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 5
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Introduction to Mathematical Thinking
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Week 3
This week we continue our analysis of language for use in mathematics. Remember, while the parts of language we are focusing have particular importance in mathematics, our main interest is in the analytic process itself: How do we formalize concepts from everyday life? Because the topics become more challenging, starting this week we have just one basic lecture cycle (Lecture -> Assignment -> Tutorial -> Problem Set -> Tutorial) each week. If you have not yet found one or more people to work with, please try to do so. It is so easy to misunderstand this material.
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Dr. Keith DevlinCo-founder and Executive Director
H-STAR institute
Here were my answers to assignment five, starting with question one.
The equation x cubed equals 27 has a natural number solution.
I got there is an x in the natural numbers such that x cubed equals 27.
For part b, a million is not the largest natural number What I said was there is
an x in n, there is a natural number x which is bigger than a million.
Part c, the natural number n is not prime, there is a natural number p and
there's a natural number q So it's that p is bigger than one, and q is bigger than
one, and n equals pq. So there are natural numbers pq, both of
them bigger than one, such that n is the product of p and q.
Notice that conjunction splits things in this way.
There's a natural order of precedence for the various relationships.
First of all, a relationship like equals, greater than, less than comes in, then
you conjoin things and disjoin them and do various other things.
And then next in line of precedence is actually conditional and biconditional,
and you read these formulas left to right.
There's a p in q, there's a q there's there's a p in n, there's a q in n, so
it's that p bigger than one and q bigger than one and n equals pq, so you read it
left right. This is why we made a big deal about that
the American Melanoma Foundation. Because in mathematics, you really have
to read things left to right, and the left to right ordering dictates the
logic. Okay?
Well, let's go on to question 2 now. Express the following goes for all the
sessions using symbols and words. Okay.
So the equation x q equals 28 does not have a natural number solution.
Or the most obvious way of expressing it using quantifier in, in, simple formulate
way. It's to say it is not the case that there
is a natural number x which satisfies the equation.
Whereas to give in terms for all assertions so this basically means
putting this in, in the sort of canonical form or positive form.
And to say it's not the case of existing x in N.
Is to say for all x in n, it's not the case that x cubed equals 28, which I'll
write in the familiar fashion x cubed is not equal to 28, okay?
I've could of written it. I've could of written the last part as
being not the case at x cube equals 28. Okay I could have done it that way.
They're both fine. Okay.
So now we've specified that there's no no natural number solution.
another way of saying it is every natural number fails to be a solution.
Okay, number, part b, zero is less than every natural number.
Let me just stress that the natural numbers do not include 0, they begin 1,
2, 3, et cetera. Historically zero was an unnatural
number, in fact originally zero wasn't a number it was just a, a zero symbol was a
circle that denoted there was nothing there, and you needed to denote that
there was nothing there when you had place value arithmetic.
So this came much later Roma Gupta in India 600 and something[UNKNOWN] was the
one who sort of came up with that. okay, I mean she was the one who wrote
about it and is today credited to have been to one that really sort of nailed
it. Okay, so zero is less than every natural
number so this is actually true. This is a true statement and here's a way
to capture it for all natural numbers x. Zero is less than x.
I could of written this as x greater than zero.
Alright which ever way I write it doesn't matter.
but I've written it. Since its expressed in terms of less than
I can use the formula for less than but I mean that still says the same thing.
okay, so I think there's anything more to say about that.
the natural number n is prime. This is really the negation of the
previous one of part c of question one where we looked at the formula that
captures the natural number n is not prime.
And so we're essentially going to negate that.
you can go back to the previous one and put a negation sign in the front and then
let the, let the negation ripple through as you sort of work your way in which
towards a positive statement. But let's just sort of jump straight in
and, and say it directly. To say that the natural number n is, is a
prime in terms of universal quantifiers. What you are going to say is if all
possible numbers that would divide into it, they can't on necessary equal to one
of the number itself okay, so here's a way of saying it whenever you try to
factor it so for all possible factors into two.
