WEBVTT 00:00:00.364 --> 00:00:02.442 - So I have a very simple equation written here 00:00:02.442 --> 00:00:06.366 a plus b is equal to c and what I want to think about 00:00:06.366 --> 00:00:09.443 if we know that a is an integer, so let's say that this 00:00:09.443 --> 00:00:14.443 is the set of all integers right over here. Integers. 00:00:15.224 --> 00:00:18.997 We know that a is an integer so let's say that's a there. 00:00:18.997 --> 00:00:21.841 We also know that b is an integer, could be the same integer 00:00:21.841 --> 00:00:24.002 but I'll draw it so it looks like it's another one, 00:00:24.002 --> 00:00:26.322 it doesn't have to be but they could be the same integer. 00:00:26.322 --> 00:00:27.599 B is an integer. 00:00:27.599 --> 00:00:29.341 If we're adding these two integers, 00:00:29.341 --> 00:00:31.989 are we definitely going to get another integer? 00:00:31.989 --> 00:00:36.120 Do we know that c is going to be an integer? 00:00:36.120 --> 00:00:38.722 And I encourage you to pause the video and think about it. 00:00:38.722 --> 00:00:39.940 And when you've thought about it, you might have 00:00:39.940 --> 00:00:40.858 come up with some cases. 00:00:40.858 --> 00:00:41.925 Well, what if a and b are both one, 00:00:41.925 --> 00:00:43.981 then for sure, c is going to be two. 00:00:43.981 --> 00:00:46.046 And c is going to be an integer. 00:00:46.046 --> 00:00:48.206 Well is that always going to be the case? 00:00:48.206 --> 00:00:50.040 Well, it is. There are ways to prove it. 00:00:50.040 --> 00:00:52.257 I'm not going to go into it now but when you have 00:00:52.257 --> 00:00:53.720 a situation like this, when you're able 00:00:53.720 --> 00:00:55.927 to take two members of a set, in this case, 00:00:55.927 --> 00:00:58.665 two members of a set and perform an operation on it, 00:00:58.665 --> 00:01:02.380 and always get another member of the set and it could be 00:01:02.380 --> 00:01:06.038 a or b again or it could be another distinct member 00:01:06.038 --> 00:01:08.684 of the set, I'll draw it as another distinct member 00:01:08.684 --> 00:01:11.377 of the set, whenever you have this, 00:01:11.377 --> 00:01:14.362 so you have some operation, in this case, it is addition. 00:01:14.362 --> 00:01:16.944 So when you're performing the addition operation 00:01:16.944 --> 00:01:20.542 on two members of the set, you get, you stay in the set. 00:01:20.542 --> 00:01:23.550 C is also going to be a member of that set 00:01:23.550 --> 00:01:25.964 so one way to think about it, we're going, 00:01:25.964 --> 00:01:27.623 we're going like this and we are 00:01:27.623 --> 00:01:29.969 performing this addition operation. 00:01:29.969 --> 00:01:32.279 Whenever you see a circumstance like this, 00:01:32.279 --> 00:01:35.518 we call, we say that this set, 00:01:35.518 --> 00:01:39.083 the set of integers is closed under addition. 00:01:39.083 --> 00:01:40.282 Let me write that down. 00:01:40.282 --> 00:01:45.282 So this means that the set of integers, integers 00:01:47.405 --> 00:01:52.405 is closed under, under addition. 00:01:54.200 --> 00:01:56.265 So once again, you're going to hear this idea 00:01:56.265 --> 00:01:58.482 of closure, it's the general term 00:01:58.482 --> 00:02:00.340 that we're talking about is closure 00:02:00.340 --> 00:02:02.476 when you take some, especially in higher 00:02:02.476 --> 00:02:03.764 mathematics course but you might even hear it 00:02:03.764 --> 00:02:05.598 some of your high school courses 00:02:05.598 --> 00:02:08.478 and it might seem like this really advanced thing 00:02:08.478 --> 00:02:10.603 but all it's saying is you have a set, 00:02:10.603 --> 00:02:12.356 you have some operation that you can perform 00:02:12.356 --> 00:02:14.154 on members of the set and if every time 00:02:14.154 --> 00:02:16.023 you perform that operation on members of the set 00:02:16.023 --> 00:02:18.903 you get, you stay in the set, the result is 00:02:18.903 --> 00:02:22.317 a member of the set, then you say that the set is closed 00:02:22.317 --> 00:02:26.078 under that operation, the set is closed, in this case, 00:02:26.078 --> 00:02:27.934 the operation is addition, you say 00:02:27.934 --> 00:02:31.197 the set is closed under addition and we can ask ourselves 00:02:31.197 --> 00:02:32.636 that same question about other things. 