0:03

Okay, let's see what we've got with problem set two.

Â By the way, this part of the course we've been looking at taking expressions in

Â everyday language, problems that might arise from the real

Â world in a certain sense, and try to make them precise.

Â 0:20

And that, of course, means that for people whose language,

Â whose native language is not English, this is even more complicated.

Â Because we have to deal with the complexities of the English language and

Â try to eliminate the ambiguity from that and make things precise.

Â When it comes to these issues about necessary and sufficiency, I actually

Â don't think it makes a lot of difference if you're a native speaker in English.

Â This is tricky when you first meet it, and it's easy to make a slip.

Â So this is one of those occasions where I don't think there's an advantage to

Â being a native English speaker.

Â We native English speakers find this difficult too.

Â 0:54

Okay, in the case of necessity,

Â a condition is necessary if it follows from this.

Â So what we're asking for is, is it the case that if 6 divides n,

Â then the condition X holds?

Â That's what we're asking.

Â Does the condition X follow from n being divisible by 6?

Â Well, let's see.

Â Is it the case for example, That if 6 divides n,

Â does it follow that 3 divides n?

Â 2:32

Because it's 6.

Â So it's not that one.

Â Okay, n squared divisible by 6, n squared divisible by 3.

Â Well let's just see. 6 divides n certainly implies 3 divides n.

Â If 6 divides it then 3 divides it.

Â And if 3 divides n, then 3 divides n squared.

Â 3:16

So 2 divides n and 3 divides n, in other words n is even and divisible by 3.

Â So it's that one.

Â So we've got parts (a), (e), and (f).

Â And this is the condition.

Â This is the way it cashes out in terms of implication.

Â And in the case of the counterexamples that we used to prove some of these false,

Â the simplest counterexample is n = 6 itself.

Â 3:54

We have to look to see if the condition implies that n is divisible by 6.

Â Okay?

Â Sufficiency means implies n is divisible by 6.

Â So for each of these statements we have to ask ourselves,

Â does it imply that n is divisible by 6?

Â Well, if n is divisible by 3.

Â In order to, if we think the answer is it's not sufficient,

Â then we have to find an example, we have to find a counterexample of a number

Â that's divisible by 3 that's not divisible by 6.

Â Well, why don't we check n = 3.

Â 3 is divisible by 3, but 3 is not divisible by 6, so

Â that's a counterexample.

Â 4:57

Let's take n = 9, that's a counterexample.

Â n is divisible by 9.

Â But if we take n = 9, then the n is divisible by 9,

Â but 9 is not divisible by 6, so that's a counterexample.

Â 7:04

And when we come to question three in a minute,

Â it's really going to be a matter of combining questions one and two.

Â So we've now got all the information we need to answer the next one.

Â Question three asks us for necessary and sufficient.

Â 7:22

So let's just see what we did in question one, we dealt with necessity, and

Â in question two, we dealt with sufficiency.

Â Let's remind ourselves of what we have.

Â Let me see now.

Â We've got necessity, that was true in (a), wasn't it,

Â it was true in (e), it was true in (f), if I remember correctly.

Â Yeah, okay.

Â Actually I'm not remembering, I'm working these out as I go through,

Â because I don't have the other one in front of me anymore.

Â Sufficiency, let's see, that was (c).

Â 8:11

with a question that's described in a puzzling way.

Â Even the experts can be called out with this kind of thing.

Â Okay, so we just have to look for the ones which have an x in

Â both columns and the only one that does is this one.

Â In all of the other ones, you've either got necessity or sufficiency, or

Â neither, in the case of part b, okay?

Â So you've got a one necessity, you've got a neither, you've got a sufficiency,

Â you've got a sufficiency, you've got a necessity.

Â The only one that's both is part f, and that takes care of number 3.

Â Well, we're moving along rapidly now.

Â Let's go ahead.

Â Well, question 4 starts out simply enough because it's a very straightforward if

Â then statement.

