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Here were my answers to assignment five, starting with question one.

Â The equation x cubed equals 27 has a natural number solution.

Â I got there is an x in the natural numbers such that x cubed equals 27.

Â For part b, a million is not the largest natural number What I said was there is

Â an x in n, there is a natural number x which is bigger than a million.

Â Part c, the natural number n is not prime, there is a natural number p and

Â there's a natural number q So it's that p is bigger than one, and q is bigger than

Â one, and n equals pq. So there are natural numbers pq, both of

Â them bigger than one, such that n is the product of p and q.

Â Notice that conjunction splits things in this way.

Â There's a natural order of precedence for the various relationships.

Â First of all, a relationship like equals, greater than, less than comes in, then

Â you conjoin things and disjoin them and do various other things.

Â And then next in line of precedence is actually conditional and biconditional,

Â and you read these formulas left to right.

Â 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

Â it's that p bigger than one and q bigger than one and n equals pq, so you read it

Â left right. This is why we made a big deal about that

Â the American Melanoma Foundation. Because in mathematics, you really have

Â to read things left to right, and the left to right ordering dictates the

Â logic. Okay?

Â Well, let's go on to question 2 now. Express the following goes for all the

Â sessions using symbols and words. Okay.

Â So the equation x q equals 28 does not have a natural number solution.

Â Or the most obvious way of expressing it using quantifier in, in, simple formulate

Â way. It's to say it is not the case that there

Â is a natural number x which satisfies the equation.

Â Whereas to give in terms for all assertions so this basically means

Â putting this in, in the sort of canonical form or positive form.

Â And to say it's not the case of existing x in N.

Â Is to say for all x in n, it's not the case that x cubed equals 28, which I'll

Â write in the familiar fashion x cubed is not equal to 28, okay?

Â I've could of written it. I've could of written the last part as

Â being not the case at x cube equals 28. Okay I could have done it that way.

Â They're both fine. Okay.

Â So now we've specified that there's no no natural number solution.

Â another way of saying it is every natural number fails to be a solution.

Â Okay, number, part b, zero is less than every natural number.

Â Let me just stress that the natural numbers do not include 0, they begin 1,

Â 2, 3, et cetera. Historically zero was an unnatural

Â number, in fact originally zero wasn't a number it was just a, a zero symbol was a

Â circle that denoted there was nothing there, and you needed to denote that

Â there was nothing there when you had place value arithmetic.

Â So this came much later Roma Gupta in India 600 and something[UNKNOWN] was the

Â one who sort of came up with that. okay, I mean she was the one who wrote

Â about it and is today credited to have been to one that really sort of nailed

Â it. Okay, so zero is less than every natural

Â number so this is actually true. This is a true statement and here's a way

Â to capture it for all natural numbers x. Zero is less than x.

Â I could of written this as x greater than zero.

Â Alright which ever way I write it doesn't matter.

Â but I've written it. Since its expressed in terms of less than

Â I can use the formula for less than but I mean that still says the same thing.

Â okay, so I think there's anything more to say about that.

Â the natural number n is prime. This is really the negation of the

Â previous one of part c of question one where we looked at the formula that

Â captures the natural number n is not prime.

Â And so we're essentially going to negate that.

Â you can go back to the previous one and put a negation sign in the front and then

Â let the, let the negation ripple through as you sort of work your way in which

Â towards a positive statement. But let's just sort of jump straight in

Â and, and say it directly. To say that the natural number n is, is a

Â prime in terms of universal quantifiers. What you are going to say is if all

Â possible numbers that would divide into it, they can't on necessary equal to one

Â of the number itself okay, so here's a way of saying it whenever you try to

Â factor it so for all possible factors into two.

Â If you have a factor, if you have a factorization then one of the numbers is

Â wrong, and the other one is N. So for all possible ways of factoring the

Â number N, necessarily one of the factors is equal to one, for all Possible

Â factors, one of them has to be equal to 1.

Â Okay, that's it. So, those are the three answers that I

Â got for number 2. Question three, everybody loves somebody.

Â here's how I interpreted that. For all people x, for all persons x,

Â there is a person y. So it's just x loves y.