If you have a factor, if you have a factorization then one of the numbers is
wrong, and the other one is N. So for all possible ways of factoring the
number N, necessarily one of the factors is equal to one, for all Possible
factors, one of them has to be equal to 1.
Okay, that's it. So, those are the three answers that I
got for number 2. Question three, everybody loves somebody.
here's how I interpreted that. For all people x, for all persons x,
there is a person y. So it's just x loves y.
For every person, there is a person they love.
The person they love, will depend on the person you start with.
The y will be different perhaps, from one x to the next x.
Give me an x I'll find a y. Give me a pairs in x.
I can find a pairs in y. So should x loves y.
Different people again not a medical melanoma foundation example.
Which of course its not a problem in real life because everybody interprets the
right way. In fact some of the people in the forum.
In the forum discussion there was, there was quite a few people.
Who said they couldn't see the problem with the the little statement.
Okay this is why we're making a big deal of this because to get to the point where
you can understand how the ordering of quantifiers can be important, you have to
be able to see what was going on with that kind of example.
Okay, let's move on to part b. Everyone is tall or short.
What that says is for every person x, either x is tall or x is short.
Everybody is one or the other. actually this, the way this works this is
the inclusive or, but of course, the, the properties themselves will, will, will
make this a disjoint, this disjoint on. You, you want to have someone that is
both tall and short although in real life tall and short will have an overlap you
know there comes a point where is someone taller?
Is someone short? So the interpreting theses is is really a
matter of interpretation I mean the height, this is a fairly clear answer.
But when you try to map the ambiguous fuzzy real world into the precision of
mathematics there's some trade offs. That's the whole point.
The point is things like this are very ambiguous when this tall turns into
short. By writing something like this, we've
changed, we've taken something that's ambiguous and fuzzy and we've forced it
to be precise. And that's the whole point of what we're
trying to do. To get precision.
To get rid of the ambiguity and the fuzziness.
Doesn't mean to say that's gone from the real world, but it's gone from our
interpretation of the real world. So this is an interpretation within
mathematics if you like. With it's precision of something in the
real world, which is not precise. Okay, part c.
Everyone is tall or everyone is short, that actually comes out a different way.
That says every person x is tall, that's one part.
And the second part, everybody x is short.
And here we've got another of these precedence rules for logic in that
universal and quantifiers or the same as two existential quantifier we'll see.
These are very tight, they, they bind whatever comes in the the, comes next.
Which means you have to use parentheses if you want to use a quantifier that
binds everything like we did here. So to make this bind these two.
By the way, I'm talking about things like for all x binds, the thing that comes
next, the thing that's in the parentheses.
We use the word binds for quantifies. That means it governs that.
It, it's, it, it restricts the xs in there.
And the, this, the, the formal terminology of that is binds.
And so, in, in the binding rules quantifies bind everything that comes
next to them, and if you want them to bind a disjunction, you have to put them
in parentheses. So that has to become a unit to be bound
by through all, same would be true for exists.
In this case, you don't need the parentheses because all you've got is one
predicate here tall x, so you've got here for all x tall x or for all x short x you
might ask yourselves do you need there parentheses here, and the answer to you
in no, I've put them in to be clear. the, generally the rule for parentheses
is you put parentheses in when you need them to disambiguate, but because this is
the first time we're running through these, I mean x with parentheses.
the goal in all cases is avoid, avoid ambiguity, and if you if you haven't got
something ambiguous, you don't need the parentheses.
Alright? So notice that these are different,
though. this is talking about add everybody is
tall or everybody is short. This is highly unlikely to be to be true.
Except in a very strange sort of a society.
this one is certainly true if we're prepared to say whether where the tall
and short changes at sort of, I don't know five feet or whatever you want to
do, okay? Okay, let's move on.
That's enough for that one. Well for nobody at home I took it as a,
as a, as a sort of a universal quantification, if you like.
