,050 So we take a look at, we examine n plus 1 cubed minus n plus 1. 665 00:39:25,050 --> 00:39:29,070 And we want to show it's a multiple of three. 666 00:39:29,070 --> 00:39:32,540 Well, let's expand that out. 667 00:39:32,540 --> 00:39:39,160 This is n cubed plus 3n squared, plus 3n plus 1. 668 00:39:39,160 --> 00:39:44,100 And I subtract off n plus 1. 669 00:39:44,100 --> 00:39:48,846 So I get n cubed plus 3n squared plus 2n. 670 00:39:54,830 --> 00:39:55,985 Is this a multiple of 3? 671 00:39:59,600 --> 00:40:04,220 I need to show that's a multiple of 3 and then I'd be all done. 672 00:40:04,220 --> 00:40:05,180 Is it a multiple of 3? 673 00:40:08,000 --> 00:40:09,950 Beats me. 674 00:40:09,950 --> 00:40:12,965 It doesn't look like a multiple of 3, necessarily. 675 00:40:16,350 --> 00:40:19,200 So maybe we need to massage it a little bit. 676 00:40:19,200 --> 00:40:21,890 I do know that this is a multiple of 3. 677 00:40:21,890 --> 00:40:23,579 I can use that fact. 678 00:40:23,579 --> 00:40:25,120 And in proofs by induction you always 679 00:40:25,120 --> 00:40:27,290 want to-- if you're not making use of that fact, 680 00:40:27,290 --> 00:40:29,390 then you're not really making use of induction. 681 00:40:29,390 --> 00:40:30,390 Sort of a warning sign. 682 00:40:30,390 --> 00:40:34,220 So I want to use this fact to prove this. 683 00:40:34,220 --> 00:40:39,520 Well, let's get a minus n in here. 684 00:40:39,520 --> 00:40:44,700 This equals n cubed minus n plus 3n squared plus-- 685 00:40:44,700 --> 00:40:49,070 if I subtracted an n I got to add an n-- plus 3n. 686 00:40:49,070 --> 00:40:50,700 So I've rewritten it. 687 00:40:50,700 --> 00:40:53,160 Now is it clear? 688 00:40:53,160 --> 00:40:54,740 Now it's clear. 689 00:40:54,740 --> 00:40:58,390 Very simple, because 3 divide this by pn. 690 00:40:58,390 --> 00:40:59,330 3 divides that. 691 00:40:59,330 --> 00:41:00,810 And 3 divides that. 692 00:41:00,810 --> 00:41:05,050 So this is a multiple of 3, because I've 693 00:41:05,050 --> 00:41:12,150 got 3 divides 3n squared, 3 divides 3n, and 3 divides n 694 00:41:12,150 --> 00:41:17,120 cubed minus n by pn. 695 00:41:17,120 --> 00:41:20,640 Or you could say, by the inductive hypothesis. 696 00:41:20,640 --> 00:41:25,680 pn is the inductive hypothesis, another name for it. 697 00:41:25,680 --> 00:41:30,820 So therefore, 3 divides that, which means 3 divides this. 698 00:41:30,820 --> 00:41:36,590 So that means 3 divides n plus 1 cubed minus n plus 1. 699 00:41:36,590 --> 00:41:42,560 And we are done with the proof by induction. 700 00:41:42,560 --> 00:41:43,768 Any questions about that one? 701 00:41:47,680 --> 00:41:50,600 So the key steps in induction are always the same. 702 00:41:50,600 --> 00:41:53,490 You write down "proof by induction." 703 00:41:53,490 --> 00:41:55,570 You identify your predicate. 704 00:41:55,570 --> 00:41:58,610 You do the base case, usually n equals 0, 705 00:41:58,610 --> 00:42:00,140 but it could be something else. 706 00:42:00,140 --> 00:42:03,850 And then you do your inductive step. 707 00:42:03,850 --> 00:42:06,115 Now in general, you could start your induction-- 708 00:42:06,115 --> 00:42:08,240 you don't have to start it at 0, you could start it 709 00:42:08,240 --> 00:42:11,336 at some value b, some integer b. 710 00:42:16,610 --> 00:42:17,810 Let's take a look at that. 711 00:42:25,050 --> 00:42:29,900 So you could have for the base case, 712 00:42:29,900 --> 00:42:35,130 you could have p of b is true, not p of 0. 713 00:42:35,130 --> 00:42:40,730 And then for your induction step you 714 00:42:40,730 --> 00:42:47,440 would have for all n bigger than or equal to b, pn implies 715 00:42:47,440 --> 00:42:50,000 pn plus 1. 716 00:42:50,000 --> 00:42:57,040 And then your conclusion is that for all n bigger than 717 00:42:57,040 --> 00:43:00,790 or equal to b, pn is true. 718 00:43:00,790 --> 00:43:03,630 So inductions don't always have to start at 0. 719 00:43:03,630 --> 00:43:05,790 You can pick where you start. 720 00:43:05,790 --> 00:43:10,150 Just make sure that you verify the starting point 721 00:43:10,150 --> 00:43:12,940 and you verify the implication for all n 722 00:43:12,940 --> 00:43:14,772 at that starting point and beyond. 723 00:43:20,680 --> 00:43:25,950 All right, let's now do a false proof using induction. 724 00:43:35,090 --> 00:43:42,130 We're going to prove that all horses are the same color. 725 00:43:54,600 --> 00:43:57,270 So let's go through the process. 726 00:43:57,270 --> 00:44:04,750 So the proof, we write down "by induction." 727 00:44:04,750 --> 00:44:08,320 Now we need our induction hypothesis, the predicate. 728 00:44:08,320 --> 00:44:12,550 So we're going to let that be-- we can't let it be this. 729 00:44:12,550 --> 00:44:14,950 Why can't this be our predicate? 730 00:44:14,950 --> 00:44:16,340 All horses are the same color. 731 00:44:16,340 --> 00:44:18,756 What's wrong with making that be the induction hypothesis? 732 00:44:20,917 --> 00:44:22,250 You can't plug anything into it. 733 00:44:22,250 --> 00:44:23,900 There's no number here. 734 00:44:23,900 --> 00:44:27,050 I've got to have a number, n, to induct on. 735 00:44:27,050 --> 00:44:37,000 So what I'm going to say is that in any set of n horses-- 736 00:44:37,000 --> 00:44:42,180 and let's make n bigger than or equal to 1-- 737 00:44:42,180 --> 00:44:48,920 the horses are all the same color. 738 00:44:55,600 --> 00:44:58,610 All right, now if I prove this is true for all n, 739 00:44:58,610 --> 00:45:00,280 well then all horses are the same color. 740 00:45:00,280 --> 00:45:02,738 Because in any set of n horses, they're all the same color. 741 00:45:02,738 --> 00:45:05,860 So all horses must be the same color. 742 00:45:05,860 --> 00:45:07,020 What's the next thing I do? 743 00:45:07,020 --> 00:45:09,150 What's the next step? 744 00:45:09,150 --> 00:45:11,870 Base case. 745 00:45:11,870 --> 00:45:16,790 Now, what am I going to use as my base case here? 746 00:45:16,790 --> 00:45:21,580 One horse, OK, or n equals 1. 747 00:45:21,580 --> 00:45:22,920 So p of 1. 748 00:45:22,920 --> 00:45:26,650 That would say that any set of one horses the horses 749 00:45:26,650 --> 00:45:28,510 are all the same color. 750 00:45:28,510 --> 00:45:29,050 That's true. 751 00:45:29,050 --> 00:45:29,841 I've got one horse. 752 00:45:29,841 --> 00:45:32,180 It's the same color as itself. 753 00:45:32,180 --> 00:45:33,970 So that's easy. 754 00:45:33,970 --> 00:45:38,310 It's true since just one horse. 