Â For every person, there is a person they love.

Â The person they love, will depend on the person you start with.

Â The y will be different perhaps, from one x to the next x.

Â Give me an x I'll find a y. Give me a pairs in x.

Â I can find a pairs in y. So should x loves y.

Â Different people again not a medical melanoma foundation example.

Â Which of course its not a problem in real life because everybody interprets the

Â right way. In fact some of the people in the forum.

Â In the forum discussion there was, there was quite a few people.

Â Who said they couldn't see the problem with the the little statement.

Â Okay this is why we're making a big deal of this because to get to the point where

Â you can understand how the ordering of quantifiers can be important, you have to

Â be able to see what was going on with that kind of example.

Â Okay, let's move on to part b. Everyone is tall or short.

Â What that says is for every person x, either x is tall or x is short.

Â Everybody is one or the other. actually this, the way this works this is

Â the inclusive or, but of course, the, the properties themselves will, will, will

Â make this a disjoint, this disjoint on. You, you want to have someone that is

Â both tall and short although in real life tall and short will have an overlap you

Â know there comes a point where is someone taller?

Â Is someone short? So the interpreting theses is is really a

Â matter of interpretation I mean the height, this is a fairly clear answer.

Â But when you try to map the ambiguous fuzzy real world into the precision of

Â mathematics there's some trade offs. That's the whole point.

Â The point is things like this are very ambiguous when this tall turns into

Â short. By writing something like this, we've

Â changed, we've taken something that's ambiguous and fuzzy and we've forced it

Â to be precise. And that's the whole point of what we're

Â trying to do. To get precision.

Â To get rid of the ambiguity and the fuzziness.

Â Doesn't mean to say that's gone from the real world, but it's gone from our

Â interpretation of the real world. So this is an interpretation within

Â mathematics if you like. With it's precision of something in the

Â real world, which is not precise. Okay, part c.

Â Everyone is tall or everyone is short, that actually comes out a different way.

Â That says every person x is tall, that's one part.

Â And the second part, everybody x is short.

Â And here we've got another of these precedence rules for logic in that

Â universal and quantifiers or the same as two existential quantifier we'll see.

Â These are very tight, they, they bind whatever comes in the the, comes next.

Â Which means you have to use parentheses if you want to use a quantifier that

Â binds everything like we did here. So to make this bind these two.

Â By the way, I'm talking about things like for all x binds, the thing that comes

Â next, the thing that's in the parentheses.

Â We use the word binds for quantifies. That means it governs that.

Â It, it's, it, it restricts the xs in there.

Â And the, this, the, the formal terminology of that is binds.

Â And so, in, in the binding rules quantifies bind everything that comes

Â next to them, and if you want them to bind a disjunction, you have to put them

Â in parentheses. So that has to become a unit to be bound

Â by through all, same would be true for exists.

Â In this case, you don't need the parentheses because all you've got is one

Â predicate here tall x, so you've got here for all x tall x or for all x short x you

Â might ask yourselves do you need there parentheses here, and the answer to you

Â in no, I've put them in to be clear. the, generally the rule for parentheses

Â is you put parentheses in when you need them to disambiguate, but because this is

Â the first time we're running through these, I mean x with parentheses.

Â the goal in all cases is avoid, avoid ambiguity, and if you if you haven't got

Â something ambiguous, you don't need the parentheses.

Â Alright? So notice that these are different,

Â though. this is talking about add everybody is

Â tall or everybody is short. This is highly unlikely to be to be true.

Â Except in a very strange sort of a society.

Â this one is certainly true if we're prepared to say whether where the tall

Â and short changes at sort of, I don't know five feet or whatever you want to

Â do, okay? Okay, let's move on.

Â That's enough for that one. Well for nobody at home I took it as a,

Â as a, as a sort of a universal quantification, if you like.

Â I took, I read this as for every person x, x is not at home.

Â you might have gone a different way, you might have said, it's not the case that

Â division x choose at home x, okay? That would be fine.

Â you may think that, you might think that this is actually a closer rendering of

Â this one. it depends whether you regard that as

Â some kind of universal quantifier or not. you know, so I'm interpreting that as

Â saying something about a sort of, a negative universal quantifier almost.