I took, I read this as for every person x, x is not at home.
you might have gone a different way, you might have said, it's not the case that
division x choose at home x, okay? That would be fine.
you may think that, you might think that this is actually a closer rendering of
this one. it depends whether you regard that as
some kind of universal quantifier or not. you know, so I'm interpreting that as
saying something about a sort of, a negative universal quantifier almost.
I'm saying that this for all x, it's not the case the x at all.
But you could equally well argue, I think, to say that that the most natural
one is to say there is no person who's at home.
Okay? in an, in natural language is that a sort
of a universal quantification, or is this an existent, a sort of a negation of an
existential quantification? depends how you interpret it, but both of
them are correct. These are equivalent.
As we've seen, these are equivalent assertions so it's just a matter of
choice as to which we think more accurately reflects the nuances of of
English language. Alrighty, part e.
If John comes, all the women will leave. If John comes then, so this is I think
very clearly an if statement, and then there's a conclusion, so this is the
antecedent, and then we've got a consequence.
So John comes, then it's the case that all the women will leave.
For any x, and our variable x, it ranges over people So I have to say for every x.
If x is a woman then x leaves. So that's our way of saying all women
leave. For all x if x is and I've put brackets
here. Because I want to make sure that the
universal quantifier applies to this. It's the same x, that if x is a woman,
then x leaves. All right.
Number f, part f, if a man comes, all the woman, all the women will leave.
In this case, it's not a single person John, it's any old man, so we'll have to
say, if there is an x, so here's the if part.
If there is an x, who is a man, and who comes, then we got the consequence that
every women leaves. This is the same as the previous thing.
So, this still says every woman leaves, but instead of saying John comes, I’m
saying there is an x who is a man and who comes.
Well, notice that number 4 isn’t about whether these things are true or false.
It’s simply about expressing them in, in a formal fashion.
in this case, using the quantifiers that range over the set of reels and the
natural numbers. This actually is a very common situation
when you're looking at real analysis, the theory behind the Calculus.
You have quantifiers that range over the real numbers.
and either the natural numbers or the integers.
the positive and negative numbers. The whole numbers.
so it's, but it's very common to have this kind of scenario.
Okay the equation x squared plus a equals 0 has a real root for any number a, for
any real number a. So here's how I, how I wrote it.
For all real numbers a there's a real number x which satisfies the equation.
Notice that here the quantifier Over a comes first, here it came at the end of
the sentence now this is because this is a very natural way of writing it in
English. When we said we have to be careful we
don't make that[INAUDIBLE] foundation mistake of getting quantifiers in wrong
order. In English, it's fine, because we, we
know how to read these things. In mathematics, it's crucial, absolutely
crucial, that the first thing that comes here is that for all a.
If the, if this guy comes first it's going to be wrong.
It's not going to capture it. So, when we take a sentence in English,
even when, if the quantifier comes at the end and here the quantifier.
For any real number a, this is a universal quantifier.
Even though that comes at the end of the sentence, which is fine in English,
wasn't fine for the American Melanoma Found, actually it was because everyone
reads that Melanoma example correctly, but it it, it causes it causes
mathematicans some amusement every time. But in this case, the order is absolutely
crucial And here's why. Because the x that we get depends on the
a. Okay?
In fact, for some a's, you're not going to get an x as we know.
I mean, this is only true if a is negative.
And we'll look at that case in a minute. So this isn't even true for all a's.
But even if you take the negative a's, the x that solves it depends on the a.
So you have to have an a before you can find the x.
One way of reading this is to say, if you give me any a, I will find an x that
solves the equation. You give me an a, I'll find an x,
depending on the a, that solves the equation.
Alrighty? Quantify order is crucial in this kind of
expression. part b actually brings it to the previous
one, but except we're really talking about for any negative real number, and
this is actually going to be true. So there's a quantifier for any real For
any negative real number and we're going to capture that as follows, we're
going to say because we don't have a set of negative real numbers, we've got the
set of real numbers so we have to say for all real numbers a, if it's the case that
a is negative then, so there's an if there.
