2:00

That's this one. So that's correct.

Â No doubt really does capture it. There is a time when they, when they play

Â together and as a partner, as a partnership and they win.

Â Now, now you know, you can look at some sort of gent will say that sometimes this

Â is used exclusively to mean more than one.

Â I mean, I don't think that's the case. I mean you'll find people that say that

Â you know, language is flexible. In any case, in mathematics, we always

Â interpret things like some. sometimes, as at least one.

Â you know? That's the whole point about the way we

Â set things up. We, we, we, eliminate these ambiguities

Â by being specific. And we're specific to say that, whenever

Â you're asserting something exists. Sometime, some game or whatever.

Â You mean at least one. Okay in which case you've got existence

Â from, existent to quantify that means there is at least one tennis game were

Â they play together and they win, okay, so that one is okay.

Â whenever Rosario plays with Antonio, she wins the match.

Â Well that's really again, that's for all t wt.

Â So it's not that one. Rosario and Antonio win exactly one match

Â where they are partners. Well, that one isn't going to work,

Â right? Because that says exactly one match.

Â There's no specification here of exactly one match.

Â If you wanted to do that, there's a, a notation.

Â This notation exists a unique t. Such that wt.

Â You can't say it, and you can say it other way.

Â I mean, you can, this is just an abbreviation for, for for an expression.

Â You know, we've seen that in the problem sets.

Â Hm, yes actually one of the assignments, so you can't capture it but this doesn't

Â capture it, this just says is it at least one it doesn't say that's exactly one we

Â think when, okay. Rosario and Antonio win at least one

Â match when they are partners, that's it that's another one that's fine this one

Â when they are partners Okay. If Rosario wins the match, she must be

Â partnering with Antonio. Well, first of all, there's a, there's a

Â universal quantifier floating around here, I think.

Â Well it's here. because it's saying whenever she wins the

Â match, she must be partnered. So there's a universal quantifier here.

Â But it's even worse, because the universal quantifier is actually for all

Â of X. For all matches.

Â Okay? For all doubles tennis matches where she

Â wins something or other. So this one here is false.

Â So this action here what's generally known as a scope problem.

Â In this statement the quantification Is actually over something different.

Â It's over all possible double tennis matches, not just the ones where she's

Â partnered. So not only does this not capture it, it,

Â it, there's actually another issue. There's a scope issue involved.

Â Okay. Because here the T ranges only over games

Â where they've played together. In this case we're looking at games where

Â Rosario plays with whoever she's playing with.

Â Okay? So there are a couple of things that

Â prevent this one being the, being the right answer.

Â Okay? Well I, you know, as I said at the

Â beginning, from a mathematical perspective, this is actually very clear.

Â It's definitely B and it's definitely E. And the reason I can be so definitive

Â about that is becuase I'm familar the way that we've set up the meaning of this in

Â mathematics to correspond to mean at least one and we interpret in mathematics

Â we intepret anything that exerts an existance to mean exists one, at least

Â one. And so things like sometimes, some of

Â these, some of those [NOISE]. they're all interpreted to mean at least

Â one. Okay?

Â Let's look at the next one. Well, same setup as in question 1.

Â The only difference is now we're talking about for all t, w of 2.

Â So let's, let's run through this one. Rosario and Antonio win every match where

Â they are partners. So, every match where they are partners.

Â That would be for all t. Because that's what t captures.

Â T is the doubles tennis matches where they partner.

Â And they win. Oh, that is that.

Â So that one's okay. What about this one?

Â [LAUGH] this has really got nothing to do with that as it is.

Â Rosario has always got nothing to do with winning.

Â It's just saying she always plays together.

Â that would essentially say that that X and t are, are the same variable in fact.

Â And is just saying that this, there's no distinction between X and T.

Â so I think rather than cross out those, say that it's wrong.

Â I'll say this isn't even a candidate. I mean, it's got nothing to do with

Â winning. Okay.

Â Let's look at part c. Whenever Rosario partners with Antonio,

Â she wins the match. Whenever she partners, that's for all t.

Â She wins. She wins, they wins.

Â It's all the same in doubles tennis. So that's okay.

Â What about this one? Sometimes, Rosario wins the match.

Â No. I mean, first of all.

Â it's, it's not about t, it's about x. Sometimes she wins with whoever she's

Â playing with. and it's an existential one.

Â So it's essentially of the form, exist x, wfx.

Â That's really what it means. Sometimes Rosario is in the winning team.

Â She wins. Well, that's not that.

Â It's a different quantifier. And it's over x, not t.

Â So, well that one. I won't cross it out, because at least it

Â talk about winning, so it's a candidate, but not the right candidate.

Â OK? Rosario wins the match whenever she

Â partners, this is whenever she partners, that's essentially for all T, and

Â whenever that happens, she wins. Okay?