Â I'm saying that this for all x, it's not the case the x at all.

Â But you could equally well argue, I think, to say that that the most natural

Â one is to say there is no person who's at home.

Â Okay? in an, in natural language is that a sort

Â of a universal quantification, or is this an existent, a sort of a negation of an

Â existential quantification? depends how you interpret it, but both of

Â them are correct. These are equivalent.

Â As we've seen, these are equivalent assertions so it's just a matter of

Â choice as to which we think more accurately reflects the nuances of of

Â English language. Alrighty, part e.

Â If John comes, all the women will leave. If John comes then, so this is I think

Â very clearly an if statement, and then there's a conclusion, so this is the

Â antecedent, and then we've got a consequence.

Â So John comes, then it's the case that all the women will leave.

Â For any x, and our variable x, it ranges over people So I have to say for every x.

Â If x is a woman then x leaves. So that's our way of saying all women

Â leave. For all x if x is and I've put brackets

Â here. Because I want to make sure that the

Â universal quantifier applies to this. It's the same x, that if x is a woman,

Â then x leaves. All right.

Â Number f, part f, if a man comes, all the woman, all the women will leave.

Â In this case, it's not a single person John, it's any old man, so we'll have to

Â say, if there is an x, so here's the if part.

Â If there is an x, who is a man, and who comes, then we got the consequence that

Â every women leaves. This is the same as the previous thing.

Â So, this still says every woman leaves, but instead of saying John comes, Iâ€™m

Â saying there is an x who is a man and who comes.

Â Well, notice that number 4 isnâ€™t about whether these things are true or false.

Â Itâ€™s simply about expressing them in, in a formal fashion.

Â in this case, using the quantifiers that range over the set of reels and the

Â natural numbers. This actually is a very common situation

Â when you're looking at real analysis, the theory behind the Calculus.

Â You have quantifiers that range over the real numbers.

Â and either the natural numbers or the integers.

Â the positive and negative numbers. The whole numbers.

Â so it's, but it's very common to have this kind of scenario.

Â Okay the equation x squared plus a equals 0 has a real root for any number a, for

Â any real number a. So here's how I, how I wrote it.

Â For all real numbers a there's a real number x which satisfies the equation.

Â Notice that here the quantifier Over a comes first, here it came at the end of

Â the sentence now this is because this is a very natural way of writing it in

Â English. When we said we have to be careful we

Â don't make that[INAUDIBLE] foundation mistake of getting quantifiers in wrong

Â order. In English, it's fine, because we, we

Â know how to read these things. In mathematics, it's crucial, absolutely

Â crucial, that the first thing that comes here is that for all a.

Â If the, if this guy comes first it's going to be wrong.

Â It's not going to capture it. So, when we take a sentence in English,

Â even when, if the quantifier comes at the end and here the quantifier.

Â For any real number a, this is a universal quantifier.

Â Even though that comes at the end of the sentence, which is fine in English,

Â wasn't fine for the American Melanoma Found, actually it was because everyone

Â reads that Melanoma example correctly, but it it, it causes it causes

Â mathematicans some amusement every time. But in this case, the order is absolutely

Â crucial And here's why. Because the x that we get depends on the

Â a. Okay?

Â In fact, for some a's, you're not going to get an x as we know.

Â I mean, this is only true if a is negative.

Â And we'll look at that case in a minute. So this isn't even true for all a's.

Â But even if you take the negative a's, the x that solves it depends on the a.

Â So you have to have an a before you can find the x.

Â One way of reading this is to say, if you give me any a, I will find an x that

Â solves the equation. You give me an a, I'll find an x,

Â depending on the a, that solves the equation.

Â Alrighty? Quantify order is crucial in this kind of

Â expression. part b actually brings it to the previous

Â one, but except we're really talking about for any negative real number, and

Â this is actually going to be true. So there's a quantifier for any real For

Â any negative real number and we're going to capture that as follows, we're

Â going to say because we don't have a set of negative real numbers, we've got the

Â set of real numbers so we have to say for all real numbers a, if it's the case that

Â a is negative then, so there's an if there.