755 00:46:03,240 --> 00:46:08,100 All right, what's the next step of the proof? 756 00:46:08,100 --> 00:46:10,200 What's the next thing I do? 757 00:46:10,200 --> 00:46:11,180 Inductive step. 758 00:46:15,440 --> 00:46:28,060 So I'm going to assume that pn is true to prove pn-- and show 759 00:46:28,060 --> 00:46:32,030 pn plus 1 is true. 760 00:46:32,030 --> 00:46:34,630 All right, so I'm going to assume 761 00:46:34,630 --> 00:46:36,310 that in any set of n horses, the horses 762 00:46:36,310 --> 00:46:38,680 are all the same color that I start with. 763 00:46:38,680 --> 00:46:41,050 And now I look at a set of n plus 1 horses. 764 00:46:44,160 --> 00:46:50,500 So we consider a set of n plus 1 horses. 765 00:46:53,290 --> 00:47:00,060 And let's call those horses h1, h2, hn plus 1. 766 00:47:03,820 --> 00:47:08,400 What do I know about horses h1 to hn? 767 00:47:13,740 --> 00:47:17,340 They're the same color, because pn is true. 768 00:47:17,340 --> 00:47:19,290 There are a set of n horses. 769 00:47:19,290 --> 00:47:21,140 By p1 they're all the same color. 770 00:47:31,810 --> 00:47:42,221 Also, what do I know but h2, h3 and hn plus 1? 771 00:47:42,221 --> 00:47:44,470 All the same color, because they're a set of n horses. 772 00:47:48,430 --> 00:47:51,170 All right. 773 00:47:51,170 --> 00:48:01,760 Well, since the color of h1 equals 774 00:48:01,760 --> 00:48:09,430 the color of those guys, h2 to hn, 775 00:48:09,430 --> 00:48:12,560 I know h1 is the same color as these guys. 776 00:48:12,560 --> 00:48:15,620 I also know that hn plus 1 is the same color as those guys. 777 00:48:22,990 --> 00:48:27,440 That means that h1 is the same color as hn plus 1. 778 00:48:27,440 --> 00:48:29,260 And all n plus 1 are the same color. 779 00:48:38,410 --> 00:48:39,700 And that's pn plus 1. 780 00:48:42,460 --> 00:48:45,660 That implies pn plus 1. 781 00:48:45,660 --> 00:48:46,460 And I'm done. 782 00:48:49,820 --> 00:48:54,720 Now a few years ago we assigned this problem as homework. 783 00:48:54,720 --> 00:48:56,410 And we asked students to figure out what 784 00:48:56,410 --> 00:48:57,860 went wrong with the problem. 785 00:48:57,860 --> 00:48:59,740 Why doesn't this work? 786 00:48:59,740 --> 00:49:03,230 And the responses were a little discouraging. 787 00:49:03,230 --> 00:49:06,950 Half the class responded, effectively as follows, 788 00:49:06,950 --> 00:49:09,980 this example just goes to show that induction doesn't always 789 00:49:09,980 --> 00:49:12,560 work. 790 00:49:12,560 --> 00:49:15,430 A third of the class said, I always 791 00:49:15,430 --> 00:49:17,390 knew that you can't trust mathematics. 792 00:49:17,390 --> 00:49:21,140 This example just proves it. 793 00:49:21,140 --> 00:49:23,540 That really hurt. 794 00:49:23,540 --> 00:49:26,930 And most of the rest were something similar to that. 795 00:49:26,930 --> 00:49:29,180 Not exactly what we were looking for in the homework. 796 00:49:29,180 --> 00:49:31,030 So now we don't leave it to homework. 797 00:49:31,030 --> 00:49:34,110 We do it in class. 798 00:49:34,110 --> 00:49:37,150 What's the flaw here? 799 00:49:37,150 --> 00:49:38,248 What was it? 800 00:49:38,248 --> 00:49:39,060 AUDIENCE: P of n. 801 00:49:39,060 --> 00:49:39,810 PROFESSOR: P of n? 802 00:49:39,810 --> 00:49:40,873 What's wrong with p of n? 803 00:49:40,873 --> 00:49:41,748 AUDIENCE: [INAUDIBLE] 804 00:49:45,690 --> 00:49:47,380 PROFESSOR: No, this is a real assumption 805 00:49:47,380 --> 00:49:52,280 in any set of n horses-- depends on n-- the horses are 806 00:49:52,280 --> 00:49:53,150 all the same color. 807 00:49:53,150 --> 00:49:55,730 That is a proposition. 808 00:49:55,730 --> 00:49:58,340 Because it depends on n, it's a predicate. 809 00:49:58,340 --> 00:50:00,990 It's true or it's false. 810 00:50:00,990 --> 00:50:04,840 Now we know it's false, because you could get a set of a couple 811 00:50:04,840 --> 00:50:06,670 horses with different color. 812 00:50:06,670 --> 00:50:07,740 But it is a proposition. 813 00:50:07,740 --> 00:50:10,640 Yeah, way back there? 814 00:50:10,640 --> 00:50:13,840 AUDIENCE: [INAUDIBLE] for like a certain set of horses. 815 00:50:13,840 --> 00:50:15,940 So even though it's the same number of horses, 816 00:50:15,940 --> 00:50:20,638 so it was the same number as a different set of horses 817 00:50:20,638 --> 00:50:23,077 it's not something you can assume anymore. 818 00:50:23,077 --> 00:50:24,410 PROFESSOR: That's a great point. 819 00:50:24,410 --> 00:50:28,450 I did gloss over something here, because in the predicate 820 00:50:28,450 --> 00:50:30,860 I've got "in any set." 821 00:50:30,860 --> 00:50:34,724 So there's really a "for all sets" sitting out here. 822 00:50:34,724 --> 00:50:36,140 So I've going to be careful that I 823 00:50:36,140 --> 00:50:41,080 establish that when I'm trying to prove pn implies pn plus 1. 824 00:50:41,080 --> 00:50:43,520 So to establish pn plus 1 here, I 825 00:50:43,520 --> 00:50:45,710 assume pn which means in any set of n horses 826 00:50:45,710 --> 00:50:46,860 they're all the same color. 827 00:50:46,860 --> 00:50:47,910 That I'm given. 828 00:50:47,910 --> 00:50:48,840 That's pn. 829 00:50:48,840 --> 00:50:52,500 I've got to look at any set of n plus 1 horses. 830 00:50:52,500 --> 00:50:58,270 So consider any set, not a set, any set of n plus 1 horses. 831 00:50:58,270 --> 00:51:00,060 So you pick any set you want. 832 00:51:00,060 --> 00:51:03,870 Call them this, h1 through hn plus 1. 833 00:51:03,870 --> 00:51:08,900 Well then, the first n are a set of n horses, so I can apply pn, 834 00:51:08,900 --> 00:51:11,290 therefore they have the same color. 835 00:51:11,290 --> 00:51:14,860 The last n horses in the set are a set of n horses, 836 00:51:14,860 --> 00:51:16,590 so I can imply pn. 837 00:51:16,590 --> 00:51:19,750 So these guys all have the same color, and I'm on my merry way 838 00:51:19,750 --> 00:51:20,810 here. 839 00:51:20,810 --> 00:51:23,800 So you're right, I had to do a little bit more work, 840 00:51:23,800 --> 00:51:26,660 but I can do that work and I still get a proof. 841 00:51:26,660 --> 00:51:28,114 Yeah? 842 00:51:28,114 --> 00:51:28,989 AUDIENCE: [INAUDIBLE] 843 00:51:28,989 --> 00:51:30,030 PROFESSOR: Good question. 844 00:51:30,030 --> 00:51:31,950 Is there a problem with the base case? 845 00:51:31,950 --> 00:51:37,270 So my base case was here, n equals 1. 