Â And when you've got an if then statement, identifying the antecedent is pretty

Â straightforward, because it's the part that goes with the if, okay?

Â So it's this one.

Â 9:11

If the apples are red, then they're ready to eat.

Â Okay, so this was an easy start, but

Â these things are going to get a little bit more tricky as we move forward.

Â Let's have a look at b, okay?

Â Well, we're talking about sufficiency, and

Â sufficiency is the thing that does the implying, okay?

Â 9:41

Well, f being differentiable.

Â That's the one that does the implying, because the differentiability

Â of a function implies that it's continuous, okay?

Â Sufficiency does the implying, so

Â sufficiency is the thing we're looking for.

Â So the antecedent is a sufficiency which is differentiability.

Â 10:06

Well, I said these become a little bit tricky as soon as you get into them,

Â even though in one sense this is straightforward.

Â My experience is that many students, including myself when I was a student, and

Â occasionally today, if I don't really put my mind to it,

Â I have to think a little bit to just flesh these out.

Â All righty, let's move on to part c.

Â 10:36

This case again, this actually is fairly straightforward,

Â because it doesn't matter whether you put the if clause first or second,

Â it's the thing that goes with the if, so long as it's an if not an only if.

Â The thing that just goes with a naked if is the thing that's the antecedent.

Â So in this case, it's this guy, it's f is integrable,

Â that's the thing that it does implying.

Â 11:02

And whenever actually, it's sort of the same as f, it's another way of saying if.

Â This tells us the condition under which something happens.

Â This is bounded under the conditions that S is convergent,

Â whenever S is convergent, or if S is convergent, so it's that guy, okay?

Â So this is the antecedent, this is the antecedent.

Â Let's move on to part e.

Â 11:45

So we have to ask ourselves, what is it that's following in this case, okay?

Â And the answer is, This thing, okay, so this is the antecedent.

Â 12:53

The team wins only when Karl is playing.

Â So if you know that the team wins, you can conclude

Â that Karl is playing, because they only win when Karl is playing.

Â So that's just another way of saying that Karl is playing is a consequence of

Â the team winning,, okay?

Â So this is the antecedent.

Â 13:37

Well, these two are a little more straightforward than the last one or two,

Â because the when is really almost the same as if.

Â That just tells you the condition in which something is.

Â So the thing that just goes with the when, without an only combined with it,

Â is the antecedent, okay?

Â So the antecedent in this case is that Karl is playing,

Â because it's when he's playing that the team wins.

Â If you know that Karl is playing, then you can conclude that

Â the team's going to win on the basis of this statement.

Â So that's the antecedent there.

Â And it doesn't matter whether the when clause comes first or

Â second as was the case with an if clause.

Â It can come first or it can come second, that's still the antecedent, okay?

Â Being the antecedent is not directly related to whether you're the first clause

Â or the second clause in a sentence,

Â it gets to go with which word you're combined with.

Â If it's an if or if it's a when, they're not combined with the antecedent.

Â If it's an only if or an only when, then that flips it around and

Â then you're dealing with the consequence, okay?

Â Well, that takes care of those kind of examples.

Â And the last three parts of this question,

Â of this problem set, were a little bit different.

Â 15:31

So that's what this boils down to, this simplification.

Â Okay, we know there's two implications here, so if and only if, so

Â that means equivalence.

Â It means the implication holds in both directions, and

Â one implication is certainly true.

Â If m and n are even, then mn is even.

Â And so it boils down to the question if the product is even,

Â then are the two numbers necessarily even?

Â And once you get it down to that stage, All you need to do is the counter,

Â there are many counterexamples.

Â We could take m=2, n=3 then, mn=6.

Â So, here we've got A product that's even.

Â 16:35

This is a statement about any pairs of integers.

Â And if we found one pair of integers that makes it fail then,

Â the whole statement fails.

Â So, it's a counterexample that we need to find and we found one, m=2, n=3.

Â The product is even but it's not the case of both numbers are even.

Â Okay, now, let's move on to number 6.

