Well question one in problem set five asks us to judge whether this is a valid proof or not. Well certainly the algebra looks correct, right, if m and n are odd, then by definition an odd number is simply one that's not even and an even number is one that's a multiple of two. So an odd number has to be of the form 2p+1, so that's okay for some p and for some q. So it's odd integers, search that. In which case multiplying them together you get that. The object was correct so mn is odd. Everything so far until this part. Because it doesn't complete the proof. What we've shown is if. We've shown that mn is odd if m and n are odd. And as we'll I'll just improve the implication that way, okay, if m and n are odd. Then mn is odd. So in this statement we've proved the right left implication. And we need to show the only part. We need to prove the only if part. In other words the implication from left to right. It's if and only if. So we've proved the if part from right to left. We need to prove the only part from left to right. Ie we need to show, let's write it out in full if it's not the case. That mn outward. Then mn is odd. Is not odd, yeah, is not odd. Okay, there are various ways you can talk about this. Okay, ie. But if it's not the case that mn are odd. That should have been a comma there. Okay, if it's not the case that m and n are odd. Than m times n is not odd. Okay, I, I got the tens correct. The, the, the chloraility, that was our [UNKNOWN] . Okay, if at least one of m and n is even, than m times n is even. Well, if m equals 2k, it doesn't matter which one we assume is going to be even, because multiplication is going to be commutative. If m equals 2k, then mn equals 2k times n. Which is even. So this was how, this is how to set the well on fire, right? I mean there's nothing terribly deep about this. This is extremely elementary. And it's not that you don't know this and understand it. The question is can you write it down in a proof? and the point here is that we have to start with the defenitions. The definitions are, n is even if and only if, there's an integer k, such that n equals 2k. And n is odd, if it is not even. So starting from these two definitions, this is the proof we give. The first part that was actually given to us, that was fine from those definitions, we started with the definitions. Going in the other direction, [UNKNOWN] the definition. So the question is simply, all this really involves is reducing this to the definition and then the argument itself is very simple. that was a slightly more complicated bit of algebra, but not much more. So it's not about the complexity of the algebraic manipulations, you've been able to do that for years I know that. The question is, can you reduce the statement to something coming from the, the definitions. it, it, because in many cases you simply would, this would be so simple compared with most proofs in mathematics. You wouldn't go into these details. But the focus here isn't on the, this stage. Isn't on the, the complexity of the argument. The focus is on the structure of the argument. How are you able to get the logical structure right? And experience tells me that it takes most of us, and it certainly is with me, a long time to get the sense of what do I have to prove in order to prove something. that's not something, that's not cookie cutter. You've got to build up experience in what constitutes a proof. And be able to judge whether a proof is correct or not. Okay, let's go ahead and look at number two now. Let me jump first of all straight to the to the correct answer. And the correct answer is a. And this says there are people and there are times. That which you can fool those people. So, you can fool some of the people. Some of the time, remember existence quantifies what we use in mathematics to capture the word sum, or at least one. little bit a typical in terms of what the word sum means, in, in, in, Apedel language. Okay, and here the way I've expressed it is there's a person and there is a time. So, so, you cannot fool that person at that time. Okay, there's a person and a time. So, so you cannot fool that person at that time. Which is actually equivalent to saying you can't fool all the people, all the time. Formally, you could take that part and you could re-write it, it's not the case that for all x. And for all t, fx t. It's not the case that you can fool all the people all of the time. Okay? But, but I wrote it this way because we're only asking for the equivalent. I'm not saying which is the closest way of capturing it. I'm just saying which one is equivalent. Okay. let's look at part b. What does this. well if the first part were the same in all of these three, so the distinction is in the second part. So let's see what this one says. Can we express that in English? well this is tricky to express in English because of these American Melanoma Foundation type issues. I mean you could say something like oh, let's say you can't fool everyone let's try something like this, at sometime or other. For everyone it's the case that you can't fool them at sometime or other. and really, what we're saying is, is, it's not the case for, for every person. there's a time when you can fool them. Okay. So you can't fool everyone at some time over that. And and