846 00:51:37,270 --> 00:51:42,060 In any set of one horse, the horses-- let's say in the set 847 00:51:42,060 --> 00:51:47,960 just to be really careful-- are all the same color. 848 00:51:47,960 --> 00:51:51,100 No, in any set of one horses there's only one horse, 849 00:51:51,100 --> 00:51:53,080 so it's the same color as itself. 850 00:51:53,080 --> 00:51:54,096 Yeah? 851 00:51:54,096 --> 00:51:54,971 AUDIENCE: [INAUDIBLE] 852 00:51:57,780 --> 00:52:00,622 PROFESSOR: n plus 1 is not the same as n, yeah. 853 00:52:00,622 --> 00:52:02,088 AUDIENCE: [INAUDIBLE]. 854 00:52:02,088 --> 00:52:04,896 So you can't have the same assumption 855 00:52:04,896 --> 00:52:06,768 because it's not a set of n horses. 856 00:52:06,768 --> 00:52:08,395 It's a set of n plus 1? 857 00:52:08,395 --> 00:52:09,520 PROFESSOR: Well, let's see. 858 00:52:09,520 --> 00:52:12,700 So you're arguing I made a mistake here, right? 859 00:52:12,700 --> 00:52:16,230 Well, I've got horses-- horse 2, 3, up to n plus 1. 860 00:52:16,230 --> 00:52:18,000 How many horses are in this set here? 861 00:52:20,640 --> 00:52:21,802 AUDIENCE: Oh, in that set? 862 00:52:21,802 --> 00:52:22,760 PROFESSOR: In this set. 863 00:52:22,760 --> 00:52:24,400 How many horses are there? 864 00:52:24,400 --> 00:52:24,900 AUDIENCE: N. 865 00:52:24,900 --> 00:52:28,270 PROFESSOR: N. So I can apply p of n to this set just the same 866 00:52:28,270 --> 00:52:31,290 as that set, because p of n is any set of n horses. 867 00:52:31,290 --> 00:52:35,097 Well, I'm picking this one now and applying pn to it. 868 00:52:35,097 --> 00:52:36,930 And so therefore they're all the same color. 869 00:52:39,760 --> 00:52:41,646 Yeah? 870 00:52:41,646 --> 00:52:44,060 AUDIENCE: H1, h2, dot, dot, dot. 871 00:52:44,060 --> 00:52:45,450 PROFESSOR: Yes. 872 00:52:45,450 --> 00:52:48,414 AUDIENCE: Because it does work it there's two or more, 873 00:52:48,414 --> 00:52:52,370 but you didn't prove it with just one-- or from one to two. 874 00:52:52,370 --> 00:52:53,175 PROFESSOR: Exactly. 875 00:52:53,175 --> 00:52:54,758 Remember I told you that dot, dot, dot 876 00:52:54,758 --> 00:52:57,740 is so reasonable, so easy to use. 877 00:52:57,740 --> 00:52:58,540 Everybody uses it. 878 00:52:58,540 --> 00:53:00,440 It was going to catch us up. 879 00:53:00,440 --> 00:53:02,420 The bug is in the dot, dot, dot. 880 00:53:02,420 --> 00:53:08,775 And in particular, it has to do with the case n equal 2. 881 00:53:08,775 --> 00:53:10,400 So there's two ways to look at the bug, 882 00:53:10,400 --> 00:53:13,580 dot, dot, dot, or we didn't completely 883 00:53:13,580 --> 00:53:16,090 do all the inductive steps. 884 00:53:16,090 --> 00:53:19,680 So let's look at the case-- actually it's the case n 885 00:53:19,680 --> 00:53:25,090 equals 1, here, in this inductive step. 886 00:53:25,090 --> 00:53:30,780 Did I prove p1 implies p2? 887 00:53:30,780 --> 00:53:33,160 Let's just double check that worked for n equals 1. 888 00:53:40,630 --> 00:53:42,066 n equals 1. 889 00:53:42,066 --> 00:53:46,030 I've got a set of two horses. 890 00:53:46,030 --> 00:53:48,360 This becomes 2. 891 00:53:48,360 --> 00:53:51,542 So I've got h1, h2. 892 00:53:51,542 --> 00:53:52,750 Nothing in the dot, dot, dot. 893 00:53:52,750 --> 00:53:55,310 It's just h1 and h2. 894 00:53:55,310 --> 00:54:01,570 Then this becomes h1. 895 00:54:01,570 --> 00:54:04,260 So sure enough, h1 is the same color as itself. 896 00:54:04,260 --> 00:54:07,170 This becomes-- what does this become, this set of horses? 897 00:54:07,170 --> 00:54:08,990 What is it really? 898 00:54:08,990 --> 00:54:11,590 h2. 899 00:54:11,590 --> 00:54:13,190 All right, so I've got the color of h1 900 00:54:13,190 --> 00:54:16,550 equals the color of-- oh man, this 901 00:54:16,550 --> 00:54:18,850 is so hard to see this bug. 902 00:54:18,850 --> 00:54:23,550 I wrote down h2 dot, dot, dot, to hn because-- take out 903 00:54:23,550 --> 00:54:26,410 h1, what's left? 904 00:54:26,410 --> 00:54:28,550 What's left is h2 through hn. 905 00:54:28,550 --> 00:54:29,830 But this set is only h1. 906 00:54:29,830 --> 00:54:30,830 What's really left here? 907 00:54:30,830 --> 00:54:31,600 What is this set? 908 00:54:34,400 --> 00:54:35,190 What is this set? 909 00:54:39,980 --> 00:54:41,962 h2, h1? 910 00:54:41,962 --> 00:54:43,450 No. 911 00:54:43,450 --> 00:54:45,870 You see, what is the whole set? h1, dot, dot, dot, hn. 912 00:54:45,870 --> 00:54:46,870 What is the whole thing? 913 00:54:46,870 --> 00:54:48,930 It's just h1. 914 00:54:48,930 --> 00:54:52,950 pull out h1 and look at the rest of it, 915 00:54:52,950 --> 00:54:55,600 how many horses are here? 916 00:54:55,600 --> 00:54:56,580 0 horses. 917 00:54:59,240 --> 00:55:01,120 This is the empty set. 918 00:55:01,120 --> 00:55:04,560 There are no horses here to compare to. 919 00:55:04,560 --> 00:55:08,220 Even though I got an h2, I got an hn, I got a dot, dot, dot. 920 00:55:08,220 --> 00:55:11,770 Because generally, if n is big, this 921 00:55:11,770 --> 00:55:13,870 has n minus 1 horses are here. 922 00:55:13,870 --> 00:55:15,310 There's n total. 923 00:55:15,310 --> 00:55:18,120 I got n minus 1 right here. 924 00:55:18,120 --> 00:55:20,280 But n minus 1 is 0. 925 00:55:20,280 --> 00:55:22,680 There are no horses here. 926 00:55:22,680 --> 00:55:27,220 And so this bridge in the equality totally breaks. 927 00:55:27,220 --> 00:55:31,910 I got color of h1 equals-- there's no information here-- 928 00:55:31,910 --> 00:55:33,882 equals the color of h2. 929 00:55:33,882 --> 00:55:35,590 There's no equality here, because there's 930 00:55:35,590 --> 00:55:39,380 no horses in this set, all because of that dot, dot, dot. 931 00:55:39,380 --> 00:55:43,360 Do you see where the problem is by using the dot, dot, dot? 932 00:55:43,360 --> 00:55:46,912 For the case n equals-- it was true for every other case of n. 933 00:55:46,912 --> 00:55:51,240 n equals 2, n equals 3, it's all true. 934 00:55:51,240 --> 00:55:55,360 In fact, what we proved, we proved the base case of p1 935 00:55:55,360 --> 00:55:57,030 is true. 936 00:55:57,030 --> 00:55:59,610 This is an argument that p2 implies p3. 937 00:55:59,610 --> 00:56:01,790 That is true. 938 00:56:01,790 --> 00:56:06,680 p3 implies p4, that is true. 939 00:56:06,680 --> 00:56:08,200 And so forth. 940 00:56:08,200 --> 00:56:11,050 We proved for all n bigger than or equal to 2, 941 00:56:11,050 --> 00:56:13,780 pn implies pn plus 1. 942 00:56:13,780 --> 00:56:15,440 That we proved. 943 00:56:15,440 --> 00:56:20,060 What is the one missing implication we did not prove? 