## 字幕列表 影片播放

• 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.

• Numberphile is brought to you by the Mathematical Sciences Research Institute, MSRI.

• That's the building behind me there.

• This is a place where many of the world's top mathematicians come together, for sometimes

• a semester at a time, cracking some of the hardest problems in mathematics.

• If you'd like to find out more, I've put some links in the description under the video.

What we're going to talk about today are sets of positive numbers, okay?

B1 中級

# 讓我們來談談套裝 - Numberphile(數字愛好者) (Let's Talk About Sets - Numberphile)

• 0 0
林宜悉 發佈於 2021 年 01 月 14 日