And here's a then. Then it follows that there is an x that
satisfies the equation. So let me read that once more, one more
time. For any real number a, if it happens that
a is negative, then there is an x which solves the equation.
Or, in terms of you giving me a number, and me finding an answer, if you give me
an a Then providing the a that you give me as negative I can find an x such as
pass the equation but again notice what's here the x depends on the a.
You give me a different a few minutes later I can still find an x providing the
a as negative. But it'll be a different x.
This is why the order is important, the a has to come first, because the x depends
on the a. Now that's actually to here as well, but
English allows you to say them in the opposite order and still mean the same
thing, that's because English is a rich, natural language.
Mathematics is a precise formal language, which exists kind of formulaic
mathematics. This is very precise and formal and it
has to be for doing particular things in mathematics.
I mean, most mathematicians most of the time don't write things formally this
way. They use English.
They write things like this. This kind of expression is important in
certain crucial stages of mathematics. It' not that mathematicians argue using
precise language all the time, they don't.
They aren't using everyday language, but when it counts, they pull this kind of a
thing out because it's crucial the last topic.
And in fact, the very last week of the, of the lectures, we gave, when we look at
the last the last topic. We're going to make some very, some
historically, and actually contemporary important we're going to look at
something very crucial to mathematics which developed in the late 19th, early
20th century, middle 19th century. Where it was actually crucial in order to
advance in mathematics to make one or two key definitions with this kind of
decision. Okay.
Now issue of convergence, the sequences and the continuity of functions.
Okay, part c, every real number is rational, so, every real number x.
What are we trying to say? We're trying to say, for every, this is
got it's false, right? That's this again another example, I've
been looking at. This isn't about whether things are true
or false. I mean this one is false.
This one I like it's true. This one is false.
let me just mention these things, this one is at false.
This one's true, this one's false. Literally, this isn't what we're asked to
do, we're just, we're just looking at expressing things.
The point is, you can express things in mathematics precisely whether they're
true or false. And sometimes you have to express them
formally in order to determine whether they're true or false.
But it's a separate issue. True or falsisy, truth or falsity is a
sepaprate issue from whether you can express it formally.
Okay. Every real number is rational.
That really says that you look at any real number x, so this is the every real
number part. What you want to say is that real number
x can be expressed as a quotient of two integers.
So you would probably say m over n equals x.
Or, well, these are positive numbers. The x could be negative.
So, we've got to allow for the fact, minus m over n equals x.
So, that's to say m over n equals x, or Minus m over n equals x or the x could
even be 0. Okay, so you could have 0 equals x if
we're putting the x second. So 3 possibilities, it's equal to the
ratio of 2 natural numbers, or minus the ratio of 2 natural numbers or it's equal
to 0. Okay?
Oh, incidentally, I wrote it this way just because it's sort of neater.
I mean, the aesthetic the, I've been in mathematics a long time, and you develop
an aesthetic sense. And the sense I developed was it, it's
cleaner to just write it this way in part because, when we're talking about natural
numbers, in the natural numbers you have two operations essentially.
You've got plus and you've got times. You don't have negation and you, not
always, I mean you can sometimes subtract but one number has to be bigger than the
other. And you don't always have division.
You have a property of divisibility but you don't have division.
So, when you're dealing with just natural numbers all you can do is add them or
multiply them. Now here we've got real numbers flow,
thrown in as well. So there's nothing wrong with writing it
this way, but because m and n typically when they're on their own don't involve
division I just thought it was sort of nice and elegant, to avoid writing m over
n. Okay, because this actually take you out
of the, of the integers out of the natural numbers.
Then I mean this is, this is just a, this is just a setting.
This is not question of right or wrong its just a matter of a settings.
If you do it in this way that's absolutely fine, okay.