944 00:56:20,060 --> 00:56:24,410 The missing link, p1 implies p2. 945 00:56:24,410 --> 00:56:26,080 We did not prove that. 946 00:56:26,080 --> 00:56:28,650 Now is that true? 947 00:56:28,650 --> 00:56:30,350 No. 948 00:56:30,350 --> 00:56:33,330 You can find a set of two horses are not the same color. 949 00:56:33,330 --> 00:56:35,880 And just because every horse is the same color as itself, 950 00:56:35,880 --> 00:56:37,400 does not give you that. 951 00:56:37,400 --> 00:56:39,432 That was missing in this proof. 952 00:56:39,432 --> 00:56:40,890 And so we have to be really careful 953 00:56:40,890 --> 00:56:43,510 when you're doing these proofs that you establish 954 00:56:43,510 --> 00:56:45,620 the inductive step for all n bigger 955 00:56:45,620 --> 00:56:47,032 than or equal to the base case. 956 00:56:47,032 --> 00:56:48,490 And then make sure, if you're going 957 00:56:48,490 --> 00:56:51,790 to use this really convenient dot dot dot notation, that you 958 00:56:51,790 --> 00:56:55,650 don't wind up here saying, oh this is horses h2 through hn 959 00:56:55,650 --> 00:57:00,640 when there's no horses there, because n is 1. 960 00:57:00,640 --> 00:57:02,370 Any questions about this? 961 00:57:02,370 --> 00:57:03,302 Yeah? 962 00:57:03,302 --> 00:57:04,177 AUDIENCE: [INAUDIBLE] 963 00:57:08,580 --> 00:57:09,670 PROFESSOR: Great question. 964 00:57:09,670 --> 00:57:11,970 All right, let's fix the proof. 965 00:57:11,970 --> 00:57:14,500 Start with the base case of p of 2, 966 00:57:14,500 --> 00:57:16,250 and now I've got all this done. 967 00:57:16,250 --> 00:57:19,665 So therefore, that's another proof. 968 00:57:19,665 --> 00:57:20,165 Yeah? 969 00:57:23,530 --> 00:57:24,357 Say it again. 970 00:57:24,357 --> 00:57:25,232 AUDIENCE: [INAUDIBLE] 971 00:57:28,834 --> 00:57:30,250 PROFESSOR: This is still the same. 972 00:57:30,250 --> 00:57:34,515 In any set of n bigger than or equal to two horses, 973 00:57:34,515 --> 00:57:36,390 all the horses in the set are the same color. 974 00:57:36,390 --> 00:57:37,390 That's what he saying. 975 00:57:37,390 --> 00:57:40,000 And he's saying, hey look, the proof worked here. 976 00:57:40,000 --> 00:57:41,930 The inductive step is just fine. 977 00:57:41,930 --> 00:57:44,630 p2 does imply p3. 978 00:57:44,630 --> 00:57:45,512 Yeah? 979 00:57:45,512 --> 00:57:47,220 AUDIENCE: The base case for 2 isn't true. 980 00:57:47,220 --> 00:57:48,219 PROFESSOR: That's right. 981 00:57:48,219 --> 00:57:51,060 The base case fails. 982 00:57:51,060 --> 00:57:54,450 That's why you've always got to check the base case. 983 00:57:54,450 --> 00:57:55,390 Yeah? 984 00:57:55,390 --> 00:57:59,851 AUDIENCE: What it does prove is that if you find any two horse 985 00:57:59,851 --> 00:58:02,725 and they're always going to be the same color, 986 00:58:02,725 --> 00:58:04,650 then all horses have to be the same color. 987 00:58:04,650 --> 00:58:05,770 PROFESSOR: That's correct. 988 00:58:05,770 --> 00:58:07,170 That's a great point. 989 00:58:07,170 --> 00:58:10,580 We have given a proof that if you look at any pair of horses 990 00:58:10,580 --> 00:58:14,380 and they're the same color, then all horses are the same color. 991 00:58:14,380 --> 00:58:16,620 That's true. 992 00:58:16,620 --> 00:58:17,790 That is true. 993 00:58:17,790 --> 00:58:19,370 Of course, there are pairs of horses 994 00:58:19,370 --> 00:58:21,000 that aren't the same color. 995 00:58:21,000 --> 00:58:22,560 So the base case would fail. 996 00:58:22,560 --> 00:58:24,170 So always check the base case. 997 00:58:24,170 --> 00:58:25,860 You could prove some great stuff if you 998 00:58:25,860 --> 00:58:26,950 don't check the base case. 999 00:58:30,670 --> 00:58:34,262 All right, any other questions about that? 1000 00:58:34,262 --> 00:58:37,148 Yeah? 1001 00:58:37,148 --> 00:58:39,994 AUDIENCE: Negative number [INAUDIBLE]? 1002 00:58:39,994 --> 00:58:40,660 PROFESSOR: Yeah. 1003 00:58:40,660 --> 00:58:41,806 As long as it's an integer. 1004 00:58:41,806 --> 00:58:43,790 And as long as you prove pn implies 1005 00:58:43,790 --> 00:58:46,390 pn plus 1 starting there all the way out. 1006 00:58:46,390 --> 00:58:48,990 Yeah, you can start at negative numbers if you want. 1007 00:58:48,990 --> 00:58:51,500 Usually there's not many cases where that comes up 1008 00:58:51,500 --> 00:58:52,630 you want to, but you can. 1009 00:58:52,630 --> 00:58:55,030 Nothing wrong with that. 1010 00:58:55,030 --> 00:58:57,730 Any other questions? 1011 00:58:57,730 --> 00:59:02,820 Yeah, you can see why it was a messy homework problem. 1012 00:59:02,820 --> 00:59:04,790 So far we've seen examples of how induction 1013 00:59:04,790 --> 00:59:09,220 is useful in proving the hypothesis is true, but not 1014 00:59:09,220 --> 00:59:12,920 in solving the problem, per se. 1015 00:59:12,920 --> 00:59:16,540 Or even figuring out what the hypothesis should be. 1016 00:59:16,540 --> 00:59:19,400 Now in the last example, we're going 1017 00:59:19,400 --> 00:59:21,330 to show you how induction can be used 1018 00:59:21,330 --> 00:59:24,300 to prove there is a solution to a problem, 1019 00:59:24,300 --> 00:59:25,727 and also how to find the solution. 1020 00:59:25,727 --> 00:59:28,310 So it's actually going to be a very useful, constructive thing 1021 00:59:28,310 --> 00:59:29,390 in this case. 1022 00:59:29,390 --> 00:59:31,390 Now this problem arose in the construction 1023 00:59:31,390 --> 00:59:35,110 of the status center, the building we're in now. 1024 00:59:35,110 --> 00:59:37,950 This whole building was originally supposed to cost, 1025 00:59:37,950 --> 00:59:41,120 completely furnished, under $100 million. 1026 00:59:41,120 --> 00:59:42,400 That was the goal. 1027 00:59:42,400 --> 00:59:45,000 But the first mistake they made was the first step, 1028 00:59:45,000 --> 00:59:46,260 was hiring the architect. 1029 00:59:46,260 --> 00:59:47,820 They hired Frank Gehry. 1030 00:59:47,820 --> 00:59:51,350 I think MIT is now in a lawsuit with Frank Gehry. 1031 00:59:51,350 --> 00:59:53,200 So he was the architect. 1032 00:59:53,200 --> 00:59:55,600 And costs just went nuts. 1033 00:59:55,600 --> 00:59:57,960 As you could imagine, all these slanted walls 1034 00:59:57,960 --> 01:00:00,850 and crazy things happening actually are expensive. 1035 01:00:00,850 --> 01:00:05,862 And the cost quickly got over $300 million, literally. 