Â Bingo. That matches that.

Â That's correct. And finally, if Rosario wins a match, she

Â must be partnering with Antonio. Well essentially, you've got something

Â like for all x here etcetera. So as before, as in question 1.

Â We've got a scope issue here. this is actually about all possible

Â matches, not just the ones where she is, she is partnering.

Â And, thus, the conclusion is that she is partnering on somebody and so, so this

Â doesn't, I mean just talk about winning but it's seems doesn't conclude because

Â there is, there is a scope problem, the way I'm presuming different things and

Â okay so it's not that one. So in this case we've got 'a', we've got

Â 'c' And we've got E. It's kind of unusual to have one of these

Â multiple choice things where three of them are correct, but there you go,

Â sometimes that, sometimes that happens. Okay?

Â Let's move on to question three. On question 3, if you look through these,

Â looking for something that seems to say There's no largest prime.

Â I think you quickly end up looking at, at this 1d.

Â Which says that, for any number x, there is a number y, which is prime and bigger

Â than x. So that certainly says there's no largest

Â prime. Now the question is, do any of these.

Â Say the same in a different way. Well let's just look at them in turn.

Â Let's, that says there don't exist any Xs and Ys for which X is a prime, Y is not a

Â prime, and X is less than one. Well there are plenty of pairs X and Y

Â that satisfy that. So this is actually false.

Â >> So, I mean, we weren't, we weren't ask to say whether things are true or

Â false. But this is false, and we do actually

Â know there was no largest prime. That's Euclid's theorem, that the primes

Â are infinite. and in, in, and the list of primes goes

Â on forever. so it can't be this, because this is

Â actually false, but in any case, it doesn't mean the same.

Â It just means something just, nonsensical.

Â of course, there exists pairs x and y. With x a prime, and y not a prime.

Â And x less than y. So, to say it's not the case is, is, is,

Â is clearly wrong. What does this say?

Â for every x, there is a y. Such that x.

Â Well, first of all, that would say, for all x, x is prime.

Â That would say every number is a prime. So that's false as well.

Â So that can't possibly be it, that can't be it, that can't be it.

Â What does this one say? For all x and for all y, x is, what that

Â says as well, every number is a prime. That's false.

Â 12:46

And this part says it's the only thing that satisfies phi.

Â Because if you look at all the possible y's that could satisfy phi.

Â The only one that does so is y equals x. So this actually captures, there is a

Â unique x. So, we said 5x.

Â And these 4. they're, they're not on.

Â If you, if you try to figure out what they say.

Â If they say anything vaguely sensible at all.

Â They turn out to be just nonsensical and they certainly don't capture that.

Â Incidently, this symbol is a moderately common symbol in mathematics.

Â It's not something I made up for the exercise.

Â The exists with the exclamation point does actually mean there is a unique x.

Â You'll, you'll find that quite a bit in mathematics.

Â Often. You need to be able to say there is a

Â unique solution to something. now this exercise tells you that you

Â don't need to have a separate quantifier to mean there is a unique one because you

Â can accurately define it in terms of the [INAUDIBLE] existential quantifier.

Â And the universal quantifier, so this is in fact just an abbreviation.

Â But it's a useful abbreviation and so you'll often see it.

Â Okay. Well for question five let me observer

Â that this symbol, I mean you do see this symbol sometimes in computer science,

Â very rarely in mathematics except in a situation like this...

Â Well I'm using this to refer to some arbitrary but unspecified binary

Â operation. So this isn't a particular operation, I

Â just mean there is some, some operation which I'll call x upper arrow y, and we

Â need to be able to express the fact that that's not communicative.

Â so this doesn't have a particular interpretation, I'm just using it to mean

Â any particular, any unspecified binary operation.

Â And so what we need to do is, is ask ourselves which one of these means there

Â its not commutative, well commutative lets write down what it means to be

Â commutative, commutative means for all x and for all y, x power y equals y power x

Â So that's what being commutated means. Which one of these negates that?

Â Well, when you negate universal quantifiers, you get existential

Â quantifiers. And things that are true become false.

Â So the equality becomes an inequality. And so if you skip through these, you

Â find, yep. Here it was c, that says there is an x,

Â there is a y for which they're unequal. There is an x there is a y for which

Â they're unequal. So that's certainly the negation.

Â Do either, do any of the other ones fall in this as being a negation?

Â Not really and probably not even close because when you negate both the foils

Â become a [INAUDIBLE] They don't remain for all so it's not that one.

Â they don't remain for all so it isn't for all the.

Â So for a variety of reasons none of this three qualify for that.

Â So there's no there's no possibility of having two possible expressions.

Â That's the only one. Okay.

Â 15:58

Okay. Question six.

Â evaluating this proof. and this is, very typical of the kind of,

Â work you'll see from students who are beginning to look at proofs.