I'm, that's sort of, I can't think of a way of saying this that really, that reads well in English. [LAUGH] And yet, it really captures this. but hopefully, it's clear what the thing means. It's not the case for every person there's a time when you can feel them. And, okay, and to me that says the same. But, but, you know, you may interpret that one different because this is one of these thing like American Melanoma Foundation. it's, it's ambiguous, as natural language so often is. Okay, let's see again. And in the case of this one, what does this one say? It's not the case that the reason x number is a t, so you should convert. This one I think is easy to say in English. Okay, it basically says you can never full anyone. Okay. It's not that you cannot find a single person at a single time so that you can fool that person at that time. You can never fool anyone. So this one I'm very happy with. Okay, that, that one really captures it. and the smiley doesn't indicate that I think that's true. It indicates that I think that's a really clean interpretation of what that one means. One and number four doesn't arise because we've already found something correct. With questions three, four, and five, which is truth table arguments. That was essentially a revision material. So I won't go through those, those here. let's just take a look at question six which when you first meet it, it looks as though it might be to be deep, especially since we've, we've seen this now after we looked at things like square root of root two and square root of root three. But actually, this one, when you step back and think about what it says, tends out to be very simple. we simply observe that whenever n is a perfect square, in other words. It's a square of some integer, any integer and there infinitely many integers, so there are infinitely many perfect squares. Then of course the square root of n is just k. Which is rational. And that's all there is to that one. And I'm not only is it rational, it, it's a whole number. The point is their infinitely many numbers in for which the square root of n is a whole number in Hen's rational. Namely all the perfect squares. Okay, that's only still out. You don't really need to say anymore. That's it, now here's that it's true and it tells you why it's true. And that's the proof. because you won't ask really to sort of, say whether it's true or false. But it was this that I was interested in. Can you construct a proof of its negation? Of it, or its negation. That that was really what it's about, it's all about proving things here, okay? Well, finally question seven is one of these fallacious proofs. There's there's a whole range of these things. quite amusing. usually they depend upon dividing by zero in some disguise form. this one's different, okay? So let's follow it through, because normally they, the point is just to find the mistake. We're clear that there's going to be a mistake because one doesn't equal two. and indeed one of the things that I asked you to do was to find a mistake. But, the focus here isn't so much on finding the mistake. It's how would you grade this as a proof? And remember when we're grading proofs logical correctness is certainly part of it, so there's going to be some losses of grade here. But also it's about communication, okay. So let's go through this one and just see how we would grade the thing. Okay. Well for example, is it, is it clear? Well it's absolutely clear, so we're going to have to give four mark for, for clarity, this is very clear. There's a good strong opening, we start with the identity, 1 minus 3 equals 4 minus 6, okay, both sides are equal to 2. So four marks for giving the, the opening. the conclusion is stated. So it's 4 marks for stating the conclusion. And reasons given, yes absolutely. Adding 9 plus 4, 9 over 4 to both sides is just completing the square to, to, to, give you perfect squares. Then we're saying that it factors, which it does. That's why we added 9 over 4 to both sides. Now I'm taking square root of both sides. Absolutely! Reasons, reasons, reasons. So in terms of the structure, this is wonderful. above all I'm going to have to give it a 0, right, and that's clear, because the thing is plain false. the question is, you know, what do I give for logical correctness. Well, first of all we need to identify the the evidence where the errors occurring, right? and the errors here. it's in this line. Because when you check square roots, then of course there are positive negative square roots that we had. And in fact, the correct solution is to see that, then, then, minus on the left hand side the negative root on the left hand side, minus 1 minus 3 over 2 and it's a positive root you take on the right hand side when you take square roots. So that's what you should have. In other words, 3 over 2 minus 1 equals 2 minus 3 over 2, in other words, a half. It equals a half. Okay, so now we dig f, I mean not so. That's nice, the world still exists. a half is equal to a half, even though one doesn't equal two. So the issue here was when your taking square roots there are two, two possible signs. And and the one that makes it valid is is is the negative on the left and the positive on the right. Okay. So, this is where the mistake is. The only question is, do we give