Â And here's a then. Then it follows that there is an x that

Â satisfies the equation. So let me read that once more, one more

Â time. For any real number a, if it happens that

Â a is negative, then there is an x which solves the equation.

Â Or, in terms of you giving me a number, and me finding an answer, if you give me

Â an a Then providing the a that you give me as negative I can find an x such as

Â pass the equation but again notice what's here the x depends on the a.

Â You give me a different a few minutes later I can still find an x providing the

Â a as negative. But it'll be a different x.

Â This is why the order is important, the a has to come first, because the x depends

Â on the a. Now that's actually to here as well, but

Â English allows you to say them in the opposite order and still mean the same

Â thing, that's because English is a rich, natural language.

Â Mathematics is a precise formal language, which exists kind of formulaic

Â mathematics. This is very precise and formal and it

Â has to be for doing particular things in mathematics.

Â I mean, most mathematicians most of the time don't write things formally this

Â way. They use English.

Â They write things like this. This kind of expression is important in

Â certain crucial stages of mathematics. It' not that mathematicians argue using

Â precise language all the time, they don't.

Â They aren't using everyday language, but when it counts, they pull this kind of a

Â thing out because it's crucial the last topic.

Â And in fact, the very last week of the, of the lectures, we gave, when we look at

Â the last the last topic. We're going to make some very, some

Â historically, and actually contemporary important we're going to look at

Â something very crucial to mathematics which developed in the late 19th, early

Â 20th century, middle 19th century. Where it was actually crucial in order to

Â advance in mathematics to make one or two key definitions with this kind of

Â decision. Okay.

Â Now issue of convergence, the sequences and the continuity of functions.

Â Okay, part c, every real number is rational, so, every real number x.

Â What are we trying to say? We're trying to say, for every, this is

Â got it's false, right? That's this again another example, I've

Â been looking at. This isn't about whether things are true

Â or false. I mean this one is false.

Â This one I like it's true. This one is false.

Â let me just mention these things, this one is at false.

Â This one's true, this one's false. Literally, this isn't what we're asked to

Â do, we're just, we're just looking at expressing things.

Â The point is, you can express things in mathematics precisely whether they're

Â true or false. And sometimes you have to express them

Â formally in order to determine whether they're true or false.

Â But it's a separate issue. True or falsisy, truth or falsity is a

Â sepaprate issue from whether you can express it formally.

Â Okay. Every real number is rational.

Â That really says that you look at any real number x, so this is the every real

Â number part. What you want to say is that real number

Â x can be expressed as a quotient of two integers.

Â So you would probably say m over n equals x.

Â Or, well, these are positive numbers. The x could be negative.

Â So, we've got to allow for the fact, minus m over n equals x.

Â So, that's to say m over n equals x, or Minus m over n equals x or the x could

Â even be 0. Okay, so you could have 0 equals x if

Â we're putting the x second. So 3 possibilities, it's equal to the

Â ratio of 2 natural numbers, or minus the ratio of 2 natural numbers or it's equal

Â to 0. Okay?

Â Oh, incidentally, I wrote it this way just because it's sort of neater.

Â I mean, the aesthetic the, I've been in mathematics a long time, and you develop

Â an aesthetic sense. And the sense I developed was it, it's

Â cleaner to just write it this way in part because, when we're talking about natural

Â numbers, in the natural numbers you have two operations essentially.

Â You've got plus and you've got times. You don't have negation and you, not

Â always, I mean you can sometimes subtract but one number has to be bigger than the

Â other. And you don't always have division.

Â You have a property of divisibility but you don't have division.

Â So, when you're dealing with just natural numbers all you can do is add them or

Â multiply them. Now here we've got real numbers flow,

Â thrown in as well. So there's nothing wrong with writing it

Â this way, but because m and n typically when they're on their own don't involve

Â division I just thought it was sort of nice and elegant, to avoid writing m over

Â n. Okay, because this actually take you out

Â of the, of the integers out of the natural numbers.

Â Then I mean this is, this is just a, this is just a setting.

Â This is not question of right or wrong its just a matter of a settings.

Â If you do it in this way that's absolutely fine, okay.