00:02:32.636 --> 00:02:37.636 Is the set of, is the set of integers closed under, 00:02:41.517 --> 00:02:44.788 closed under, actually, let's take some examples. 00:02:44.788 --> 00:02:49.788 Is it closed under multiplication, multi-plication. 00:02:50.272 --> 00:02:53.418 And I encourage you to pause the video and think about it. 00:02:53.418 --> 00:02:56.738 And then think about, is it closed under division? 00:02:56.738 --> 00:03:01.738 Is it closed under division? Is it closed under subtraction? 00:03:02.114 --> 00:03:05.300 Is it closed under subtraction? 00:03:05.300 --> 00:03:05.908 I encourage you to think about 00:03:05.908 --> 00:03:07.568 all these different operations. 00:03:07.568 --> 00:03:09.119 Let's take it one by one. Multiplication. 00:03:09.119 --> 00:03:11.836 If you take two integers and you were to multiply it, 00:03:11.836 --> 00:03:14.552 you actually always will get another integer 00:03:14.552 --> 00:03:19.115 so the set of integers is closed under multiplication. 00:03:19.115 --> 00:03:20.554 Now, what about division? 00:03:20.554 --> 00:03:22.319 Well, it's very easy to prove a case, in which, 00:03:22.319 --> 00:03:26.370 I'm dividing two integers and not getting another integer. 00:03:26.370 --> 00:03:30.677 For example, I could say, let's just do a very simple one. 00:03:30.677 --> 00:03:34.382 One divided by two. One and two are both integers 00:03:34.382 --> 00:03:35.855 but what's that going to give me? 00:03:35.855 --> 00:03:39.918 That's going to give me 1/2 which is not, another integer. 00:03:39.918 --> 00:03:42.136 This is a rational number but not an integer so if we were 00:03:42.136 --> 00:03:44.318 to extend our sets right over here. 00:03:44.318 --> 00:03:46.036 All integers are also rational but 00:03:46.036 --> 00:03:48.885 all rationals are not integers necessarily 00:03:48.885 --> 00:03:52.344 so this is rational, rational numbers here. 00:03:52.344 --> 00:03:56.234 This case of one divided by two so that's one 00:03:56.234 --> 00:03:59.066 and this is two when we performed division, 00:03:59.066 --> 00:04:02.363 we went out of the set, so when we performed division, 00:04:02.363 --> 00:04:05.288 we out of that set to something that's not an integer 00:04:05.288 --> 00:04:08.051 but it is a rational number so the set of integers 00:04:08.051 --> 00:04:12.428 is not closed under division. Subtraction? Sure! 00:04:12.428 --> 00:04:13.925 And once again, I'm not going to rigorously 00:04:13.925 --> 00:04:16.108 prove it here but take any two integers, 00:04:16.108 --> 00:04:19.209 you indeed are going to get another, 00:04:19.209 --> 00:04:22.137 you are going to get another integer. 00:04:22.137 --> 00:04:24.320 So the whole idea here is to just give you 00:04:24.320 --> 00:04:27.930 a brief introduction to the notion of closure. 00:04:27.930 --> 00:04:31.100 The idea that if you have a set and if you perform 00:04:31.100 --> 00:04:33.119 an operation on members of that set 00:04:33.119 --> 00:04:36.033 and you always get a member of your original set 00:04:36.033 --> 00:04:40.375 then that set is closed under that operation. 00:04:40.375 --> 00:04:42.171 What matters is that the set that you're talking about 00:04:42.171 --> 00:04:43.949 in this case, we talked a lot about integers 00:04:43.949 --> 00:04:45.492 and we talked about addition, multiplication, 00:04:45.492 --> 00:04:46.966 division, subtraction but you could think 00:04:46.966 --> 00:04:50.042 about the set of polynomials and whether they are 00:04:50.042 --> 00:04:52.899 closed under addition, which they actually are 00:04:52.899 --> 00:04:55.778 or the set of rational numbers. 00:04:55.778 --> 00:04:59.284 Are they closed under exponentiation? 00:04:59.284 --> 00:05:00.979 There's all sorts of things we can think about 00:05:00.979 --> 00:05:02.268 but hopefully, this gives you a good sense 00:05:02.268 --> 00:05:05.019 of what we mean when we say closure 00:05:05.019 --> 00:05:08.003 or for a set being closed under some operation.