1036 01:00:05,862 --> 01:00:06,820 That that's parts true. 1037 01:00:06,820 --> 01:00:10,180 Now I'm going to fabricate a little bit. 1038 01:00:10,180 --> 01:00:13,562 Now actually, fund raising became a huge priority 1039 01:00:13,562 --> 01:00:15,020 once they're more than $200 million 1040 01:00:15,020 --> 01:00:17,670 over budget they haven't even bought the furniture yet. 1041 01:00:17,670 --> 01:00:21,000 So some pretty radical ideas were proposed. 1042 01:00:21,000 --> 01:00:24,390 And one of them was to build a large 2 to the n by 2 1043 01:00:24,390 --> 01:00:27,080 to the n courtyard and put a statue 1044 01:00:27,080 --> 01:00:31,990 of a wealthy, potential donor in the center of the courtyard. 1045 01:00:31,990 --> 01:00:35,390 So let's draw this. 1046 01:00:47,370 --> 01:00:50,780 So the courtyard-- and of course, 1047 01:00:50,780 --> 01:00:55,380 you know, it's computer science, so it has to be a power of 2. 1048 01:00:55,380 --> 01:00:59,760 So it's 2 to the n by 2 to the n. 1049 01:00:59,760 --> 01:01:04,060 And I've drawn here the case n equals 2. 1050 01:01:04,060 --> 01:01:07,960 And we've got to get the statue of the wealthy guy 1051 01:01:07,960 --> 01:01:08,614 in the middle. 1052 01:01:08,614 --> 01:01:10,280 And I'm not supposed to reveal his name. 1053 01:01:10,280 --> 01:01:11,970 So we're just going to call him Bill. 1054 01:01:15,980 --> 01:01:18,130 So Bill's got to go in the center. 1055 01:01:18,130 --> 01:01:20,920 Now this would be fine, except to that nothing 1056 01:01:20,920 --> 01:01:22,870 was easy with Frank Gehry, everything 1057 01:01:22,870 --> 01:01:25,310 was some weird, weird thing going on. 1058 01:01:25,310 --> 01:01:28,860 And he insisted on using l-shaped tiles 1059 01:01:28,860 --> 01:01:30,660 for the courtyard. 1060 01:01:30,660 --> 01:01:34,560 So the tiles that we're going to use looked like this. 1061 01:01:34,560 --> 01:01:38,260 So it's almost a two by two, except you're 1062 01:01:38,260 --> 01:01:41,220 missing that piece. 1063 01:01:41,220 --> 01:01:42,930 So what you need to figure out how to do 1064 01:01:42,930 --> 01:01:47,000 is tie all this courtyard perfectly leaving 1065 01:01:47,000 --> 01:01:50,060 one spot for Bill using tiles like this, 1066 01:01:50,060 --> 01:01:51,125 these l-shaped tiles. 1067 01:01:57,400 --> 01:01:58,650 And this is 2 by 2 here. 1068 01:02:01,460 --> 01:02:03,150 So that's the task. 1069 01:02:03,150 --> 01:02:07,030 So let's see if we can do that for n equals 2 here. 1070 01:02:11,090 --> 01:02:15,040 Let's see, we can do a tile here. 1071 01:02:15,040 --> 01:02:19,100 We can do a tile here. 1072 01:02:19,100 --> 01:02:20,870 A tile here. 1073 01:02:20,870 --> 01:02:22,520 A tile here. 1074 01:02:22,520 --> 01:02:24,730 And a tile there. 1075 01:02:24,730 --> 01:02:25,320 So we can. 1076 01:02:25,320 --> 01:02:28,530 In this case we can tile the courtyard perfectly 1077 01:02:28,530 --> 01:02:32,440 using these L shaped tiles, leaving one square 1078 01:02:32,440 --> 01:02:36,819 in the Center for the statue. 1079 01:02:36,819 --> 01:02:39,110 All right, everyone understand what we're trying to do? 1080 01:02:39,110 --> 01:02:40,790 The goal? 1081 01:02:40,790 --> 01:02:42,398 Now I want to do it for n. 1082 01:02:46,070 --> 01:02:50,999 So let's start proving it by induction, 1083 01:02:50,999 --> 01:02:52,790 even though we don't know how to do it yet. 1084 01:02:52,790 --> 01:02:55,123 Because we're going to see how induction's going to help 1085 01:02:55,123 --> 01:02:58,745 us show it's possible and maybe even show us how to do it. 1086 01:03:03,594 --> 01:03:04,635 So let's state a theorem. 1087 01:03:07,970 --> 01:03:18,470 For all n there exists away to tile a 2 to the n by 2 1088 01:03:18,470 --> 01:03:28,840 to the n region, or courtyard, with a center square missing 1089 01:03:28,840 --> 01:03:29,480 for Bill. 1090 01:03:43,222 --> 01:03:44,680 And the proof will be by induction. 1091 01:03:52,670 --> 01:03:56,250 And our induction hypothesis, the predicate, 1092 01:03:56,250 --> 01:03:58,930 pn is going to be what we're trying to prove. 1093 01:04:01,450 --> 01:04:03,280 So this is the induction hypothesis. 1094 01:04:05,960 --> 01:04:07,680 Almost always when you do the induction 1095 01:04:07,680 --> 01:04:11,380 you want to start out with that as your hypothesis. 1096 01:04:11,380 --> 01:04:14,350 What's the next step? 1097 01:04:14,350 --> 01:04:15,040 Base case. 1098 01:04:15,040 --> 01:04:18,830 Never ever forget the base case or you'll 1099 01:04:18,830 --> 01:04:22,660 be thinking all horses are the same color. 1100 01:04:22,660 --> 01:04:25,350 p0. 1101 01:04:25,350 --> 01:04:32,190 Well, the courtyard for n equals 0 is just 1 square. 1102 01:04:32,190 --> 01:04:35,150 And that's for Bill. 1103 01:04:35,150 --> 01:04:36,230 So you're done. 1104 01:04:36,230 --> 01:04:38,790 There's no tiles at all to worry about. 1105 01:04:38,790 --> 01:04:40,500 That's easy. 1106 01:04:40,500 --> 01:04:43,070 So that's true. 1107 01:04:43,070 --> 01:04:44,905 And then we do the inductive step. 1108 01:05:11,270 --> 01:05:13,965 So inductive step. 1109 01:05:19,650 --> 01:05:22,200 For n bigger than or equal to 0, got 1110 01:05:22,200 --> 01:05:25,030 to remember to keep track of that now, 1111 01:05:25,030 --> 01:05:31,240 we assume pn to verify the inductive hypothesis, 1112 01:05:31,240 --> 01:05:34,010 or to prove pn plus 1. 1113 01:05:34,010 --> 01:05:35,930 A lot of ways to write this down, 1114 01:05:35,930 --> 01:05:40,650 but you always want to say what you're assuming and why. 1115 01:05:40,650 --> 01:05:46,190 So we need to show pn plus 1 is true. 1116 01:05:52,940 --> 01:05:56,280 Well, so let's look at a 2 to the n by-- 2 to the n plus 1 1117 01:05:56,280 --> 01:05:57,600 by 2 to the n plus 1 courtyard. 1118 01:06:10,600 --> 01:06:11,745 So let's draw it out here. 1119 01:06:14,450 --> 01:06:16,990 2 to the n plus 1 by 2 to the n plus 1. 1120 01:06:20,210 --> 01:06:23,400 What are we going to do to use our inductive hypothesis? 1121 01:06:23,400 --> 01:06:23,995 Yeah? 1122 01:06:23,995 --> 01:06:24,870 AUDIENCE: [INAUDIBLE] 1123 01:06:39,580 --> 01:06:41,170 PROFESSOR: For the-- yeah. 1124 01:06:41,170 --> 01:06:43,920 So you're on a good track there, for sure. 1125 01:06:43,920 --> 01:06:46,720 But I've got to apply-- I'm not going from 1 to 2, 1126 01:06:46,720 --> 01:06:49,336 I want to get-- I want to use pn here. 1127 01:06:49,336 --> 01:06:51,960 So I've got a 2 to the n plus 1 by 2 to the n plus 1 courtyard. 