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.
Tutorial for Assignment 6
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From the course by Stanford University
Introduction to Mathematical Thinking
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From the lesson
Week 4
This week we complete our analysis of language, putting into place the linguistic apparatus that enabled, mathematicians in the 19th Century to develop a formal mathematical treatment of infinity, thereby finally putting Calculus onto a firm footing, three hundred years after its invention. (You do not need to know calculus for this course.) It is all about being precise and unambiguous. (But only where it counts. We are trying to extend our fruitfully-flexible human language and reasoning, not replace them with a rule-based straightjacket!)
Meet the Instructors
Dr. Keith DevlinCo-founder and Executive Director
H-STAR institute
Well for assignment 6 I'm just going to do the last 3 questions, number 7, 8 and 9.
I think you should be able to do the first 6 on your own and check it on your own or
with other students. So in number 7 we have to negate
statements and put them in positive form. Part A was for all x in N, there was a y
in N X plus Y equals 1. The negation for that is, and I think
you'll get the same answer as I do, if you get it right.
There is an X and N, such that for all Y and N, X plus Y is not equal to 1.
You, you could have written that last part as not X plus Y equals one if you like,
that's equally correct, I just choose this way of writing it.
Okay, and in this one, for "X" that's greater than zero, there's a "Y" that's
less than zero, "x" plus "y" equals zero, the negation of that, in positive form, is
an "X" greater than zero and it's greater than zero, okay.
For all "Y" less than zero, these don't change around.
They stay the same way, because these simply tell us what Xs and Ys we're
looking at. And, we get X plus Y not equal to 0 here.
Okay, for part C, there is an X, greater, there is an X, so epsilon greater than 0,
negative epsilon less than X, less than epsilon.
And when you negate that you get for all x epsilon greater than 0.
But in this case you going to have to sort of split this into 2.
One of 2 effects is not between negative epsilon and plus epsilon then either x is
less than or equal to negative epsilon or x is greater or equal to epsilon I didn't
put parenthesis around either of these,[UNKNOWN] I could have done if I
wanted to be ultra clear but this takes precedents over the logical operations,
arithmetic's expressions are, inequalities and so forth.
Take precedents over logical connectors. Because these are connectors.
They connect Statements about mathematics. And this is a statement about mathematics,
and that's a statement about mathematics. But if you wanted, you could have put
parentheses around that. And you could have put parentheses around
that. That would been, that would have been
another way of doing it. Moving onto the last part.
For all x and n, for all y and n, there a z and n .
X plus Y equals z squared, we're not talking about whether these or true or
not, we're just writing them down and negating them, so you got exists an X and
N, exists a Y and N, for all Z and N, and I've just written this again, X plus Y is
not equal to Z squared, you could simply put a negation sign in front of that
expression. Okay, well that's number seven.
It';s fairly straightforward, let's move on to number eight.
Question eight is our old friend Abraham Lincoln, and this famous statement of it,
or allegedly famous statement of his, and when it's negated[INAUDIBLE] that famous
sentence. So, let's let fxt mean you can fool person
p at time t. And then his statement is this one.
You can fool all of the people some of the time.
There were some times when you could fool everybody.
You can fool some of the people all of the time.
So, there are some people that can be fooled for all ti-, at all times, but you
cannot fool all of the people all of the time.
Okay, let's just mechanically go through and negate that.
Looking at the formulas then, and then we'll try to interpret, the answer and
express the answer in terms of English. Okay.
So, negate exists, it becomes for all when it for all, it becomes exists, FPT becomes
not FPT. Okay, conjunction becomes a disjunction.
Again, I'm not putting these in parantheses.
And I'm not going to do it here because this is a, this is a, is a whole that
quantifies binding the, the very tight binding.
And then we have this junction conjunction here and then this junction's here the,
their less tightly binding. So looking at this one, exist becomes for
all, for all becomes exist we get negation.
Conjunction becomes disjunction and the negation here just disappears.
So I put a positive statement. Okay so in terms of the formalism, this is
fairly routine. In fact, this is such a routine thing that
this is essentially algorithmic. I just went through and applied, the, the
patterns, the rules that I've observed happening.
The interesting part of this, I think, and you wont have to do it, but let's do this.
Let's see how we can express this, in English language.
And, We're not going to get something quite as, as nice as this, I don't think.
Because we need to try and be, to avoid ambiguity as much as possible.
So the first part would be the best I can think of at the moment is, let's see.
At anytime, there is someone you can't fool, and, oops or, because we've negated,
or okay. Let's have a look at this one.
I can't really think of anything better than to say the following: for every
person, you can't always fool them. Well, no.
I think if you try and swap it 'round. You run into sort of a American melanoma
type problem. So, to try and avoid that, I think I
would, I would want to write it this way. And then, the final clause.
But that's easy, of course, let's just say, you can fool all the people all the
time. Ok.
I'm, I'm moderately happy with that, you might say I have different opinions on the
best way to write that. [inaudible], But, number 2 is a bit ugly.
But I was, I'm not sure we can make a better job of that one.
Okay, in any case, we've, we've, we've, we've negated the thing.
And, and this was nice and clean. Okay, now, now, let's look, look at number
9. Well, number 9 involves one of the most,
important and most famous, some might say infamous formulas.
Of, advanced mathematics. Of university level mathematics.
It's this definition. When students entering math-, entering
university to study mathematics. Therefore, a math major.
It's this definition that, that usually causes them the most problems during their
first year. In fact I think it's this definition that
probably is responsible for more math majors giving up uh,mathematics in their
first year at university more than anything else.