Â 17:11

You almost certainly realize the reason,

Â you know reason why because there were two facts.

Â We know that odd x odd, I'm going to say, is not equals is odd.

Â 17:34

So, if you take two odd numbers and multiply them together,

Â you get an odd number.

Â If you take a pair of numbers, any one of which is even, and multiply them together,

Â you get even.

Â And when you combine those two, this falls out of it.

Â 17:49

And if you don't see that, I'm going to leave it to you actually.

Â If you want to sort of give a little bit more detail and

Â express it as an implication in both directions, that's fine.

Â I'm sure you could discuss this endlessly on the forums, and

Â that would be a good idea if you want to.

Â But I'm just going to leave it with the observation that it's really

Â just these two facts that give you this result, okay.

Â This tells you how the power t even and odd works.

Â 18:14

And once you know that, you know that but have fun with this one and

Â discuss it amongst yourselves and settled

Â to your own satisfaction what comes to choose a rigorous proof of this thing.

Â Remember, there's actually no sort of goal, standard of what is or

Â is not a rigorous proof.

Â It depends on the experience of the audience,

Â a proof in many ways involves audience design you've got to cast that

Â proof at level of detail and precision that matches the audience.

Â 18:46

Typically, a professional mathematician would simply say, And

Â you see this actually in books and in papers,

Â the mathematician might very well say, this is trivial, okay.

Â And that would be a proof.

Â In advanced works on mathematics, you often see remarks, the proof is trivial.

Â 19:10

It has to be said that a beginner might take several days

Â to see why something's trivial.

Â When mathematicians use that kind of expression, they're doing it with

Â a particular audience in mind, namely, other professional mathematicians.

Â So, if you read that and it doesn't seem trivial to you,

Â it doesn't mean you're stupid.

Â It just means you haven't spent many years working as a professional mathematician.

Â It's just the way we classify things.

Â 19:48

Well, one way to do all of these is by truth tables.

Â And if you work up a truth table, you find that a is true, that b is true,

Â c is not true, d is true, e is true, and f is true.

Â So, simply by using truth tables.

Â 20:08

You could answer this.

Â This one, we've already seen in the lectures and discussions,

Â we've sort of looked at these equivalence.

Â This one and where is it?

Â That one, those are examples of what's known as,

Â 20:30

De Morgan's Laws, after a mathematician, Augustus De Morgan.

Â And if you take a negation with a disjunction,

Â you end up with a conjunction of the two negations.

Â And if you take a negation of a conjunction,

Â you end up with a disjunction of the two negations, okay.

Â 20:51

So, that was a basic factor about implication that a conditional is

Â true if either the antecedent is false or the consequence is true.

Â So, that was the truth table that we went out for the conditional.

Â And those are De Morgan's Laws, that one was not the case anyway.

Â 21:44

And the same too for this one, you could reason it out.

Â This one I think is usually not that one because of the fewer symbols to do with.

Â This says that, if you have an assumption and

Â then, you have another assumption, you can make a conclusion from it.

Â 22:06

Well, do you do it in two steps, you assume P.

Â And then, on the basis of that showed that if Q is true then, R is true.

Â Or do you simply assumed that both P and Q is true and then, use R.

Â Those two things would be equivalent.

Â 22:21

This is so if doing it in two steps and this is combining the two assumptions.

Â They both really tell us that there were two assumptions.

Â There's one assumption then, there's another assumption.

Â And in this case, we've explicitly said there were two assumptions and

Â in both cases, it's the R that's following from them.

Â Okay, so, the question is, how and when do you get from P and Q, to a conclusion R?

Â So, this you could reason it out and if you feel uneasy about that,

Â you could just work out the truth table.

Â Okay, but since we spent a lot of time on truth tables in the assignments,

Â my recommendation would be that you would go through these and

Â actually try to reason them out in terms of what they mean.

Â Truth tables are good if you are a computer.

Â 23:05

But people are not computers, they're much more interesting creatures than that.

Â And I think we have the power of reasoning.

