Okay, let's see what we've got with problem set two. By the way, this part of the course we've been looking at taking expressions in everyday language, problems that might arise from the real world in a certain sense, and try to make them precise. And that, of course, means that for people whose language, whose native language is not English, this is even more complicated. Because we have to deal with the complexities of the English language and try to eliminate the ambiguity from that and make things precise. When it comes to these issues about necessary and sufficiency, I actually don't think it makes a lot of difference if you're a native speaker in English. This is tricky when you first meet it, and it's easy to make a slip. So this is one of those occasions where I don't think there's an advantage to being a native English speaker. We native English speakers find this difficult too. Okay, in the case of necessity, a condition is necessary if it follows from this. So what we're asking for is, is it the case that if 6 divides n, then the condition X holds? That's what we're asking. Does the condition X follow from n being divisible by 6? Well, let's see. Is it the case for example, That if 6 divides n, does it follow that 3 divides n? Well the answer is yes. If 6 is divides it, then 3 is divides it. Okay, so that one is necessary. What about this one? Well notice that n = 6 itself satisfies this condition. Okay, if n = 6 then 6 divides n. But is 6 divisible by 9? The answer's no. Again, n = 6 satisfies the condition of being divisible by 6. But does it then follow that n is divisible by 12? In other words, is 6 divisible by 12? No, it's not. What about n = 24? Well again n = 6 gives us an example of a number that's divisible by 6, but that number's not necessarily 24, right? Because it's 6. So it's not that one. Okay, n squared divisible by 6, n squared divisible by 3. Well let's just see. 6 divides n certainly implies 3 divides n. If 6 divides it then 3 divides it. And if 3 divides n, then 3 divides n squared. So that one's okay. Again if 6 divides n Then of course, 2 certainly divides n because 6 does, and if 6 divides n then 3 divides n. So 2 divides n and 3 divides n, in other words n is even and divisible by 3. So it's that one. So we've got parts (a), (e), and (f). And this is the condition. This is the way it cashes out in terms of implication. And in the case of the counterexamples that we used to prove some of these false, the simplest counterexample is n = 6 itself. Okay, well that takes care of number one. Let's move on to number two. Well, in the case of sufficiency We have to look to see if the condition implies that n is divisible by 6. Okay? Sufficiency means implies n is divisible by 6. So for each of these statements we have to ask ourselves, does it imply that n is divisible by 6? Well, if n is divisible by 3. In order to, if we think the answer is it's not sufficient, then we have to find an example, we have to find a counterexample of a number that's divisible by 3 that's not divisible by 6. Well, why don't we check n = 3. 3 is divisible by 3, but 3 is not divisible by 6, so that's a counterexample. So that one is not sufficient for being divisible by 6, that doesn't imply x is divisible by 6. Because in particular, 3 is divisible by 3 but not divisible by 6. What about part (b)? Let's take n = 9, that's a counterexample. n is divisible by 9. But if we take n = 9, then the n is divisible by 9, but 9 is not divisible by 6, so that's a counterexample. n is divisible by 12. Is it the case that if 12 divides into n, does it follow that 6 divides into n? The answer is yes. So, that one's okay. What about n = 24? Is it the case that if n = 24 Does it follow that 6 divides n? In other words, does 6 Divide 24? Again the answer is yes, so that one's true. n squared is divisible by 3. Does it automatically follow that n is divisible by 6? Well, why don't we take the same counterexample we did in part a? Let's take n = 3. Then certainly 3 divides n squared, okay? But does 6 divide n? Does 6 divide 3? The answer is no. 6 does not divide n. So 3 divides n squared, 6 does not divide n, so it can't be that one. Finally, n is even and divisible by 3. I could write that as saying n is even is to say that 2 divides n. And it's divisible by 3. So, I can just rewrite this statement to say that 2 divides n and 3 divides n. But if 2 divides n and 3 divides n, then it’s the case that 6 divides n, okay? So that one’s true. And when we come to question three in a minute, it's really going to be a matter of combining questions one and two. So we've now got all the information we need to answer the next one. Question three asks us for necessary and sufficient. So let's just see what we did in question one, we dealt with necessity, and in question two, we dealt with sufficiency. Let's remind ourselves of what we have. Let me see now. We've got necessity, that was true in (a), wasn't it, it was true in (e), it was true in (f), if I remember correctly. Yeah, okay. Actually I'm not remembering, I'm working these out as I go through, because I don't have the other one in front of me anymore. Sufficiency, let's see, that was (c). It was d and it was f, okay? By the way, if I'm making this look fluent it's because I've been doing this for many years. Although you can still catch me out with a question that's described in a puzzling way. Even the experts can be called out with this kind of thing. Okay, so we just have to look for the ones which have an x in both columns and the only one