There was[INAUDIBLE] notice that the m and the N exist, depend on the x.
You give me an x, I will find an m and an n which satisfy that.
Or at least I would if that happened to be true.
It's not so I can't always find them. But the order of the quantifiers is
crucial here as well. If you get the order sum in the m and the
n it doesn't matter with those two. This is symmetrical.
But if that guy comes in here somewhere then it's not going to capture it.
Okay, well part d is just a, sort of like a negation of part c if you'd like.
Okay, and here's how I wrote it. There is an irrational number, so there
is a number with the following property. That for all pairs of natural numbers m
and n, m divided by n is not equal to x and negative m divides n is not equal to
x. Okay as before I have to look up both
possibilities, to allow for negative numbers.
Previously we had this junction here because we were talking about a positive
thing. And I really just negated what came
before. Notice that I'm not putting brackets
around this whole thing because I don't need them.
Because when you read left to right. There's only one way to read this because
of, of the, of, of the parenthesis I've got in.
There is an x in R. This is going to be the irrational number
we're asserting to exist. With the property that for all m in N and
for all n in N, Now I've got brackets in because this part Has to be bought
together. M is not equal to nx nor is m equal to
negative nx. M divided by n is not equal to x, nor is
minus m divided by n, equal to x, okay? So I suggest you compare this one with
part c that we just did to see how I've, I've just taken my answer to part c,
taken my negation, and made it an existing statement.
Part e well I mentioned when I formulated the question that this one looks quite
complicated, it does. I wish you could do, you do either one R
here. There are two ways of I, I entered these
two ways one of said well hm, what I could say is it, for all real numbers R,
for all real numbers y. There's a real number that's bigger than
y, which is not rational. So, given a real number, I can find a
bigger real number. So, given a real number You can find a
bigger real number which is not equal to the quotient of two integers m and n.
Notice that I didn't bother with the, the negative part here, because I'm going
bigger and bigger. And if you're trying to say that there's
no allowed irrational number you're really going, you, you're in the positive
range. You, you're going up into the positive
range. So what happens in the, to the left of
zero, on the, on the real line is irrelevant.
So, I don't need the negative part because I'm talking about there being no
largest one. But I'm saying that for given, given any
real number There's a number bigger than that.
Which is not the quotient of two integers.
And again because I'm now restricting myself to a natural numbers.
I think its cleaner to write it in this way.
But you don't have to because we have globally we're talking about real
numbers. In which means that we could in fact
here. Just write m over n not equal to x that
would be fine. It was fine before I'm just trying to
draw the, draw attention to the point that the natural numbers themselves only
have addition and multiplication, but we are after all talking about real numbers,
so, so there's no problem writing it this way.
wasn't a problem in the previous questions either.
Okay. You could, however, say that this is
really a statement about just the irrational numbers.
Given any irrational number, there's an even bigger irrational number.
Then it gets extremely complicated because you have to say, given any real
number r If that number is irrational then, there's an even bigger number
that's irrational. So the difference is, in this one, I've
simply said given any real number, there's a bigger number that's irrational
In the second I have said given any irrational number there is a bigger
number that's irrational now arguably the second one is a much closer
interpretation of what this means because you could say what this is really seen
is. It’s making a statement about
irrationals. So, you’re really only going to mention
irrationals. And that’s the second version.
Given any irrational there’s a big irrational.
So, this is a closer that’s a closer answer.
The first one is equivalent to it, because c rationals and e rationals make
up the bills, there is sort of in just best and if you can find any irrational
bigger than any real number, then you can find an irrational bigger than any
irrational number, that's a. Either are correct.
It's just a matter of, of how we interpret them, and exactly how much of
this statement you are capturing in the formalism.
Once you've got something formal, it's, it's, it's not ambiguous, so long as you
expressed it correctly with the parenthesis and so forth, it' not
ambiguous. It's absolutely precise This thing has
exactly one interpretation. This thing has exactly one
interpretation. That's the whole point.