Â There was[INAUDIBLE] notice that the m and the N exist, depend on the x.

Â You give me an x, I will find an m and an n which satisfy that.

Â Or at least I would if that happened to be true.

Â It's not so I can't always find them. But the order of the quantifiers is

Â crucial here as well. If you get the order sum in the m and the

Â n it doesn't matter with those two. This is symmetrical.

Â But if that guy comes in here somewhere then it's not going to capture it.

Â Okay, well part d is just a, sort of like a negation of part c if you'd like.

Â Okay, and here's how I wrote it. There is an irrational number, so there

Â is a number with the following property. That for all pairs of natural numbers m

Â and n, m divided by n is not equal to x and negative m divides n is not equal to

Â x. Okay as before I have to look up both

Â possibilities, to allow for negative numbers.

Â Previously we had this junction here because we were talking about a positive

Â thing. And I really just negated what came

Â before. Notice that I'm not putting brackets

Â around this whole thing because I don't need them.

Â Because when you read left to right. There's only one way to read this because

Â of, of the, of, of the parenthesis I've got in.

Â There is an x in R. This is going to be the irrational number

Â we're asserting to exist. With the property that for all m in N and

Â for all n in N, Now I've got brackets in because this part Has to be bought

Â together. M is not equal to nx nor is m equal to

Â negative nx. M divided by n is not equal to x, nor is

Â minus m divided by n, equal to x, okay? So I suggest you compare this one with

Â part c that we just did to see how I've, I've just taken my answer to part c,

Â taken my negation, and made it an existing statement.

Â Part e well I mentioned when I formulated the question that this one looks quite

Â complicated, it does. I wish you could do, you do either one R

Â here. There are two ways of I, I entered these

Â two ways one of said well hm, what I could say is it, for all real numbers R,

Â for all real numbers y. There's a real number that's bigger than

Â y, which is not rational. So, given a real number, I can find a

Â bigger real number. So, given a real number You can find a

Â bigger real number which is not equal to the quotient of two integers m and n.

Â Notice that I didn't bother with the, the negative part here, because I'm going

Â bigger and bigger. And if you're trying to say that there's

Â no allowed irrational number you're really going, you, you're in the positive

Â range. You, you're going up into the positive

Â range. So what happens in the, to the left of

Â zero, on the, on the real line is irrelevant.

Â So, I don't need the negative part because I'm talking about there being no

Â largest one. But I'm saying that for given, given any

Â real number There's a number bigger than that.

Â Which is not the quotient of two integers.

Â And again because I'm now restricting myself to a natural numbers.

Â I think its cleaner to write it in this way.

Â But you don't have to because we have globally we're talking about real

Â numbers. In which means that we could in fact

Â here. Just write m over n not equal to x that

Â would be fine. It was fine before I'm just trying to

Â draw the, draw attention to the point that the natural numbers themselves only

Â have addition and multiplication, but we are after all talking about real numbers,

Â so, so there's no problem writing it this way.

Â wasn't a problem in the previous questions either.

Â Okay. You could, however, say that this is

Â really a statement about just the irrational numbers.

Â Given any irrational number, there's an even bigger irrational number.

Â Then it gets extremely complicated because you have to say, given any real

Â number r If that number is irrational then, there's an even bigger number

Â that's irrational. So the difference is, in this one, I've

Â simply said given any real number, there's a bigger number that's irrational

Â In the second I have said given any irrational number there is a bigger

Â number that's irrational now arguably the second one is a much closer

Â interpretation of what this means because you could say what this is really seen

Â is. Itâ€™s making a statement about

Â irrationals. So, youâ€™re really only going to mention

Â irrationals. And thatâ€™s the second version.

Â Given any irrational thereâ€™s a big irrational.

Â So, this is a closer thatâ€™s a closer answer.

Â The first one is equivalent to it, because c rationals and e rationals make

Â up the bills, there is sort of in just best and if you can find any irrational

Â bigger than any real number, then you can find an irrational bigger than any

Â irrational number, that's a. Either are correct.

Â It's just a matter of, of how we interpret them, and exactly how much of

Â this statement you are capturing in the formalism.