that does is this one. In all of the other ones, you've either got necessity or sufficiency, or neither, in the case of part b, okay? So you've got a one necessity, you've got a neither, you've got a sufficiency, you've got a sufficiency, you've got a necessity. The only one that's both is part f, and that takes care of number 3. Well, we're moving along rapidly now. Let's go ahead. Well, question 4 starts out simply enough because it's a very straightforward if then statement. And when you've got an if then statement, identifying the antecedent is pretty straightforward, because it's the part that goes with the if, okay? So it's this one. If the apples are red, then they're ready to eat. Okay, so this was an easy start, but these things are going to get a little bit more tricky as we move forward. Let's have a look at b, okay? Well, we're talking about sufficiency, and sufficiency is the thing that does the implying, okay? So what does the implying here? Well, f being differentiable. That's the one that does the implying, because the differentiability of a function implies that it's continuous, okay? Sufficiency does the implying, so sufficiency is the thing we're looking for. So the antecedent is a sufficiency which is differentiability. Well, I said these become a little bit tricky as soon as you get into them, even though in one sense this is straightforward. My experience is that many students, including myself when I was a student, and occasionally today, if I don't really put my mind to it, I have to think a little bit to just flesh these out. All righty, let's move on to part c. Okay, remember, we'll still looking for the antecedent, the thing that does the implying. This case again, this actually is fairly straightforward, because it doesn't matter whether you put the if clause first or second, it's the thing that goes with the if, so long as it's an if not an only if. The thing that just goes with a naked if is the thing that's the antecedent. So in this case, it's this guy, it's f is integrable, that's the thing that it does implying. And whenever actually, it's sort of the same as f, it's another way of saying if. This tells us the condition under which something happens. This is bounded under the conditions that S is convergent, whenever S is convergent, or if S is convergent, so it's that guy, okay? So this is the antecedent, this is the antecedent. Let's move on to part e. When the case of necessity, necessity is the thing that follows. So a necessary condition is the thing that follows, okay? So we have to ask ourselves, what is it that's following in this case, okay? And the answer is, This thing, okay, so this is the antecedent. Because n being prime is necessary, so n being prime is the thing that's the consequence of the antecedent. So it's this, this implies that. So this is the antecedent, all righty? Let's move on to the part f. How did you do with that one? Well, this is where we start combining words like if and when with an only. And because it says only when Karl is playing, this guy is the consequence. The team wins only when Karl is playing. So if you know that the team wins, you can conclude that Karl is playing, because they only win when Karl is playing. So that's just another way of saying that Karl is playing is a consequence of the team winning,, okay? So this is the antecedent. And as I mentioned at the beginning of this discussion, even if you're a native English speaker, these typically cause people a lot of trouble. It's just the way the human mind works. It's not to do with the native language. It's to do with the way the mind works. Well, these two are a little more straightforward than the last one or two, because the when is really almost the same as if. That just tells you the condition in which something is. So the thing that just goes with the when, without an only combined with it, is the antecedent, okay? So the antecedent in this case is that Karl is playing, because it's when he's playing that the team wins. If you know that Karl is playing, then you can conclude that the team's going to win on the basis of this statement. So that's the antecedent there. And it doesn't matter whether the when clause comes first or second as was the case with an if clause. It can come first or it can come second, that's still the antecedent, okay? Being the antecedent is not directly related to whether you're the first clause or the second clause in a sentence, it gets to go with which word you're combined with. If it's an if or if it's a when, they're not combined with the antecedent. If it's an only if or an only when, then that flips it around and then you're dealing with the consequence, okay? Well, that takes care of those kind of examples. And the last three parts of this question, of this problem set, were a little bit different. Okay, well, in this case, it's certainly true, That if m and n are even, Then mn is even, we know that. So the question is, is it the case that if mn is even, Then m and n are even? So that's what this boils down to, this simplification. Okay, we know there's two implications here, so if and only if, so that means equivalence. It means the implication holds in both directions, and one implication is certainly true. If m and n are even, then mn is even. And so it boils down to the question if the product is even, then are the two numbers necessarily even? And once you get it down to that stage, All you need to do is the counter, there are many counterexamples. We could take m=2, n=3 then, mn=6. So, here we've