1128 01:06:51,960 --> 01:06:54,610 How do I use pn? 1129 01:06:54,610 --> 01:06:56,310 2 to the n by 2 to the n courtyard. 1130 01:06:56,310 --> 01:06:56,965 Yeah? 1131 01:06:56,965 --> 01:06:58,006 AUDIENCE: Oh, never mind. 1132 01:06:58,006 --> 01:06:58,964 I don't think it works. 1133 01:06:58,964 --> 01:07:01,576 I was about to say that would be as if 2 to the n 1134 01:07:01,576 --> 01:07:04,282 would divide that into four blocks. 1135 01:07:04,282 --> 01:07:05,770 PROFESSOR: Good idea, yeah. 1136 01:07:05,770 --> 01:07:07,830 Let's divide our courtyard into four blocks. 1137 01:07:07,830 --> 01:07:09,410 That's a great idea. 1138 01:07:09,410 --> 01:07:13,550 And now each of these is 2 to the n by 2 to the n. 1139 01:07:13,550 --> 01:07:15,210 Right? 1140 01:07:15,210 --> 01:07:19,040 And I can apply the inductive hypothesis there. 1141 01:07:19,040 --> 01:07:19,915 AUDIENCE: [INAUDIBLE] 1142 01:07:25,480 --> 01:07:26,490 PROFESSOR: Yeah. 1143 01:07:26,490 --> 01:07:28,444 Hm. 1144 01:07:28,444 --> 01:07:30,110 Yeah, something-- yeah, it doesn't quite 1145 01:07:30,110 --> 01:07:34,250 work because Bill wants to be here. 1146 01:07:34,250 --> 01:07:36,280 Got a little square for Bill there. 1147 01:07:36,280 --> 01:07:39,780 But I can't use my inductive hypothesis to tile 1148 01:07:39,780 --> 01:07:42,980 this, because there's no square missing. 1149 01:07:42,980 --> 01:07:46,190 In fact, even if there wasn't a square missing, I'm in trouble. 1150 01:07:46,190 --> 01:07:50,480 If I've got-- say this is size 4 by 4 here. 1151 01:07:50,480 --> 01:07:55,390 Can I tile a 4 by 4 region with L shaped tiles? 1152 01:07:55,390 --> 01:07:56,379 No, they're size 3. 1153 01:07:56,379 --> 01:07:57,420 3 doesn't divide into 16. 1154 01:07:57,420 --> 01:07:59,282 Yeah? 1155 01:07:59,282 --> 01:08:01,889 AUDIENCE: [INAUDIBLE] There's one tile missing 1156 01:08:01,889 --> 01:08:06,067 from each of those blocks at the corner that [INAUDIBLE]. 1157 01:08:06,067 --> 01:08:07,400 PROFESSOR: There's a great idea. 1158 01:08:07,400 --> 01:08:10,030 All right, we're making progress now. 1159 01:08:10,030 --> 01:08:12,640 Take a corner out of each of them. 1160 01:08:12,640 --> 01:08:16,529 Put my l-shaped tile here. 1161 01:08:16,529 --> 01:08:19,200 Now I can use the inductive hypothesis 1162 01:08:19,200 --> 01:08:21,319 to tile each one of these. 1163 01:08:24,750 --> 01:08:26,643 Yeah? 1164 01:08:26,643 --> 01:08:28,137 Yeah? 1165 01:08:28,137 --> 01:08:31,623 AUDIENCE: Well, why can't you put a 4 in the center 1166 01:08:31,623 --> 01:08:34,113 and then you have a bunch of 2s on the side? 1167 01:08:34,113 --> 01:08:35,689 PROFESSOR: A 4 in the center-- 1168 01:08:35,689 --> 01:08:37,784 AUDIENCE: Like a 2 by 2 in the center. 1169 01:08:37,784 --> 01:08:38,700 PROFESSOR: I got that. 1170 01:08:38,700 --> 01:08:40,790 I got Bill and the tile. 1171 01:08:40,790 --> 01:08:44,147 AUDIENCE: No, but like make it bigger. 1172 01:08:44,147 --> 01:08:47,393 Not just like 4 single tiles, but like-- so you 1173 01:08:47,393 --> 01:08:49,991 have something like that over there, right? 1174 01:08:49,991 --> 01:08:50,965 Put that in the center. 1175 01:08:50,965 --> 01:08:52,913 PROFESSOR: Put that in the center, OK. 1176 01:08:52,913 --> 01:08:56,529 AUDIENCE: And the rest of them will have like [INAUDIBLE]. 1177 01:08:56,529 --> 01:08:58,720 PROFESSOR: Well, the rest of them aren't 2 by 2, 1178 01:08:58,720 --> 01:09:00,399 because this is n. 1179 01:09:00,399 --> 01:09:01,600 I've got to use pn here. 1180 01:09:01,600 --> 01:09:04,869 And pn says that I can-- in a region 2 to the n by 2 1181 01:09:04,869 --> 01:09:09,970 to the n with a square out of the center, I can tile it. 1182 01:09:09,970 --> 01:09:11,310 Am I good here so far? 1183 01:09:11,310 --> 01:09:13,020 I claimed I was sort of done. 1184 01:09:13,020 --> 01:09:14,271 Yeah? 1185 01:09:14,271 --> 01:09:18,207 AUDIENCE: No, because p of n tells us-- well, 1186 01:09:18,207 --> 01:09:21,355 assume p of n tells us that you can set up a 2 to the n by 2 1187 01:09:21,355 --> 01:09:23,140 to the n courtyard with a centerpiece of-- 1188 01:09:23,140 --> 01:09:24,200 PROFESSOR: Yeah. 1189 01:09:24,200 --> 01:09:25,684 AUDIENCE: [INAUDIBLE] 1190 01:09:25,684 --> 01:09:26,350 PROFESSOR: Yeah. 1191 01:09:26,350 --> 01:09:28,100 And what's the problem for this 2 to the n 1192 01:09:28,100 --> 01:09:30,579 by 2 to the n region? 1193 01:09:30,579 --> 01:09:31,620 Bill's not in the center. 1194 01:09:31,620 --> 01:09:33,170 He wants to be in the center. 1195 01:09:33,170 --> 01:09:35,529 We put Bill in the corner. 1196 01:09:35,529 --> 01:09:40,130 So you can't use the inductive hypothesis here. 1197 01:09:40,130 --> 01:09:41,899 So I went a little too fast, here. 1198 01:09:41,899 --> 01:09:44,450 This is not a proof so far. 1199 01:09:44,450 --> 01:09:45,970 I've got a problem. 1200 01:09:45,970 --> 01:09:47,137 Yeah? 1201 01:09:47,137 --> 01:09:48,012 AUDIENCE: [INAUDIBLE] 1202 01:09:55,002 --> 01:09:55,710 PROFESSOR: Great. 1203 01:09:55,710 --> 01:10:00,060 OK, let's then change the inductive hypothesis. 1204 01:10:00,060 --> 01:10:00,750 Good. 1205 01:10:00,750 --> 01:10:03,620 Let's say there exists a way to tile a 2 to the n by 2 1206 01:10:03,620 --> 01:10:10,870 to the n region with a corner square missing for Bill. 1207 01:10:14,050 --> 01:10:19,459 All right, but now I got to put-- oh, 1208 01:10:19,459 --> 01:10:20,500 but I can make this work. 1209 01:10:20,500 --> 01:10:20,770 Yeah. 1210 01:10:20,770 --> 01:10:20,930 Yeah. 1211 01:10:20,930 --> 01:10:22,760 We can make something-- we can prove this now. 1212 01:10:22,760 --> 01:10:24,218 The inductive step is going to work 1213 01:10:24,218 --> 01:10:28,320 because I'll put Bill-- say he's in one of the four regions. 1214 01:10:28,320 --> 01:10:31,510 Then use the l down here. 1215 01:10:31,510 --> 01:10:32,850 Put the l here. 1216 01:10:32,850 --> 01:10:34,900 And now I've got a corner out of each one. 1217 01:10:34,900 --> 01:10:37,810 And now I'm done by induction. 1218 01:10:37,810 --> 01:10:41,315 I have proved there's a way to tile any 2 to the n 1219 01:10:41,315 --> 01:10:46,050 by 2 to the n region with a corner square missing. 1220 01:10:46,050 --> 01:10:48,426 I think that proof works, Yeah? 1221 01:10:48,426 --> 01:10:54,330 AUDIENCE: [INAUDIBLE] with Bill in the middle, 1222 01:10:54,330 --> 01:10:58,266 because to prove that you can put Bill in the corner in a 2 1223 01:10:58,266 --> 01:11:01,218 by 2 case, you can just blow that square off. 1224 01:11:01,218 --> 01:11:05,646 [INAUDIBLE] just becomes just another square, which is 4 by 4 1225 01:11:05,646 --> 01:11:07,770 and then you can put him in the middle. 