字幕列表 影片播放 列印英文字幕 What we're going to talk about today are sets of positive numbers, okay? And a set is basically just a collection of positive integers. And what we're going to do is try to add sets together. So let's say we have a set that's called A. And we'll write the notation, and I'll give and an example. This set is 2, 3, 7. Let's have another example, we'll call it B. And we'll say that B is 1, 5, 8, and 9. What we want to do is add these two sets together. And so what do I mean by that? What we're going to do is add each number in the set pairwise. So I'll say that A plus B is equal to 1 plus 2, 1 plus 3, 8 plus 3, plus 2, 9 plus 3, and 9 plus 7. RIght, we have a new set, and we would like to write out the sums so that we actually know what our set is. So let's say 12, and finallly 16. Well, we have some repeated numbers here, so we would like to write the set in it's actual form. So I just struck out the repeated numbers that we had. So, another thing worth noting here is that in our set A we had 3 members, in our set B had 4 members, and our set sum of A plus B, we had 9 members. And this is something that will come up again. We can think of a special type of set that I'll call arithmetic progression. So, arithmetic progression could be P equal 4, 8, 12, 16, 20. We add 4 each time to the number before it and we have a progression And it's called an arithmetic progression because we added the numbers. What we'll do is we'll add P to itself, and we'll see what happens. And something kind of interesting kind of happens. And so again, we have 4 plus 4, 4 plus 8, we have this large set of sums here, and in fact, the actual set will be much smaller. 8, 12, 16, 36 and 40. Brady: "That's the final?" That's the final set. Brady "It came down a lot, didn't it? Is it normal for sets to come down that much?" No, it's not normal. So if we have a set of integers, we add all of the sums, we have, you saw we had a lot of sums. In most cases, maybe there's not too much overlap. And in that case, we have something close to what we had originally. This whole large grid of sums. The nice thing about an arithmetic progression added to itself is that the size of the sum set will be less than two times the size of the original set. The sum set will never double the size of the original set. What you'll end up having is that the size of P plus P is always less than or equal to 2 times the size and that's special. We can do another example where we get pretty much the exact opposite. Our size of P plus P is actually closer to what we would expect with no overlap. So what we're gonna do next are geometric sequences. I'll call it G this time. And this set will be 2, 4, 8 and 16. This is 2 to the 1, 2 squared, 2 cubed, and this is 2 to the 4th. So I'll add G to itself, it's a geometric sequence to itself, and what's special about geometric sequences it that their pairwise sums we have some overlap. We have some overlap here. But we don't have overlap as much as we do in the case of the arithmetic progressions. So, the size of G, originally is size 4, the size of G plus G is now equal to 10. Brady: "So that's more than doubled." Yes. Definitely more than double, right? So now we have the size of G plus G greater than 2 times the size of G. And if we were to take a really large geometric sequence, we would actually see that G plus G is about G squared. The first thing is a result from Paul Erdős, and it's about, it's a little bit about sum sets. Okay. So let's say we take a set, 2, 3 and 7. We can define the notion of a set being 'sum free'. And what do I mean by 'sum free'? I mean that if you take any pair of numbers in your set, including the number itself, add it to itself, then the sum of that pair of numbers is not in the set. So let's look at A plus A. So this will be 4, 5, 9, 10 and 14. You see that no numbers are shared between these two sets, okay. So A is sum free. Brady: "Are sum free sets common, or are they like a special rarity? "Are the like prime numbers, or are sum free sets easy to find?" You would think they would be maybe uncommon. But in fact, they're actually more common than you would imagine. So let's say you have any set, okay? Any set of positive integers, what Paul Erdős said is that there exists a subset of that arbitrary set that you picked that is sum free, and that subset is greater than a third of the size of the original set. Yeah, I have a favorite, a fractal set, let's say, it's inside of an interval of real numbers. So let's say you have an interval of real numbers, 0 to 1, Brady: "So we're not dealing with integers any more?" We're not dealing with integers any more. We have a set of real numbers. What I would like to, I guess, construct, is something that's called the Cantor middle third set. It's an iterative process. In the first step of the Cantor middle third set is, you take the middle third out of the interval 0, 1, okay? So you find the middle third, 0, 1, and so this goes from one third to two thirds, and you erase that segment of the interval. What I have left is this set, okay? Brady: "Is it still infinitely big?" Yeah. I mean, if you just count the elements in the set, it's huge, it's uncountable. So now, the second step is to take the middle thirds out of the segments that we have left. So, we take a third of this set, and a third, and so we end up with something a little smaller You do it again, and you do it infinitely many times, I suppose. And the last, the limit set that you end up with is called the Cantor middle third set. Brady: "But you never get there. How would you ever get there?" Well, you approximate it, basically. You say that it's, you know, the thing that, if you get close enough, or you take an arbitrarily large step in our progression, that'll be the closest set to it. That'll be the limit. Brady: "How many numbers are in the Cantor middle third set?" Well, there are uncountably many, okay? And so, in some sense, you know, you can say that it's, we haven't taken the size down any, at any point. But there is a way to measure how much we've taken away. In general, we'll say that, the interval, I suppose, is just a line, and if we want to measure the dimension of the interval, we can say that dimension is 1, right? So the dimension, just like the dimension of this piece of paper should be well, it's not actually 2, right? Because it has some thickness, but in some fantasy world, it's 2, right? And we're in 3 dimensions, right? So the dimension of the line should just be 1, or the interval should just be 1, and there's some way to measure the dimension of this Cantor one third set. And, what, I'll just say it here, and I'll call it dimension, in quotations. And I'll say that the dimension of Cantor set, log 2 over log 3. 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