got A product that's even. But it's not the case of those numbers are even. So, this is a counterexample. So, the answer to the question, is no. This is a statement about any pairs of integers. And if we found one pair of integers that makes it fail then, the whole statement fails. So, it's a counterexample that we need to find and we found one, m=2, n=3. The product is even but it's not the case of both numbers are even. Okay, now, let's move on to number 6. So, number 6 ask us, is it the case that mn is odd iff m and n are odd? Well, I'm sure you all realize that the answer is yes. You almost certainly realize the reason, you know reason why because there were two facts. We know that odd x odd, I'm going to say, is not equals is odd. And we know that even x anything, any number odd or even is even. So, if you take two odd numbers and multiply them together, you get an odd number. If you take a pair of numbers, any one of which is even, and multiply them together, you get even. And when you combine those two, this falls out of it. And if you don't see that, I'm going to leave it to you actually. If you want to sort of give a little bit more detail and express it as an implication in both directions, that's fine. I'm sure you could discuss this endlessly on the forums, and that would be a good idea if you want to. But I'm just going to leave it with the observation that it's really just these two facts that give you this result, okay. This tells you how the power t even and odd works. And once you know that, you know that but have fun with this one and discuss it amongst yourselves and settled to your own satisfaction what comes to choose a rigorous proof of this thing. Remember, there's actually no sort of goal, standard of what is or is not a rigorous proof. It depends on the experience of the audience, a proof in many ways involves audience design you've got to cast that proof at level of detail and precision that matches the audience. Typically, a professional mathematician would simply say, And you see this actually in books and in papers, the mathematician might very well say, this is trivial, okay. And that would be a proof. In advanced works on mathematics, you often see remarks, the proof is trivial. It has to be said that a beginner might take several days to see why something's trivial. When mathematicians use that kind of expression, they're doing it with a particular audience in mind, namely, other professional mathematicians. So, if you read that and it doesn't seem trivial to you, it doesn't mean you're stupid. It just means you haven't spent many years working as a professional mathematician. It's just the way we classify things. And proofs involve a lot of audience design when you write and formulate proofs, okay. Well, one way to do all of these is by truth tables. And if you work up a truth table, you find that a is true, that b is true, c is not true, d is true, e is true, and f is true. So, simply by using truth tables. You could answer this. This one, we've already seen in the lectures and discussions, we've sort of looked at these equivalence. This one and where is it? That one, those are examples of what's known as, De Morgan's Laws, after a mathematician, Augustus De Morgan. And if you take a negation with a disjunction, you end up with a conjunction of the two negations. And if you take a negation of a conjunction, you end up with a disjunction of the two negations, okay. So, that was a basic factor about implication that a conditional is true if either the antecedent is false or the consequence is true. So, that was the truth table that we went out for the conditional. And those are De Morgan's Laws, that one was not the case anyway. This one, you could probably reason this one out in terms of implication. Just think in terms of on to which circumstances can you start with an assumption P and did use two conclusions? And then, see when that doesn't happen, okay. And you should end up with this. So, it's possible to reason this one out. Just in terms of implication. And the same too for this one, you could reason it out. This one I think is usually not that one because of the fewer symbols to do with. This says that, if you have an assumption and then, you have another assumption, you can make a conclusion from it. Well, do you do it in two steps, you assume P. And then, on the basis of that showed that if Q is true then, R is true. Or do you simply assumed that both P and Q is true and then, use R. Those two things would be equivalent. This is so if doing it in two steps and this is combining the two assumptions. They both really tell us that there were two assumptions. There's one assumption then, there's another assumption. And in this case, we've explicitly said there were two assumptions and in both cases, it's the R that's following from them. Okay, so, the question is, how and when do you get from P and Q, to a conclusion R? So, this you could reason it out and if you feel uneasy about that, you could just work out the truth table. Okay, but since we spent a lot of time on truth tables in the assignments, my recommendation would be that you would go through these and actually try to reason them out in terms of what they mean. Truth tables are good if you are a computer. But people are not computers, they're much more interesting creatures than that. And I think we have the power of reasoning. And so, I would suggest you go through these and try to reason them out in terms of what these things mean. because that after all is really what this