1226 01:11:07,770 --> 01:11:08,770 PROFESSOR: That is true. 1227 01:11:08,770 --> 01:11:10,140 So you've jumped ahead. 1228 01:11:10,140 --> 01:11:14,842 We have successfully now proved this is true for p of n. 1229 01:11:14,842 --> 01:11:16,550 But Bill didn't want to be in the corner. 1230 01:11:16,550 --> 01:11:18,008 We're trying to prove there's a way 1231 01:11:18,008 --> 01:11:19,844 to do it with Bill in the center, which 1232 01:11:19,844 --> 01:11:20,760 is not what we proved. 1233 01:11:20,760 --> 01:11:23,218 And you've come up with a way-- yeah, that might be doable. 1234 01:11:23,218 --> 01:11:25,690 First prove you can put Bill in the corner. 1235 01:11:25,690 --> 01:11:31,420 And do it in 2 to the n by 2 to the n's. 1236 01:11:31,420 --> 01:11:34,300 I don't know about this, actually. 1237 01:11:34,300 --> 01:11:35,890 I got Bill in the corner. 1238 01:11:35,890 --> 01:11:38,010 There's some way to do it. 1239 01:11:38,010 --> 01:11:39,630 But it might have involved doing this. 1240 01:11:39,630 --> 01:11:41,400 And now I can't rotate that. 1241 01:11:41,400 --> 01:11:43,900 I don't think we have-- I don't think that proof necessarily 1242 01:11:43,900 --> 01:11:45,115 works. 1243 01:11:45,115 --> 01:11:54,520 AUDIENCE: [INAUDIBLE] prove that he could be here. 1244 01:11:54,520 --> 01:11:57,490 But likewise, we could prove that if the block just 1245 01:11:57,490 --> 01:11:59,212 rotates you can [INAUDIBLE]. 1246 01:11:59,212 --> 01:12:01,170 PROFESSOR: Yes, I agree with you and I liked it 1247 01:12:01,170 --> 01:12:02,086 when I first heard it. 1248 01:12:02,086 --> 01:12:03,860 And you might still convince me. 1249 01:12:03,860 --> 01:12:08,310 But all we proved is this, there's a 2 to the n by 2 1250 01:12:08,310 --> 01:12:10,586 to the n-- any 2 to the n by 2 to the n region, 1251 01:12:10,586 --> 01:12:12,085 we could tile it with a corner, Bill 1252 01:12:12,085 --> 01:12:15,480 in a corner, any corner square missing. 1253 01:12:15,480 --> 01:12:18,910 Good, so now you want to say, I can get Bill in the middle. 1254 01:12:18,910 --> 01:12:20,510 And what you want to say is OK, I 1255 01:12:20,510 --> 01:12:30,549 tile this region with Bill here, in a 2 to the n by-- maybe 1256 01:12:30,549 --> 01:12:31,090 you're right. 1257 01:12:31,090 --> 01:12:35,150 So then I would apply the theorem here, with Bill here. 1258 01:12:35,150 --> 01:12:35,940 Oh, Bill here. 1259 01:12:35,940 --> 01:12:37,860 No, I think I like it now. 1260 01:12:37,860 --> 01:12:39,546 And then I would take these out. 1261 01:12:39,546 --> 01:12:45,044 AUDIENCE: [INAUDIBLE] just takes up up three blocks. 1262 01:12:45,044 --> 01:12:49,780 And that big square is just like a zoomed out version 1263 01:12:49,780 --> 01:12:52,660 of a little square. 1264 01:12:52,660 --> 01:12:55,980 And since you know that little square fits in a 2 by 2-- 1265 01:12:55,980 --> 01:13:04,240 like for example in the 4 by 4 case. 1266 01:13:04,240 --> 01:13:06,470 We proved the 2 by 2 case. 1267 01:13:06,470 --> 01:13:11,235 In the 4 by 4 case you have a big size. 1268 01:13:11,235 --> 01:13:14,770 It's just like a-- you have four squares, right? 1269 01:13:14,770 --> 01:13:18,919 And each of those four squares has [INAUDIBLE]. 1270 01:13:18,919 --> 01:13:23,965 So you take that top right square and you put Bill 1271 01:13:23,965 --> 01:13:28,470 within the smaller square exactly where you 1272 01:13:28,470 --> 01:13:29,430 want him to be. 1273 01:13:29,430 --> 01:13:33,410 And then you fill those other-- all those empty spaces. 1274 01:13:33,410 --> 01:13:34,342 PROFESSOR: I agree. 1275 01:13:34,342 --> 01:13:35,300 You could make a proof. 1276 01:13:35,300 --> 01:13:37,860 So in this case you'd make a Lemma 1277 01:13:37,860 --> 01:13:40,770 that uses induction that says you can do it 1278 01:13:40,770 --> 01:13:42,350 with Bill in the corner. 1279 01:13:42,350 --> 01:13:44,080 And then as a corollary or a theorem 1280 01:13:44,080 --> 01:13:48,980 you'd take that and apply it to four sub-squares. 1281 01:13:48,980 --> 01:13:50,970 Put Bill in here. 1282 01:13:50,970 --> 01:13:51,595 Take these out. 1283 01:13:51,595 --> 01:13:54,011 And then now you'd have your result of Bill in the center. 1284 01:13:54,011 --> 01:13:55,340 So that is is a way to do it. 1285 01:13:55,340 --> 01:13:56,860 It ends up being more complicated. 1286 01:13:56,860 --> 01:13:58,300 There is a simpler approach. 1287 01:13:58,300 --> 01:13:59,870 But that is a way that works. 1288 01:13:59,870 --> 01:14:03,490 It is a natural thing you would do if you had this on homework. 1289 01:14:03,490 --> 01:14:05,980 Is you'd think of a different thing to prove by induction, 1290 01:14:05,980 --> 01:14:09,540 then use that as a Lemma to get you where you wanted to go. 1291 01:14:09,540 --> 01:14:14,290 There's another way to do it without having that first step. 1292 01:14:14,290 --> 01:14:16,870 And that's-- yeah? 1293 01:14:16,870 --> 01:14:20,750 AUDIENCE: That courtyard where n equals 2. 1294 01:14:20,750 --> 01:14:24,630 Divide each cell into a 2 by 2-- 1295 01:14:28,154 --> 01:14:28,820 PROFESSOR: Yeah. 1296 01:14:28,820 --> 01:14:31,180 AUDIENCE: Something. 1297 01:14:31,180 --> 01:14:33,780 PROFESSOR: Well, yeah. 1298 01:14:33,780 --> 01:14:35,400 You want to do it bottoms up. 1299 01:14:35,400 --> 01:14:39,450 You want to take it and make that your inductive step, 2 1300 01:14:39,450 --> 01:14:40,310 by 2's inside. 1301 01:14:40,310 --> 01:14:42,272 I haven't thought about that approach. 1302 01:14:42,272 --> 01:14:55,310 AUDIENCE: [INAUDIBLE] So if you could 1303 01:14:55,310 --> 01:14:58,262 do that then you're all set except for the 2 1304 01:14:58,262 --> 01:15:04,176 by 2 where Bill was. 1305 01:15:04,176 --> 01:15:05,800 PROFESSOR: I'm worried about a lot of 2 1306 01:15:05,800 --> 01:15:08,883 by 2's with a corner missing if I do it that way. 1307 01:15:08,883 --> 01:15:11,620 I'll think about that. 1308 01:15:11,620 --> 01:15:17,372 There is-- let me get to another approach here. 1309 01:15:17,372 --> 01:15:19,080 We couldn't make it work with the center. 1310 01:15:19,080 --> 01:15:20,580 We could make it work with a corner. 1311 01:15:20,580 --> 01:15:22,960 But then we had to do more work after. 