Â Once you've got something formal, it's, it's, it's not ambiguous, so long as you

Â expressed it correctly with the parenthesis and so forth, it' not

Â ambiguous. It's absolutely precise This thing has

Â exactly one interpretation. This thing has exactly one

Â interpretation. That's the whole point.

Â This has several interpretations. I mean, here are two different ones.

Â They're equivalent. They're obviously equivalent, for very

Â obvious reasons. But we've cashed them out in different

Â ways. Well question five is our old friend

Â about domestic cars and fallen cars so C is a set of all cars, Dx means x is

Â domestic, Mx means x is badly made. All domestic cars are badly made for all

Â cars. If that car is domestic, then it's badly

Â made. For all cars, if the car is domestic,

Â then it's badly made. I think that's a fairly straightforward

Â one. I I wouldn't expect you to come up with

Â anything significantly different from that.

Â In fact I wouldn't expect you to come up with anything different from that at all.

Â But you might of done. Okay.

Â All foreign cars are badly made but we don't have a predicate for foreign so we

Â have to take foreign as meaning not domestic.

Â In which case we take the previous one. And we just replace the Dx by negative Dx

Â and so, but not Dx, so for all x and C the effects is not domestic then it's

Â badly made. And here I'm making use of the fact that

Â negation is a very tight binding, has a very tight binding negation like

Â quantifies applies to whatever comes next.

Â So if you want to negate a whole bunch of things.

Â You have to sort of join them together and put parenthesis around in order to

Â make sure that the negation applies to the whole thing.

Â So negation applies to whatever comes next.

Â 32:28

There is a domestic car that is not badly made.

Â There is an x which is a car, which is domestic and is not badly made.

Â Notice by the way that when we are capturing universal quantifiers, the

Â statements that involve if-then, then we have A conditional, an implication.

Â For all x, if the k-, if it's, well for all x that are in the cars, for all cars,

Â if the car is badly made, then it's domestic.

Â In the case of an existential quantifier, if there is a car, we're saying there is

Â a car which is domestic and not badly made.

Â So, when we have universal quantifier, we typically will compare it with a

Â conditional, with an implication. When we have a x as existential quantify,

Â we will typically compare or compare it with a conjunction in case so.

Â To all go with an implication, exist go with a conjunction.

Â At least in these kinds of cases. the,the we, the trick, it's not a trick.

Â The rule we have to remember is to try and understand what these things say.

Â so it's never a good idea to look for symbolic rules that can lead you astray

Â if, if the formulation is a little bit different.

Â There is a foreign car that's badly made. There is an x in set of cars.

Â Which is foreign namely nondomestic and it's badly made.

Â And because negation binds tightly, I didn't bother putting parenthesis around

Â this. If I had wanted to say not Dx and m of x

Â I would have put parenthesis around there but the negation just combines what comes

Â next to it. And what comes next to it, it's not

Â domestic okay, it's foreign. All right?

Â Well that takes care of the foreign cars, let's move on to question six.

Â Well question six is about the same kind of things that we've already been looking

Â at in some of the earlier ones, namely whether things are rational whether

Â they're ordered in set and where, whether they're a biggest rationals or whatever,

Â but we've got a different set of restrictions.

Â Here we're asked to, to simply use quantifiers for real numbers logical

Â connectives, the order relation, but we do have a symbol q of x, meaning x is

Â rational. So we're going to get a different

Â expression. So the focus in what we're trying to make

Â precise has been, has been put elsewhere. Okay, we're allowed to assume what it is

Â to be rational that's not part of what we're trying to explicate.

Â So in this case we don't have to talk about things being in reals on it or

Â whatever because that's all we've got. So I don't have to say for all x and all

Â y I can say for all x and y. And I've, I've, left of parenthesis,

Â because there's no ambiguity, that just means for all x, for all[INAUDIBLE]

Â whatever comes next to it, so for all x and for all y.

Â Now I do have brackets because there's a bunch of things going to come next, that

Â depend on the 4x and 4y. So for all x and y If x is less than y,

Â then, there's a z which is rational, and lies between x and y.

Â So reading left to right, for all pairs of real numbers x and y, with x less than

Â y, There's a z, which is rational, and lies between them.