Â Because what this person has done. Is if identifying the key idea.

Â This is absolutely the key mathematical idea behind this.

Â the factor is that this, this is not prime.

Â and if I'll look, what I'll do, let me just give you the proof that the person

Â should have done. Okay, and then I'll, then I'll discuss

Â why, why there's a problem with writing this down.

Â Okay, so what the person should have done is, is something like the following, he

Â began by saying, the claim is logically equivalent to the following statement For

Â any positive integer N, N squared plus 4N plus 3 is not prime It's logically

Â equivalent to that. To say that it doesn't take just an

Â integer for which that's prime is logically equivalent to saying that for

Â any positive integer it's not prime. And the person then was going, which

Â should maybe prove this is, is true. Okay.

Â We prove this this statement so we are proving the logically equivalent

Â statement, okay so let N be a positive integer and I am doing this one in

Â [UNKNOWN] detail because I am trying to get maximum points for this one, okay

Â Then by basic algebra N squred plus 4N plus 3 equals N plus 1, N plus 2, N plus

Â 3 But N plus 1 and N plus 3 are positive integers, each greater than 1.

Â Okay N plus 1 is at least 2, N plus 3 is at least 4 so these are positive integers

Â greater than 1, so by definition N squared plus 4N Plus 3 is not prime

Â because its a product of two positive integers each greater than 1, okay so

Â that's what a person should have done, now lets go back to what was here this is

Â the key algebraic heart of this thing already But it's not a proof.

Â And the reason students often does this kind of thing.

Â Is, they're used, from high school. They're used to the fact that algebra is

Â all about algebraic manipulation. And indeed, it is.

Â But we're talking about proofs here. And a proof is much more than getting

Â the, the algebraic manipulation right. if the algebraic manipulation is not

Â right, you don't have a valid proof. But the proof is all about giving reasons

Â and making a, giving an explanation. It's a story.

Â A story with a beginning, a middle, and an end.

Â you know, most, you know there's a, there's a, I mean there's a joke about a

Â Woody Allen in the Woody Allen movie where Woody Allen, in the character Woody

Â Allen character says, he's been reading a book it was "War and peace" and he

Â summarizes by saying, it was about some Russians.

Â Well, this is like saying it's about this.

Â obviously War and Peace is just about some Russians, but there's much more than

Â that. And that's really what we're doing here.

Â We're looking for the, the full story. this one I think is going to be actually

Â fairly difficult to grade. I'm going to put four for logical

Â correctness here. Because, logically, this is the heart of

Â it. Once you realize that that's the logical

Â heart. So, that's, that's fine.

Â Okay. clarity, I'm going to have to give a

Â zero. just, there's just no clarity about this.

Â Because there's no explanations, nothing. Okay?

Â this is really nothing, nothing valuable here.

Â opening. there isn't an opening.

Â It just jumps straight in. let's see.

Â Stating conclusion. Well the person did state the conclusion.

Â So I'm going to give that, you have to give full marks.

Â The conclusion is stated. that is, that is how you're supposed to

Â end. okay?

Â As I did here, that's not prime. reasons.

Â There are no reasons given. Okay?

Â I mean, that's, that's just going to be a zero.

Â and then, let's see. What have I got here.

Â overall evaluation, zero. I mean, as, as a proof, I can't really

Â give anything for that. Okay.

Â Well I don't, let me think. you know, actually now I'm going to be a

Â bit generous here. I'm going to give 2 for that I think.

Â Because this is, this is key. I mean this, you, I've got, I'm going to

Â give some credit for this. I mean I It is the key part of this.

Â and it was the, the setting that was wrong.

Â So I'm going to give 2 for that. So that means I go for a, I've got 10 for

Â that one. yeah.

Â A little bit generous, maybe, as a proof. the person certainly has the, has the

Â algebraic ability. And seeing this is key.

Â I mean that really is, yeah, okay, I was, I was, I think that's okay.

Â I think that's actually a good mark to give for that one.

Â Okay, but these are not easy to do. you're making value judgements, you're

Â trying to sort of asses a whole bunch of different things.

Â the way we've structured this course is you'll be seeing a lot of exercises like

Â this And the intention is that by the time we get to the end of the course

Â you'll have go the general gist of how to do this.

Â I mean all instructors differ about their own methods and I'm, I'm just giving you

Â mine as an example but with something that is sort of essentially qualitative

Â as as grading proofs, it's like grading essays.

Â you know, people, people end up with different, you know, there were, there

Â were stricter graders and less strict graders.

Â my goal is always when I'm grading, at, at this kind of a context is to, is to,

Â is to look for reasons to give people marks because I want to give the credit

Â for what they're doing, but at the same time point out the things that still need

Â to be done. Okay alrighty, well that was the end of

Â the problem set three.

Â