entire course is about. >> In Question 15. >> We make use of the cost evaluation rubric. This is the first of many questions that you'll be asked to do, making use of this rubric in order to evaluate purported mathematical proofs. Well, here's the first one. The claim is that for any two propositions P and Q, not P conjoined with not Q is equivalent to not P and Q. Yeah, [INAUDIBLE] fairly short. In fact, all of the ones I'm going to be using as examples are short. That's why they're good as examples In particular, I want to be able to give the solutions on a single slide, as I'm going to do here. So in a way, these are not typical. In fact, if you do test flights at the end of the course, you will almost certainly see proofs presented by other students, which are much more complicated. And much longer and perhaps have all sorts of mistakes in them. The arguments I'm giving you, they all began as arguments that were produced by students over the years, when I've taught this material. But I've done is I've picked particular aspects of proofs where students typically go wrong. And so the examples will have one or two common mistakes embedded in them. Just so you get used to looking at proofs from the different perspectives, as captured by the rubric, and seeing how they work and don't work. As I mention in the description of the use of the rubric on the course website. The way it will work is that because I'm using short examples in each particular example, some of these features won't really apply, in which case you have to sort of default and give 4 or 0 depending on how the student handles it. Typically you would end up giving full marks because it just doesn't really apply. Okay, well let's take a look at how this one was done. So I'll pretend that it was done by a single student, even though this is a composite of the kind of things that students have done over the years, okay? Well, it's an equivalence. So we have to prove it in two directions. We have to prove that this implies that, and that implies that. So there were two equivalences to prove here. So let's just check the left-right implication. So suppose that not P and not Q is true. Then a conjunction is true, if and only if, the two conjuncts are true. So if the conjunction is true, then both not P and not Q are true, okay, that's correct. That's what conjunction means. If not P is true, that means P is false, that's what negation means. If not Q is true, that means Q is false. So the truth of these two negations means the falsity of P and Q. But if P and Q are both false, then again because of the way a conjunction works, P and Q is false. Hence not P and Q is true. Again because of the way negations works. So that's absolutely okay. Okay, left to right was proved, great. What about right to left? Well in this case, the student makes an attempt to be fairly sophisticated, by saying the other argument, the argument in the other direction works the other way. If it does work the other way, it's absolutely okay to simply say that. There's no requirement that you would have to repeat something that's obviously the case. But we'd better check that it is obviously the case, because we're actually evaluating whether this is the case. Okay so let's spell out what this person didn't do. In other words, we'll try to do the same as here, going the other direction. So we're going to assume it's not the case that P and Q, okay? Let me just say we'll assume that's true. And that's sort of redundant, but I want to talk about truth and falsity. So we'll assume that not P and Q is true. That means that P and Q is false. Well, what does that mean? That means at least one of P and Q is false, Okay? Because you only need one of them false, to make the conjunction false. But the other one, whichever it is, Could be true, it doesn't have to be, but there's nothing to rule out the fact that the other one's true. So, one of P and Q is true, that means one of not P and not Q, could be false, okay? At least one of them is false, so the other one could be true. That means one of those could actually be false. One of these guys, not P and Q, could be false. That means, if one of them could be false, it means not P and not Q, could be false. That's not ruled out. In other words, that implication, Does not work. Just because not P and Q is true, it does not necessarily follow that not P and not Q is true. Of course, it could be false. We're not saying it is, we're saying it could be. So there isn't an implication. There is no implication from right to left because you could have that without having that. That thing, could in fact, be false. In other words, the original claim is a false claim. It's simply not true. So this statement, though it was a nice attempt to be somewhat sophisticated, it didn't work because in fact the argument does not work the other way. I mean, it is essentially the same kind of argument, and if it had been correct, that would have been fine. You wouldn't need to give this. But it's not correct. So now we're going to have to sort of put some numbers in here that capture what we've just said, okay? Well, what am I going to say. There are four marks available for logical correctness. The left to right part was absolutely correct. This is logically correct. So I'm just going to give half marks for that, I want to say two captures the fact that left to right was correct. But remember, logical correctness is just one aspect of proofs. There were other factors, too, and that's what the rubric items are doing. So let's look