Â Now I didn't write exists z in Q, I didn't write that, because the Q isn't a

Â set. In this case, we've got a predicate

Â Meaning that x is rational. Okay, so I'm distinguishing, I mean, what

Â I'm doing with a lot of these examples, actually is distinguishing between the

Â various formulaism we have to express things within mathematics.

Â Sometimes we use sets, sometimes we use predecates.

Â Sometimes we'll use quantify restrictions.

Â Sometimes we use predicate restrictions. They're all different ways of getting

Â getting formality and precision into statements.

Â in, in mathematics, we, we pick the one that's that's most relevant.

Â In fact, in most mathematics, whenever we use these formulas which is in this

Â detail. this is much smaller thing to, to, to, to

Â things we have to write down in computer signs.

Â Look we're actually trying to unravel or to, to learn how to unravel the formal

Â structures beneath the statements we make.

Â Okay. Well that was number six, let's press on

Â to number seven. Okay, and number seven is this famous

Â quotation which is allegedly made by Abraham Lincoln, there is some So dispute

Â on, on websites as to who would have said this.

Â But it's it's commonly and popularly described to as Abraham Lincoln.

Â for our purpose, for our purpose, we're not worry about who said it.

Â and the issue is can we can we make it precise within the mathematical language.

Â You may fool all of the people some of the time, you can even fool some of the

Â people all of the time. Where you cannot fool all of the people

Â all of the time. The reason I like this example is that

Â the quantifiers in English are used in the way we use quantifiers in English.

Â the quantifiers come at the end of, of the clauses.

Â But in mathematics we read strictly left to right and so the quantifiers have to

Â come first. So let's say f, x,t mean you can fool

Â person Woops. P that should of been a P shouldn't it.

Â So let me just change that now. Okay.

Â Alright Fpt meaning you can fool person p at time t.

Â Then lets do these one by one. You may fool all of the people some of

Â the time. That means there are some times, when you

Â can fool all of the people. There are some times when you can, the

Â point is taking this clause, the people Depend on the time, there are sometimes

Â when you can fool all of the people, there was sometime when you can fool all

Â of the people. So we have to say, there were sometimes

Â when you can fool all of the people. Then there's a second clause, and this is

Â going to be a conjunction because these are different things we're saying so

Â there's an and if you'd like. There's a understated and here.

Â You may fool all the people some of the time and you can fool some of the people

Â all of the time, that's the second part. So, and there are some people, and

Â because quantifiers are these extreme of cases, we only have there exists and for

Â all, at least those are the, the common ones.

Â And so some, in, in, in, in, in the case when we use quantifiers, some is captured

Â simply by there is at least one. And strictly speaking, that's really the

Â logical heart of some. If we say something then strictly

Â speaking we are saying there is at least one.

Â Now arguably you can say in English, when you use the word some is an implication

Â that there's more than one. Okay well you could follow that but it

Â turns out than in mathematics It's really more, more, let's just say more

Â efficient, to just focus down on one. So there's a, there's a, a specialization

Â occurs in mathematics, that when we say some things we usually mean there's at

Â least one. And, and we can capture that by the

Â quantifier exists. You can fool some of the people all of

Â the time. That means there are some people, there's

Â at least 1 person, who can be fooled all of the time.

Â And in this clause, see what's been captured that's important.

Â I mean you can argue about. you know, whether, whether some is really

Â well captured by exists. but, but, but really what's going on here

Â is that the people that fooled can change from one time to another.

Â Okay? No, I said, well, we won't, didn't I?

Â What's really going on here, is that all of the time you'll be able to, so at, for

Â any time you can find some people. Oh, is that what I was saying?

Â Let me think about it for a minute. Some of the people, all of the time.

Â 41:21

Yep. I think that's correct.

Â Okay? You can fool some of the people all of

Â the time. So you can find some people who can be

Â fooled all of the time. You can find some people Who can be

Â fooled all of the time. Okay?

Â I thought for a moment that, when I worked this out a little while ago, I got

Â it wrong. But now I think that's exactly what it

Â means. There are some people who can be fooled

Â for all of the time. Okay?

Â So I'm happy with that one. I'm happy with that one.