Â And so, I would suggest you go through these and

Â try to reason them out in terms of what these things mean.

Â because that after all is really what this entire course is about.

Â 23:39

Well, here's the first one.

Â The claim is that for any two propositions P and

Â Q, not P conjoined with not Q is equivalent to not P and Q.

Â Yeah, [INAUDIBLE] fairly short.

Â In fact, all of the ones I'm going to be using as examples are short.

Â That's why they're good as examples In particular,

Â I want to be able to give the solutions on a single slide, as I'm going to do here.

Â So in a way, these are not typical.

Â In fact, if you do test flights at the end of the course, you will almost certainly

Â see proofs presented by other students, which are much more complicated.

Â And much longer and perhaps have all sorts of mistakes in them.

Â The arguments I'm giving you, they all began as arguments that were produced by

Â students over the years, when I've taught this material.

Â But I've done is I've picked particular aspects of proofs where students typically

Â go wrong.

Â And so the examples will have one or two common mistakes embedded in them.

Â Just so you get used to looking at proofs from the different perspectives,

Â as captured by the rubric, and seeing how they work and don't work.

Â As I mention in the description of the use of the rubric on the course website.

Â The way it will work is that because I'm using short examples in each particular

Â example, some of these features won't really apply, in which case you have

Â to sort of default and give 4 or 0 depending on how the student handles it.

Â Typically you would end up giving full marks because it just doesn't

Â really apply.

Â 25:06

Okay, well let's take a look at how this one was done.

Â So I'll pretend that it was done by a single student, even though this is

Â a composite of the kind of things that students have done over the years, okay?

Â 25:23

Well, it's an equivalence.

Â So we have to prove it in two directions.

Â We have to prove that this implies that, and that implies that.

Â So there were two equivalences to prove here.

Â 25:45

Then a conjunction is true, if and only if, the two conjuncts are true.

Â So if the conjunction is true,

Â then both not P and not Q are true, okay, that's correct.

Â That's what conjunction means.

Â If not P is true, that means P is false, that's what negation means.

Â If not Q is true, that means Q is false.

Â 26:21

Hence not P and Q is true.

Â Again because of the way negations works.

Â So that's absolutely okay.

Â Okay, left to right was proved, great.

Â 26:35

Well in this case, the student makes an attempt to be fairly sophisticated, by

Â saying the other argument, the argument in the other direction works the other way.

Â If it does work the other way, it's absolutely okay to simply say that.

Â There's no requirement that you would have to repeat something that's

Â obviously the case.

Â But we'd better check that it is obviously the case,

Â because we're actually evaluating whether this is the case.

Â 27:01

Okay so let's spell out what this person didn't do.

Â In other words, we'll try to do the same as here, going the other direction.

Â So we're going to assume it's not the case that P and Q, okay?

Â 28:27

So, one of P and Q is true,

Â that means one of not P and

Â not Q, could be false, okay?

Â 28:49

At least one of them is false, so the other one could be true.

Â That means one of those could actually be false.

Â One of these guys, not P and Q, could be false.

Â That means, if one of them could be false, it means not P and not Q,

Â could be false.

Â 29:34

We're not saying it is, we're saying it could be.

Â So there isn't an implication.

Â There is no implication from right to left

Â because you could have that without having that.

Â 29:50

In other words, the original claim is a false claim.

Â It's simply not true.

Â So this statement, though it was a nice

Â attempt to be somewhat sophisticated,

Â it didn't work because in fact

Â the argument does not work the other way.

Â I mean, it is essentially the same kind of argument, and if it had been correct,

Â that would have been fine.

Â You wouldn't need to give this.

Â 30:33

There are four marks available for logical correctness.

Â The left to right part was absolutely correct.

Â This is logically correct.

Â So I'm just going to give half marks for that,

Â I want to say two captures the fact that left to right was correct.

Â 30:59

Even here it was clear, it was wrong, but it was clear.

Â So in terms of clarity, this is very clear, so

Â I'm going to give full marks for clarity.