at what we've go. Clarity, this is absolutely clear. Even here it was clear, it was wrong, but it was clear. So in terms of clarity, this is very clear, so I'm going to give full marks for clarity. Is there an opening, insofar as there's any reason to give an opening here? Yes, it begins by stating that we're going to go from left to right. And so I'm going to give four for that. Now you could say, maybe the person should have begun by saying, to prove an equivalence you have to prove implications from left to right and right to left. Well, it's a judgment call. I would say that for someone at this stage, where they're producing arguments like this, and this is a nice argument, this is perfectly correct. For someone that's producing this, I think it's okay to say, yeah, it's obvious by the way they're doing it that they're doing that. But remember, even though the rubric does break a difficult task into smaller pieces, they're still not easy. So these are going to be judgement calls. I would say that there's enough demonstration of knowing the person, knowing what he or she is doing to just say, I'm not going to insist that they say proving an equivalence is enough to prove left to right and right to left. I think it's clear from the way they're doing it. Well, they do start out by saying assume one and prove the other one, okay? What about stating the conclusion? Absolutely, it was stated, we have implication in both directions. Well, this just justifies what I've just said, so the person has now said we have implication in both directions. So just emphasizing the fact that he or she knew exactly what they were doing. Remember also that we're really try to do formative assessments rather than summative the process is one of seeing what the person has done well and rewarding what they did well, it's not about trying to take marks off. When we do these kind of things we should always be looking for reasons to give marks to acknowledge what's been done correct, and is a point how things to improve. So we begin by adopting a positive giving my x attitude, that's the best way to do this kind of thing, so I'm always going to be looking for reasons to give my x. On the other hand, you cant give my x if it's notjustified. What about reasons? Reasons were given, but it was really the same as in the first case. The reasons for the other part that should've been given weren't because the person thought it was the same and it wasn't. So some of the reasons are here, they were given as the person was going along. So, I'm going to have to give two for that one, because it was only half a proof. The proof was this, the result was false, so this is absolutely false. So the more sucker give here is 2, because even though the reasons we were given that's not of using. That would have been that would have actually been a reason, if it was correct. If this kind of argument was correct that would have been okay, but it's not. And then overall, again, I'm just going to have to give 2, because it's basically just half a result. That's really what it's coming down to, the person has laid it out well, so he or she is going to get a lot of marks for laying it out correctly. But they're going to be losing some marks because there was only half of a thing. I'm just saying there's only half available and so I'm giving half the marks. So, the total is going to be 18. It maybe a little be generous actually, after all this thing is false. In fact, if this was a mathematics course and I was sort of grading people for doing mathematics work, I think I'd have been much more harsh. Had it probably come down at ten, 12, maybe even less, but this is about much more than mathematics. It's about mathematical thinking, it's about communication, it's about understanding proofs. In reality, the mark I've given the person is 2. I've given them two out of the four. So essentially, I've said this is worth half marks in terms of the overall evaluation. Because half the proof is a, okay, but because we are also rewarding or looking to reward, prove, structure, communication, all of those other aspects of proves, which are important, I'm going to give credit for the fact that a person has got the general idea. So I've got 18. As I said maybe we can add it as a little bit generous, I'm inclined to think as a little bit generous myself. On the other hand this is a 75%, which means 25% has been docked, if you like it or not, given. And let's look at what happened. This was essentially just one mistake. The person looked at it and didn't look closely enough, but it comes down to just one error, it's one error in logic. The person who could produce that argument almost certainly could produce the correct argument in the other direction, to show that it's false. So, it really comes down to one slip, and frankly, it seems to me that if you docked more than 25% or maybe 30% for just one error, that's kind of harsh. We are, after all, trying to turn people into better mathematicians, to make them better mathematical thinkers. So let's look at what they do right and then give due credit for that, and this person who's done a lot of things right, there was just one mistake. And it's a mistake that almost certainly this person shouldn't have made, they had the ability not to make the mistake. But, people do make mistakes. So we're not not going to give for free, but having hesitated a little bit on the 18, I've actually now talked myself back into seeing 18 is actually a pretty good grade. Well, that's the end of that problem set.