1312 01:15:22,960 --> 01:15:25,336 One general technique to use with induction, 1313 01:15:25,336 --> 01:15:27,460 when you're having struggling, what's the induction 1314 01:15:27,460 --> 01:15:29,140 hypothesis to use. 1315 01:15:29,140 --> 01:15:33,040 If what you've got doesn't work, you could pick a different one, 1316 01:15:33,040 --> 01:15:35,180 but better to pick a stronger one. 1317 01:15:35,180 --> 01:15:36,643 Yeah? 1318 01:15:36,643 --> 01:15:37,518 AUDIENCE: [INAUDIBLE] 1319 01:15:40,930 --> 01:15:42,390 PROFESSOR: Yes. 1320 01:15:42,390 --> 01:15:42,890 You could. 1321 01:15:42,890 --> 01:15:45,110 And that is a good thing to do. 1322 01:15:45,110 --> 01:15:50,460 Make the induction hypothesis be much stronger. 1323 01:15:50,460 --> 01:15:52,810 I had trouble proving there's a way to do it 1324 01:15:52,810 --> 01:15:54,880 with a center square missing. 1325 01:15:54,880 --> 01:15:57,710 Turns out to be easier to prove it 1326 01:15:57,710 --> 01:15:59,890 where you say any square could be missing. 1327 01:15:59,890 --> 01:16:01,600 Seems like this should be harder, right? 1328 01:16:01,600 --> 01:16:03,183 We had a hard enough time just showing 1329 01:16:03,183 --> 01:16:05,250 this square missing was doable. 1330 01:16:05,250 --> 01:16:07,850 But by assuming something-- by trying to prove a harder 1331 01:16:07,850 --> 01:16:10,330 problem, assuming something stronger in pn, 1332 01:16:10,330 --> 01:16:12,680 it gets easier to prove. 1333 01:16:12,680 --> 01:16:16,140 All right, now we better check the base case, p0, 1334 01:16:16,140 --> 01:16:16,900 but that's easy. 1335 01:16:16,900 --> 01:16:19,960 There's only one square Bill can be. 1336 01:16:19,960 --> 01:16:22,458 But now let's look at pn implies pn plus 1. 1337 01:16:26,014 --> 01:16:26,930 So we go back to this. 1338 01:16:26,930 --> 01:16:32,780 We've got a 2 to the n plus 1 by 2 to the n plus 1 courtyard. 1339 01:16:32,780 --> 01:16:34,940 And now Bill can be anywhere. 1340 01:16:38,030 --> 01:16:39,750 Put him here. 1341 01:16:39,750 --> 01:16:42,420 There's Bill. 1342 01:16:42,420 --> 01:16:45,660 Now I'm going to place my first l-shaped tile here, 1343 01:16:45,660 --> 01:16:49,000 in the other three regions. 1344 01:16:49,000 --> 01:16:52,710 And now I apply pn to each region. 1345 01:16:52,710 --> 01:16:54,950 A pn says I can do it with any square missing. 1346 01:16:54,950 --> 01:16:57,540 So I'll pick this one out here, this one out here, 1347 01:16:57,540 --> 01:16:59,520 that one there, that one there. 1348 01:16:59,520 --> 01:17:01,080 And now I'm done. 1349 01:17:01,080 --> 01:17:02,720 That was easy. 1350 01:17:02,720 --> 01:17:05,650 No extra steps. 1351 01:17:05,650 --> 01:17:08,340 Now, how is it possible that it was easier to prove something 1352 01:17:08,340 --> 01:17:11,370 that was harder? 1353 01:17:11,370 --> 01:17:13,394 Yeah? 1354 01:17:13,394 --> 01:17:15,602 AUDIENCE: First need to be in the center [INAUDIBLE], 1355 01:17:15,602 --> 01:17:17,518 you're putting another constraint on yourself. 1356 01:17:17,518 --> 01:17:18,840 PROFESSOR: Yes. 1357 01:17:18,840 --> 01:17:19,470 That is true. 1358 01:17:19,470 --> 01:17:22,440 But by allowing him to be anywhere, 1359 01:17:22,440 --> 01:17:25,420 I could have started-- I have to-- that's a possibility. 1360 01:17:28,220 --> 01:17:31,350 See, what I'm trying to do here, this inductive step 1361 01:17:31,350 --> 01:17:37,380 is all about proving pn implies pn plus 1 is true. 1362 01:17:37,380 --> 01:17:40,130 That's what I'm trying to show. 1363 01:17:40,130 --> 01:17:43,840 Now, how is it useful for me if pn is stronger? 1364 01:17:43,840 --> 01:17:45,826 Has more? 1365 01:17:45,826 --> 01:17:46,974 AUDIENCE: You grow it. 1366 01:17:46,974 --> 01:17:48,515 You prove that he can be in a corner. 1367 01:17:48,515 --> 01:17:51,090 But when you grow it, the corner moves. 1368 01:17:51,090 --> 01:17:52,840 But since we proved that it can be in any, 1369 01:17:52,840 --> 01:17:53,784 it's fine if it moves. 1370 01:17:53,784 --> 01:17:54,575 PROFESSOR: Exactly. 1371 01:17:54,575 --> 01:17:56,420 That's exactly right. 1372 01:17:56,420 --> 01:18:00,150 ph got more powerful, which means 1373 01:18:00,150 --> 01:18:02,560 I get more to assume here. 1374 01:18:02,560 --> 01:18:06,082 In the recursive problem, Bill can be anywhere now. 1375 01:18:06,082 --> 01:18:07,040 It gives me more power. 1376 01:18:07,040 --> 01:18:10,750 I can tile any courtyard with any square missing. 1377 01:18:10,750 --> 01:18:11,810 This is more powerful. 1378 01:18:11,810 --> 01:18:13,790 So this got more powerful. 1379 01:18:13,790 --> 01:18:15,440 So did this. 1380 01:18:15,440 --> 01:18:17,790 So what it means is that my tool set 1381 01:18:17,790 --> 01:18:20,450 is bigger with a stronger pn. 1382 01:18:20,450 --> 01:18:24,240 And what I'm trying to construct or prove got harder. 1383 01:18:24,240 --> 01:18:28,430 And sometimes, if I've got more tools, 1384 01:18:28,430 --> 01:18:31,750 it becomes easier to prove, even a harder thing. 1385 01:18:31,750 --> 01:18:33,960 And so a general rule with induction 1386 01:18:33,960 --> 01:18:36,505 is if you don't first succeed, try, try again. 1387 01:18:36,505 --> 01:18:39,130 Well, the rule with induction is if you don't succeed at first, 1388 01:18:39,130 --> 01:18:41,310 try something harder. 1389 01:18:41,310 --> 01:18:42,070 All right? 1390 01:18:42,070 --> 01:18:45,110 And it's amazing, but it actually works a lot of time, 1391 01:18:45,110 --> 01:18:47,010 as it did here. 1392 01:18:47,010 --> 01:18:49,930 If I don't assume something strong enough, 1393 01:18:49,930 --> 01:18:54,100 and pn is some little weak thing like Bill just in the center, 1394 01:18:54,100 --> 01:18:57,120 it's not enough to go anywhere. 1395 01:18:57,120 --> 01:18:58,940 But if I could assume a lot more, like Bill 1396 01:18:58,940 --> 01:19:01,680 could be anywhere he pleases. 1397 01:19:01,680 --> 01:19:05,870 Then I could prove lots of things here. 1398 01:19:05,870 --> 01:19:07,453 So it's all the art-- and you're going 1399 01:19:07,453 --> 01:19:09,244 to learn this over the next several weeks-- 1400 01:19:09,244 --> 01:19:12,650 it's all the art of what's your induction hypothesis. 1401 01:19:12,650 --> 01:19:14,110 Picking a good one, life is easy. 1402 01:19:14,110 --> 01:19:17,140 Picking the wrong one, very painful, 1403 01:19:17,140 --> 01:19:19,830 as you'll see with beaver flu. 1404 01:19:19,830 --> 01:19:22,180 OK, that's it for today.