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.