This has several interpretations. I mean, here are two different ones.
They're equivalent. They're obviously equivalent, for very
obvious reasons. But we've cashed them out in different
ways. Well question five is our old friend
about domestic cars and fallen cars so C is a set of all cars, Dx means x is
domestic, Mx means x is badly made. All domestic cars are badly made for all
cars. If that car is domestic, then it's badly
made. For all cars, if the car is domestic,
then it's badly made. I think that's a fairly straightforward
one. I I wouldn't expect you to come up with
anything significantly different from that.
In fact I wouldn't expect you to come up with anything different from that at all.
But you might of done. Okay.
All foreign cars are badly made but we don't have a predicate for foreign so we
have to take foreign as meaning not domestic.
In which case we take the previous one. And we just replace the Dx by negative Dx
and so, but not Dx, so for all x and C the effects is not domestic then it's
badly made. And here I'm making use of the fact that
negation is a very tight binding, has a very tight binding negation like
quantifies applies to whatever comes next.
So if you want to negate a whole bunch of things.
You have to sort of join them together and put parenthesis around in order to
make sure that the negation applies to the whole thing.
So negation applies to whatever comes next.
And so if you wanted to apply to a bunch of things you have to put them in
parenthesis so just as arithmetic in algebra you have to use parenthesis to
disambiguate and the rules actually follow pretty close to the rules in
arithmetic in algebra and so once you realise how negation, conjunction,
disjunction. A conditional by conditional.
Once you realize how they correspond to to formulas and to the symbols of
mathematics. Or to the symbols of algebra and
arithmetic. Then you you should be able to follow
these kinds of formulation. Okay.
It just takes a little bit of getting used to.
Just in arithmetic and algebra. Part C.
All badly made cars are domestic. For all cars, if the car is badly made,
then it's domestic. And I think that one's straightforward.
There is a domestic car that is not badly made.
There is an x which is a car, which is domestic and is not badly made.
Notice by the way that when we are capturing universal quantifiers, the
statements that involve if-then, then we have A conditional, an implication.
For all x, if the k-, if it's, well for all x that are in the cars, for all cars,
if the car is badly made, then it's domestic.
In the case of an existential quantifier, if there is a car, we're saying there is
a car which is domestic and not badly made.
So, when we have universal quantifier, we typically will compare it with a
conditional, with an implication. When we have a x as existential quantify,
we will typically compare or compare it with a conjunction in case so.
To all go with an implication, exist go with a conjunction.
At least in these kinds of cases. the,the we, the trick, it's not a trick.
The rule we have to remember is to try and understand what these things say.
so it's never a good idea to look for symbolic rules that can lead you astray
if, if the formulation is a little bit different.
There is a foreign car that's badly made. There is an x in set of cars.
Which is foreign namely nondomestic and it's badly made.
And because negation binds tightly, I didn't bother putting parenthesis around
this. If I had wanted to say not Dx and m of x
I would have put parenthesis around there but the negation just combines what comes
next to it. And what comes next to it, it's not
domestic okay, it's foreign. All right?
Well that takes care of the foreign cars, let's move on to question six.
Well question six is about the same kind of things that we've already been looking
at in some of the earlier ones, namely whether things are rational whether
they're ordered in set and where, whether they're a biggest rationals or whatever,
but we've got a different set of restrictions.
Here we're asked to, to simply use quantifiers for real numbers logical
connectives, the order relation, but we do have a symbol q of x, meaning x is
rational. So we're going to get a different
expression. So the focus in what we're trying to make
precise has been, has been put elsewhere. Okay, we're allowed to assume what it is
to be rational that's not part of what we're trying to explicate.
So in this case we don't have to talk about things being in reals on it or
whatever because that's all we've got. So I don't have to say for all x and all
y I can say for all x and y. And I've, I've, left of parenthesis,
because there's no ambiguity, that just means for all x, for all[INAUDIBLE]
whatever comes next to it, so for all x and for all y.
