One Example is Enough

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来自 University of California San Diego 的课程

What is a Proof?

173 个评分

University of California San Diego

What is a Proof?

173 个评分

课程 1（共 5 门，Specialization Introduction to Discrete Mathematics for Computer Science）

Mathematical thinking is crucial in all areas of computer science: algorithms, bioinformatics, computer graphics, data science, machine learning, etc. In this course, we will learn the most important tools used in discrete mathematics: induction, recursion, logic, invariants, examples, optimality. We will use these tools to answer typical programming questions like: How can we be certain a solution exists? Am I sure my program computes the optimal answer? Do each of these objects meet the given requirements? In the course, we use a try-this-before-we-explain-everything approach: you will be solving many interactive (and mobile friendly) puzzles that were carefully designed to allow you to invent many of the important ideas and concepts yourself. Prerequisites: 1. We assume only basic math (e.g., we expect you to know what is a square or how to add fractions), common sense and curiosity. 2. Basic programming knowledge is necessary as some quizzes require programming in Python.

从本节课中

Making Convincing Arguments

Why some arguments are convincing and some are not? What makes an argument convincing? How to establish your argument in such a way that there is no possible room for doubt left? How mathematical thinking can help with this? In this week we will start digging into these questions. We will see how a small remark or a simple observation can turn a seemingly non-trivial question into an obvious one. Through various examples we will observe a parallel between constructing a rigorous argument and mathematical reasoning.

One Example is Enough3:26

Splitting an Octagon1:24

Making Fun in Real Life: Tensegrities10:25

Know Your Rights5:49

Nobody Can Win All The Time: Nonexisting Examples8:27

与讲师见面

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Vladimir Podolskii
Associate Professor
Computer Science Department

Now we return to the simplest kind of proof, which is called existence proof.

And this is the case when one example is enough.

So if somebody sends you somewhere for

a unicorn, they should explain you how this unicorn looks like.

Otherwise, you will not recognize it even if you see it.

The same with proofs, if somebody wants you to find a proof,

first you should understand what does the proof looks like.

And of course it depends on what you want to prove.

So we are now considering something, which is called existential statement.

The claim is that an object with given properties exists.

So this is the claim.

And to prove this claim, it's enough to show an example.

And what is important is one example is enough,

if you found an example, that's all.

Now of course you should check that it's really an example and

the properties are true.

But still one example is enough.

And let's see how it works on some simple geometric problem.

So this problem is about cutting things in two congruent pieces.

Congruent is just a mathematical word which means very simple things.

It's just the pieces of the same shape and the same size.

So they can be any place.

You can rotate them.

You can move them, but the shape and size should be the same.

And the task is to cut this figure, this figure made

of three squares, then this figure should be cut into two congruent pieces.

Again, probably, you see how to do this immediately

because the figure is symmetric, there is a axis of symmetry, like this.

So you can cut along the sections and you get two symmetric things.

So, we should have a prepared slide for this, yeah?

So these are two parts, this is part one and part two, and

they are congruent because you can reflect 1 and

make it like 2.

Okay, if you asked to cut this figure into three congruent pieces.

This but

what is interesting is to cut into 4 pieces.

Can you cut this figure into 4 congruent pieces?

So what do you think?

And I can tell you some hint.

The hint is that these pieces should be also of the same L-shape, but smaller.

So if you need four pieces, then each of them has area divided by four so

it should be twice smaller in both directions.

And so the thing like this.

So you should the big L shape into four pieces of the same shape, but smaller.

So, this is the last chance to invent the cutting yourself,

I will show it, and here it is.

Yeah, so you see these pieces, 1, 2, 3, 4, and they are all congruent, so

we are done.

We don't need any other example, actually I think it doesn't exist with

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