Tutorial for Assignment 2

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Из курса от партнера Stanford University

Introduction to Mathematical Thinking

1436 оценки

Stanford University

Introduction to Mathematical Thinking

1436 оценки

Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of years. Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.

Из урока

Week 1

START with the Welcome lecture. It explains what this course is about. (It comes with a short Background Reading assignment, to read before you start the course, and a Reading Supplement on Set Theory for use later in the course, both in downloadable PDF format.) This initial orientation lecture is important, since this course is probably not like any math course you have taken before – even if in places it might look like one! AFTER THAT, Lecture 1 prepares the groundwork for the course; then in Lecture 2 we dive into the first topic. This may all look like easy stuff, but tens of thousands of former students found they had trouble later by skipping through Week 1 too quickly! Be warned. If possible, form or join a study group and discuss everything with them. BY THE WAY, the time estimates for watching the video lectures are machine generated, based on the video length. Expect to spend a lot longer going through the lectures sufficiently well to understand the material. The time estimates for completing the weekly Problem Sets (Quiz format) are a bit more reliable, but even they are just a guideline. You may find yourself taking a lot longer.

Tutorial for Assignment 12:30

Tutorial for Assignment 29:21

Tutorial for Problem Set 19:41

Познакомьтесь с преподавателями

Dr. Keith Devlin
Co-founder and Executive Director
H-STAR institute

Okay, let's see what we've got.

Well number one has been done for you.

Number two, the simple way to write that is to

say 7 less than or equal to p less than 12.

The next one we would write as 5

less than x less than seven.

The next one, well if x, let me see,

if x is less than 4 then it's automatically less than 6.

So the second conjunct here is superfluous,

we could just write that as x less than 4.

What about the next one?

Well y squared less than 9 means, let me see,

negative 3 is less than y is less than 3.

That's what the second conjunct means.

But if y satisfies this condition then certainly y is less than 4,

so the first conjunct is superfluous.

And the only one that counts is the second one.

So we could simply write that as y squared less than 9.

Okay that was just our thinking on the way.

What about this one?

Well x is greater than or equal to 0 and it's less or equal to 0,

there's only one possibility and that's x equal 0.

Well for number three,

in order to show that the conjunction is true,

what you would do is show that all of phi 1,

phi 2, etc., up to phi n are true.

And for number four, to show that a conjunction is false, is you show that

1 of phi 1, phi 2,

phi n is false.

So you have to find one of these, at least one of these, which is false.

And then it follows that the conjunction is false.

Okay, well that's that one.

Okay well with part a,if pi is bigger then 10, then it's automatically bigger than 3.

So in terms of the disjunct, pi bigger than 3 dominates.

That says more, if we draw a picture we've

got 0 we’ve got 3 we've got 10.

And the first one says that x is to the right of this point.

And the second disjunct is at x is to the right of that point.

And the disjunction will be true, if at least one of them's true.

And at least one of them will be true, if we start at 3.

First of all, only the first disjunct is correct, and

then when we get beyond this point, both disjuncts are correct.

So it's the first one that works.

This is that's either x is less than 0 or x is greater than 0.

So the simplest way to write that is to say that x is not equal to 0.

So the less 0, which is negative, or it's bigger than 0, it's positive.

This says that x equals 0 or x is greater than 0 the standard abbreviation for

that is x greater than or equal to 0.

And once you've got x greater than or equal to 0,

that's going to dominate over the first disjunct in terms of a disjunction.

So you're going to have x greater than or equal to 0.

That makes a more general claim of the two.

And for this one, let me make a note that x squared greater than 9,

this part means that either x is greater than 3 or

x is less than negative 3.

Well, one of these two disjuncts is the one that's here.

So, this one is superfluous.

So, that's the one that counts.

So, x squared greater than 9, is a simple way of saying it.

So again, as with the previous case, as with numbers one and two,

as I was going through them I actually articulated what these things are.

These were really just to prompt you to thinking about how you will

express them in English.

Okay, well that's that one.

Well in number seven, just show that if disjunction is true,

you have to show that one of them is true, at least one of them.

And for number 8 to show that it's false.

You have to show that all of phi 1,

phi 2, etc., to phi n are false.

Okay, well that's number seven and eight taken care of.

Turning to number nine now.

To say that it's not the case that pi is greater than

3.2 is to say that pi is less than or equal to 3.2.

When you negate a strict greater than you get less than or equal to.

To say that it's not the case that x is negative

is to say that x is greater than or equal to 0.

These are real numbers, from the context of the expression

we can assume that these are talking about real numbers.

And for real number, every real number has a square which is

strictly positive with one exception and that exception is 0.

So the only real number for

which it's not the case that x squared is strictly positive is x equal 0.

The standard abbreviation for not x equals 1 is x not equal to 1.

And when you negate a negation it takes you back to the original statement.

And again as I went through them I essentially answered number ten.

We're returning to question 11 now, here are the answers that I get.

Dollar and Yuan both strong.

Okay, Dollar strong and Yuan strong.

I think that one's fairly straightforward.

What about this one?

Well, with this word, despite, and but,

and in our mind, those are just nuanced forms of conjunction.

They both mean and, so, we put the Yuan weak and

there's a trade agreement, and the Dollar strong.

So the despite and the but, they sort of get at whether these things are contrary

to our expectations or consistent with our expectations.

But they still say that all three of these things hold together.

Okay, this one I think is fairly straightforward.

They can't both be strong at the same time.

So it's not the case.

The Dollar strong and the Yuan strong.

For part d, well, if that doesn't prevent that,

it means that happens and both of those fail.

So the trade agreement signed but the Dollar falls, and the Yuan falls.

The trade agreement holding doesn't prevent

the fall in the Dollar and the Yuan.

That means, the Dollar and the Yuan do fall, okay.

And then, for the last one, US-China trade agreement fails but

both currencies remain strong.

Again the but I think is, and it's conjunction so

we've got the trade agreement fairly,

currencies both remain strong, the Dollar strong and the Yuan strong.

Okay, those are my answers I would hope you get more or

less the same I mean you might end up writing things slightly differently.

But even though we had to think a little bit about what these words mean,

to my mind, there's no real dispute as to whether these are correct or not.

Okay, and how did you do on that one?

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