Now I do have brackets because there's a bunch of things going to come next, that
depend on the 4x and 4y. So for all x and y If x is less than y,
then, there's a z which is rational, and lies between x and y.
So reading left to right, for all pairs of real numbers x and y, with x less than
y, There's a z, which is rational, and lies between them.
Now I didn't write exists z in Q, I didn't write that, because the Q isn't a
set. In this case, we've got a predicate
Meaning that x is rational. Okay, so I'm distinguishing, I mean, what
I'm doing with a lot of these examples, actually is distinguishing between the
various formulaism we have to express things within mathematics.
Sometimes we use sets, sometimes we use predecates.
Sometimes we'll use quantify restrictions.
Sometimes we use predicate restrictions. They're all different ways of getting
getting formality and precision into statements.
in, in mathematics, we, we pick the one that's that's most relevant.
In fact, in most mathematics, whenever we use these formulas which is in this
detail. this is much smaller thing to, to, to, to
things we have to write down in computer signs.
Look we're actually trying to unravel or to, to learn how to unravel the formal
structures beneath the statements we make.
Okay. Well that was number six, let's press on
to number seven. Okay, and number seven is this famous
quotation which is allegedly made by Abraham Lincoln, there is some So dispute
on, on websites as to who would have said this.
But it's it's commonly and popularly described to as Abraham Lincoln.
for our purpose, for our purpose, we're not worry about who said it.
and the issue is can we can we make it precise within the mathematical language.
You may fool all of the people some of the time, you can even fool some of the
people all of the time. Where you cannot fool all of the people
all of the time. The reason I like this example is that
the quantifiers in English are used in the way we use quantifiers in English.
the quantifiers come at the end of, of the clauses.
But in mathematics we read strictly left to right and so the quantifiers have to
come first. So let's say f, x,t mean you can fool
person Woops. P that should of been a P shouldn't it.
So let me just change that now. Okay.
Alright Fpt meaning you can fool person p at time t.
Then lets do these one by one. You may fool all of the people some of
the time. That means there are some times, when you
can fool all of the people. There are some times when you can, the
point is taking this clause, the people Depend on the time, there are sometimes
when you can fool all of the people, there was sometime when you can fool all
of the people. So we have to say, there were sometimes
when you can fool all of the people. Then there's a second clause, and this is
going to be a conjunction because these are different things we're saying so
there's an and if you'd like. There's a understated and here.
You may fool all the people some of the time and you can fool some of the people
all of the time, that's the second part. So, and there are some people, and
because quantifiers are these extreme of cases, we only have there exists and for
all, at least those are the, the common ones.
And so some, in, in, in, in, in the case when we use quantifiers, some is captured
simply by there is at least one. And strictly speaking, that's really the
logical heart of some. If we say something then strictly
speaking we are saying there is at least one.
Now arguably you can say in English, when you use the word some is an implication
that there's more than one. Okay well you could follow that but it
turns out than in mathematics It's really more, more, let's just say more
efficient, to just focus down on one. So there's a, there's a, a specialization
occurs in mathematics, that when we say some things we usually mean there's at
least one. And, and we can capture that by the
quantifier exists. You can fool some of the people all of
the time. That means there are some people, there's
at least 1 person, who can be fooled all of the time.
And in this clause, see what's been captured that's important.
I mean you can argue about. you know, whether, whether some is really
well captured by exists. but, but, but really what's going on here
is that the people that fooled can change from one time to another.
Okay? No, I said, well, we won't, didn't I?
What's really going on here, is that all of the time you'll be able to, so at, for
any time you can find some people. Oh, is that what I was saying?
Let me think about it for a minute. Some of the people, all of the time.
Yep. I think that's correct.
Okay? You can fool some of the people all of
the time. So you can find some people who can be
fooled all of the time. You can find some people Who can be
fooled all of the time. Okay?