Â 31:29

Well, it's a judgment call.

Â I would say that for someone at this stage, where they're producing arguments

Â like this, and this is a nice argument, this is perfectly correct.

Â For someone that's producing this, I think it's okay to say,

Â yeah, it's obvious by the way they're doing it that they're doing that.

Â But remember, even though the rubric does break a difficult

Â task into smaller pieces, they're still not easy.

Â So these are going to be judgement calls.

Â I would say that there's enough demonstration of knowing the person,

Â knowing what he or she is doing to just say, I'm not going to insist that

Â they say proving an equivalence is enough to prove left to right and right to left.

Â I think it's clear from the way they're doing it.

Â Well, they do start out by saying assume one and prove the other one, okay?

Â What about stating the conclusion?

Â 32:23

Absolutely, it was stated, we have implication in both directions.

Â Well, this just justifies what I've just said, so

Â the person has now said we have implication in both directions.

Â So just emphasizing the fact that he or she knew exactly what they were doing.

Â Remember also that we're really try to do formative assessments rather than

Â summative the process is one of seeing what the person has done well and

Â rewarding what they did well, it's not about trying to take marks off.

Â When we do these kind of things we should always be looking for

Â reasons to give marks to acknowledge what's been done correct,

Â and is a point how things to improve.

Â So we begin by adopting a positive giving my x attitude, that's the best way to do

Â this kind of thing, so I'm always going to be looking for reasons to give my x.

Â On the other hand, you cant give my x if it's notjustified.

Â What about reasons?

Â Reasons were given, but it was really the same as in the first case.

Â The reasons for the other part that should've been given weren't

Â because the person thought it was the same and it wasn't.

Â So some of the reasons are here, they were given as the person was going along.

Â 33:49

So the more sucker give here is 2,

Â because even though the reasons we were given that's not of using.

Â That would have been that would have actually been a reason, if it was correct.

Â If this kind of argument was correct that would have been okay, but it's not.

Â And then overall, again, I'm just going to have to give 2,

Â because it's basically just half a result.

Â That's really what it's coming down to, the person has laid it out well, so he or

Â she is going to get a lot of marks for laying it out correctly.

Â 34:26

So, the total is going to be 18.

Â It maybe a little be generous actually, after all this thing is false.

Â In fact, if this was a mathematics course and I was sort of grading people for

Â doing mathematics work, I think I'd have been much more harsh.

Â Had it probably come down at ten,

Â 12, maybe even less, but this is about much more than mathematics.

Â It's about mathematical thinking,

Â it's about communication, it's about understanding proofs.

Â 35:07

Because half the proof is a, okay, but because we are also rewarding or

Â looking to reward, prove, structure, communication,

Â all of those other aspects of proves, which are important,

Â I'm going to give credit for the fact that a person has got the general idea.

Â 35:26

So I've got 18. As I said maybe we can add it as a little

Â bit generous, I'm inclined to think as a little bit generous myself.

Â On the other hand this is a 75%,

Â which means 25% has been docked,

Â if you like it or not, given.

Â And let's look at what happened.

Â This was essentially just one mistake.

Â The person looked at it and didn't look closely enough, but

Â it comes down to just one error, it's one error in logic.

Â The person who could produce that argument almost certainly

Â could produce the correct argument in the other direction, to show that it's false.

Â So, it really comes down to one slip, and frankly,

Â it seems to me that if you docked more than 25% or

Â maybe 30% for just one error, that's kind of harsh.

Â 36:22

We are, after all, trying to turn people into better mathematicians,

Â to make them better mathematical thinkers.

Â So let's look at what they do right and then give due credit for

Â that, and this person who's done a lot of things right, there was just one mistake.

Â And it's a mistake that almost certainly this person shouldn't have made,

Â they had the ability not to make the mistake.

Â 36:46

But, people do make mistakes.

Â So we're not not going to give for

Â free, but having hesitated a little bit on the 18,

Â I've actually now talked myself back into seeing 18 is actually a pretty good grade.

Â