Â Let's look at the last one. but, the but's that's just another

Â conjunction really. You can not fool all of the people all of

Â the time. I think this one is the one that's

Â easiest because there's just to all's. It's not the case that you can fool all

Â people all time, and for this one you could swap these around.

Â You could have[UNKNOWN]. And for all p.

Â You've got, when you've got 2 universal quantifiers or 2 existential quantifiers

Â it really doesn't matter which order you put them in.

Â But when you have for all and exists it really does matter which way you, you

Â write them down. So let's just recap this says there are,

Â there are times. When you can fool all of the people, a

Â lot of times, when you can fool all of the people.

Â This, the first clause is about fooling all the people.

Â And it said a lot of time, when you can do that.

Â The second clause is different. It says the right people.

Â That you can fool all of the time. It's saying there are some people, who

Â can always be fooled. You can fool all people some of the time,

Â so there are some times when you can fool all people.

Â And you can fool some people all of the time, there are some people, who can be

Â fooled all of the time. But you cannot fo, fool all of the people

Â all of the time. Okay?

Â Well I stumbled a bit here, or at least I has second thoughts for a moment, but on

Â reflection I decided that I got it right the first time and and I'm happy with

Â those 3. So, it's I hope, I hope you manage to get

Â something similar to that. This is tricky This is tricky no doubt

Â about it. That's why its a good exercise.

Â It's a really good exercise in understanding how mathematical formulism

Â capture the kind of things we will. We actually do say in the real world.

Â And this is a real world statements. One with some culture significance.

Â Okay lets move on to number eight. Okay, let's do number 8 is, is really

Â very similar to the American Melanoma Foundation example.

Â Okay, a driver is involved in an accident every 6 seconds.

Â Okay, so we've got x is a variable to denote a driver, t a variable for a 6

Â second interval. x,t means that x is an accident during

Â interval t. The headline is written if we simply take

Â a literal translation from this into the form of this, this, this language we got

Â here we would say there is a driver. So for all times t, A is involved in an

Â accident times t. You know, six second interval.

Â Now okay, so that's a literal interpretation.

Â Then we're going to rewrite it so that it's it, so that the, the English

Â expression really captures it in a literal way.

Â And then we would say, every six seconds a driver is involved in an accident.

Â And in that case we're saying, for every six second interval, there is a driver

Â involved in an accident. Notice here that the driver can change.

Â From one interval, to another. So driver can change from one interval to

Â another. For all t, there's an x, x, t These are

Â different. These are very, very different.

Â And this makes a huge difference in mathematics.

Â you know, as I mentioned, a lot of you had trouble really seeing there was any

Â problem with the American Melanoma Foundation example.

Â Then of course in everyday language There isn't a problem because we use our

Â understanding of the, of the real world in order to[INAUDIBLE] .

Â But when you start to make those things precise for doing mathematics, in order

Â to develop a language and a formalism, a way of saying things that's totally

Â precise and reliable, then the order in which you say things makes a big, big

Â difference. And that was the whole point of that

Â exercise and then these other ones. Its to make sure that the left right

Â ordering, of the various formulas captures the logical flow.

Â And there's a logic to this. This one says, there is a driver who's in

Â an accident. Every six seconds, which is nonsensical.

Â This one says for every six seconds, there's a driver in an accident, and when

Â we take the formal language, then the distinction is, is, is really, really

Â significant. Okey dokey, well, I, I guess we finished

Â exer-, we finished assignment five. Okay, quite a lot of quite a lot of

Â questions in there. This is not easy to master but once

Â you've mastered it boy you can, you can tear through mathematical arguements at a

Â much greater rate. a lot of the difficulties people have

Â following mathematics is they bring to mathematics the sort of the inherent

Â sloppiness. and ambiguity.

Â and vagueness. Of everyday language.

Â when you do that you just run into trouble.

Â The whole points about about this formalism is if you limit it solely

Â ambiguity. Because in mathematics we very often

Â don't have knowledge of the everyday world to help ourselves out.

Â As we do when we use language in the everyday world.

Â Okay. How well, how did you guys do in

Â assignment five? Wasn't easy was it?

Â