I thought for a moment that, when I worked this out a little while ago, I got
it wrong. But now I think that's exactly what it
means. There are some people who can be fooled
for all of the time. Okay?
So I'm happy with that one. I'm happy with that one.
Let's look at the last one. but, the but's that's just another
conjunction really. You can not fool all of the people all of
the time. I think this one is the one that's
easiest because there's just to all's. It's not the case that you can fool all
people all time, and for this one you could swap these around.
You could have[UNKNOWN]. And for all p.
You've got, when you've got 2 universal quantifiers or 2 existential quantifiers
it really doesn't matter which order you put them in.
But when you have for all and exists it really does matter which way you, you
write them down. So let's just recap this says there are,
there are times. When you can fool all of the people, a
lot of times, when you can fool all of the people.
This, the first clause is about fooling all the people.
And it said a lot of time, when you can do that.
The second clause is different. It says the right people.
That you can fool all of the time. It's saying there are some people, who
can always be fooled. You can fool all people some of the time,
so there are some times when you can fool all people.
And you can fool some people all of the time, there are some people, who can be
fooled all of the time. But you cannot fo, fool all of the people
all of the time. Okay?
Well I stumbled a bit here, or at least I has second thoughts for a moment, but on
reflection I decided that I got it right the first time and and I'm happy with
those 3. So, it's I hope, I hope you manage to get
something similar to that. This is tricky This is tricky no doubt
about it. That's why its a good exercise.
It's a really good exercise in understanding how mathematical formulism
capture the kind of things we will. We actually do say in the real world.
And this is a real world statements. One with some culture significance.
Okay lets move on to number eight. Okay, let's do number 8 is, is really
very similar to the American Melanoma Foundation example.
Okay, a driver is involved in an accident every 6 seconds.
Okay, so we've got x is a variable to denote a driver, t a variable for a 6
second interval. x,t means that x is an accident during
interval t. The headline is written if we simply take
a literal translation from this into the form of this, this, this language we got
here we would say there is a driver. So for all times t, A is involved in an
accident times t. You know, six second interval.
Now okay, so that's a literal interpretation.
Then we're going to rewrite it so that it's it, so that the, the English
expression really captures it in a literal way.
And then we would say, every six seconds a driver is involved in an accident.
And in that case we're saying, for every six second interval, there is a driver
involved in an accident. Notice here that the driver can change.
From one interval, to another. So driver can change from one interval to
another. For all t, there's an x, x, t These are
different. These are very, very different.
And this makes a huge difference in mathematics.
you know, as I mentioned, a lot of you had trouble really seeing there was any
problem with the American Melanoma Foundation example.
Then of course in everyday language There isn't a problem because we use our
understanding of the, of the real world in order to[INAUDIBLE] .
But when you start to make those things precise for doing mathematics, in order
to develop a language and a formalism, a way of saying things that's totally
precise and reliable, then the order in which you say things makes a big, big
difference. And that was the whole point of that
exercise and then these other ones. Its to make sure that the left right
ordering, of the various formulas captures the logical flow.
And there's a logic to this. This one says, there is a driver who's in
an accident. Every six seconds, which is nonsensical.
This one says for every six seconds, there's a driver in an accident, and when
we take the formal language, then the distinction is, is, is really, really
significant. Okey dokey, well, I, I guess we finished
exer-, we finished assignment five. Okay, quite a lot of quite a lot of
questions in there. This is not easy to master but once
you've mastered it boy you can, you can tear through mathematical arguements at a
much greater rate. a lot of the difficulties people have
following mathematics is they bring to mathematics the sort of the inherent
sloppiness. and ambiguity.
and vagueness. Of everyday language.
when you do that you just run into trouble.
The whole points about about this formalism is if you limit it solely
ambiguity. Because in mathematics we very often
don't have knowledge of the everyday world to help ourselves out.
As we do when we use language in the everyday world.
Okay. How well, how did you guys do in
assignment five? Wasn't easy was it?
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