It's a really tricky thing to understand as many of you've noticed, and I've seen
from discussions on the forum. Figuring this out is, is really hard.
This was hundreds of years of effort starting with the invention of the
calculus by Newton and Leibniz in the, in the 17th century.
It took a long time in several hundred years before mathematicians were able to
figure out the notion of continuity and come up with this definition.
This was late-19th century that this was done.
And it was it was a tricky thing. Negating it is relatively straight
forward, actually, because we've, we've got rules for doing them.
And so, when you negate it, what you get is that the for all becomes exists and
the, it's still a greater than, we're still talking about positive numbers,
epsilon. The exists becomes a for all, the for all
becomes an exists and, then there's a bunch of stuff that's in the bracket here.
This is a conditional, an implication, and so, when you negate it, you get the
antecedent conjoined with the negation of the consequent.
So, here's the antecedent. Again, I haven't put parentheses around
that because this is a, a mathematical statement, and That, that's take
precedence over the, over the conjuncts not just junction is I've just connected
there operators that bring things together.
So that's a piece on its own. That guy, conjoined with the negation of
this guy. Okay, the negation of it will be less than
F1. If it's originally F1.
Okay, so, so that's the negation and that's relatively mechanical to do that,
as long as you sort of pay attention to Preserving things that need to be
preserved. Changing for alls to exist and negating a
conditional. The interesting question is, what on earth
does this mean? You know you can read it through as I just
did for all epsilon, greater than 0, this is delta greater than 0, such that for all
x, yada, yada, yada. What does it, what does it mean?
Well, this is capturing in sym-, in, in, in a symbolic language.
In an aglebraic form-, formalism. It's capturing something geometric.
So, let's see what it's capturing. Let's look at the original definition of
continuity. It's about functions.
So let's look at a function this way. I'll draw a wavy line vertically, this is
a real line, and then I'm going to draw the real line vertically here.
So this is the real line, instead of writing horizontally as we usually do, I'm
going to write it horizontally. And the function, f, is going to take
Numbers here, to numbers here. Okay, so somewhere here we've got A.
F applies to that. And it gives me F of A.
Okay. Now we're trying to capture the notion of
continuity at a. That means when we go slightly to the left
or right of a, left or right is up and down the way I represent it, then the
numbers don't sort of have a discontinuity.
And the way to caption that it turns out and chances are very, very high that
you're not going to follow this the first time, it's going to take you weeks if you
need to, to master this. But here's what this formula says, it says
the epsilons are going to work on here, it says let's take an epsilon here and let's
look at F of a plus epsilon and f of a minus epsilon.
So I'm going to take an epsilon interval around f of a.
And what this definition says, is that, given an epsilon in 1 of these intervals,
I can find a delta. So here's a plus delta, and a minus delta.
So starting with an epsilon, which gives me an interval here, through any one of
those, I can find a delta, which gives me an interval here.
Search that. Now let's look at this.
Any x, in this region, because this says, that x is within delta of a.
So any x in this interval, gets sent to an image f of x, in here.
So it's saying, in order to make sure that all the values of the function Are in this
interval. I can find an interval around A, such that
they're all sent into here. So, if I want to hit the target, imagine
this is hitting the target, like throwing darts at a dart board.
If I want to hit the target within a specified accuracy of a, of f of a, I can
always do it by starting out within an interval of a.
So to get within a given interval around f of a, I can always find an interval around
a that does it. So everything from here gets sent into
here. And if you think about it long enough,
you'll realize that what that means is that the function is continuous at here,
there's no, no, no jumps. And to try and understand that, let's look
at what this guy means in terms of a diagram.
I'll do the same thing again, I'll draw the rail line.
And I'll draw the real line. And here's a, and here's f of a.
Now the negation says there is some epsilon.
In, in the previous case this was happening for all epsilons.
You could find a delta. In this case there's a fixed epsilon that
we can find. And we look at the interval around there,
F of A minus Epsilon, okay, what it says, is that there is an Epsilon, such that, no
matter what you take here, no matter which one you take here, here we found one of
these guys, we said Take any interval here, we could find and interval here.
Here we're saying the reason interval here for some epsilon such as no matter what
delta we take here, no matter how small you make this delta, no matter how close
you get here, that was a point that gets sent to that, you can always find a point
in here, that gets sent outside of there. Maybe sent that way, maybe sent that way.
So the sum epsilon here, that no matter what you do in here, no matter how close
you are to a, something gets sent out here.
In other words, there are points really, really, really close to a that get sent
outside this region. So there's no way that you can, you can
get all points in here. A gets sent to that, but arbitrarily close
to a There are points that get sent away. So there's discontinuity, because only a
gets sent close to a. When you get, well some other points
maybe, but no matter how close you get to a, you'll find points that are sent
further away. So that's a discontinuity.
Thing's jump, there's a sort of jump form there from there to thing here or here.
Now I, the chances are high, if it's the first time you've seen this, that
explanation is going to be hard to follow. This is really difficult to understand.
This is extremely, extremely hard. And, and the goal of this course actually
isn't for you to understand this, the goal of this course is to give you the
machinery so that you're now able to take a whole semester course on real analysis
that you're enduring, which you should be able to understand that.
So, I wouldn't worry too much if you can't understand this.
The point is if you could understand the first part and understand how to deal with
the formalism you now have the machinery you need to understand this is very deep
conceptual definition from within mathematics.
Okay, and so to have reached such stage in 4 weeks I think is pretty, is pretty
remarkable. The hard part is to come.
And for those of you that want to go on and study more mathematics sooner or later
and hopefully sooner you're going to need to master this definition and what it
means. Okey dokey?
Well as far as we're concerned we're done with this one.
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