WEBVTT 00:00:00.250 --> 00:00:02.080 The question we want to think about is 00:00:02.080 --> 00:00:04.720 how you could define who your ancestors are. The 00:00:04.720 --> 00:00:07.689 way to think about ancestors. Well, here's you. 00:00:08.850 --> 00:00:12.680 Hopefully, you're smiling, because you're enjoying learning about recursive 00:00:12.680 --> 00:00:16.670 definitions. And you had some parents. Let's assume 00:00:16.670 --> 00:00:21.100 you had two. And we won't assume whether of 00:00:21.100 --> 00:00:23.000 not your parents are smiling, but let's hope 00:00:23.000 --> 00:00:25.670 at least one of them is. And your parents 00:00:25.670 --> 00:00:28.090 were your ancestors, but they're not all 00:00:28.090 --> 00:00:30.750 of your ancestors. Your parents had parents, as 00:00:30.750 --> 00:00:34.970 well. These are your grandparents. Grandparents are always 00:00:34.970 --> 00:00:38.760 happy. So, they're all smiling. And your grandparents, 00:00:38.760 --> 00:00:40.640 well, they also had parents. Those would 00:00:40.640 --> 00:00:44.620 be your great-grandparents, and so forth. And all 00:00:44.620 --> 00:00:48.750 of these, except yourself, are your ancestors. So 00:00:48.750 --> 00:00:51.620 our goal now is to define that precisely. 00:00:51.620 --> 00:00:53.910 Can we come up with a definition that describes 00:00:53.910 --> 00:00:57.550 exactly this same relationship, all the ancestors that you 00:00:57.550 --> 00:00:59.940 have? So the question is, which of these is 00:00:59.940 --> 00:01:02.885 the best definition of ancestors? And there are three 00:01:02.885 --> 00:01:05.730 choices, the first has one rule, ancestor is replaced 00:01:05.730 --> 00:01:09.330 by parent of ancestor. The second has two rules, 00:01:09.330 --> 00:01:12.540 ancestor is replaced by parent, and ancestor is replaced 00:01:12.540 --> 00:01:16.220 by parent of the ancestor. The third has three rules. 00:01:16.220 --> 00:01:19.600 Ancestor is replaced by parent. Ancestor is replaced by parent 00:01:19.600 --> 00:01:23.430 of parent. And ancestor's replaced by parent of parent of ancestor.