partial credit for the the fact that there's logical correctness everywhere else? This is a judgement call. I would say that these steps are so simple it's just ele, very elementary algebra arithmetic. That's this level I'm not going to give particular credit for getting these bits right. And I'm going to say that this is a big mistake. This is huge. Okay, when you take square roots you have to take a plus or minus. That's some mathematical mistake. Realizing that, that, that not recognizing that you need to worry about the signs when you check square roots. So that's a big error for mathematical points of view, so this grading that I've given it. Which, which means I'm actually giving 16. Means I'm giving no points at all, on the mathematical correctness. I'm giving lots of credit on everything else. Now if this was a mathematics course, I wouldn't be using this rubric this way. I would take account of these, but they would have less weight. I mean, these have all got the same weight, a maximum of, of four. and that's because of this focus of this course is on, on proofs and reasoning. And, and common communication. And these are important parts of that. you know the assumption in this course is that you already can do some mathematics. and so I'm not really grading you on that. I'm grading you on mathematical thinking and mathematical communication. so the zero max here on the mathematics. Build this formax and everything else. now this is the reason why I did this one. This is where everything is welded out, but the thing is planned wrong. So it distinguishes between mathematical correctness and the mathematical thinking, and the communication, and then writing a proof out correctly. Having said that, in this case, it's blatantly obvious that something's gone wrong. But very often in mathematics, mathematicians, professional mathematicians make, make mistakes, buried in proofs. And because they're professionals, they can usually write things correctly. We know how, we learn this part of becoming mathematicians. We learn how to write things out, we give reasons. So in fact, this, although it looks absurd here you often find that because mathematicians know how to lay things out well, results get published even though they're absolutely wrong, whenever one gets published. Basically what that means is that the referee of the journal where it's published, has graded it and said this is, this is perf this is actually correct. So, you know, this is not an unrealistic situation in principal. When false results get proved. When false results get proved, when false results get published, that is, because they've never been proved. The false results get published when, what's going on then is that the referee has gone through it and and think, and thought, everything's okay. Okay, here it's dramatic because the answer's absurd. And these peyton, they force. But this is actually a not unrealistic scenario. And then so, giving a high grade is, isn't, you know, it's not about. I mean it's, it's, it's, I'm doing this to emphasize the distinction between the various things we're looking at here. And this one I think makes it clear. That there are other issues involved other than, other than logical correctness. But as I said, if this was a regular mathematics course, I wouldn't be giving 16 out of out of 24. In fact, I'm not even sure I'd give any marks for this, if a student came up for this. Because it really this is really a big mistake. Okay? But within this context, within the context of this course this is the kind of thing we're looking for in looking at mathematical thinking and mathematical proofs and communication. Okay. There we go! Well that's the end of the, the problem set questions. But let me leave you with one more tantalizing little puzzle. We are probably familiar with the story of Archimedes, lived in Greece about 250 B.C [UNKNOWN] who was asked by the king to determine whether a crown had been given was actually made of pure gold or not and that involved calculating the density and he knew how to calculate to find out the, the, the, the massively, the weights of the crown. But the question is how do you calculate its volume? now, now Archimedes knew lots of mathematics for calculus in volumes, indeed he'd invented a lot of that mathematics. He was able to calculate areas of circle and volumes of spheres and various other shapes like boxes and rectangles and pyramids and so forth. He knew all of that stuff. So he had a lot of mathematics at his disposal that he could have applied. But it didn't seem to work for something irregular, like a crown, or at least not easily. But then one day when he's taking a bath, this is sort of the story goes, when he is taking a bath, he has this amazing insight. He says to himself, if I immerse the crown in water, it will displace some water. In fact, the amount of water it will displace is exactly equal to the volume of the crown. So if I collect the water when it spills out of the bath, and if I, when I put a crown inside it, into, into the water, then I'll be able to just measure the volume of the water in, in a standard way and I'll know the volume of the crown. And as that story goes, he was so impressed and tickled about his solution that he jumped out of the bath and ran stark naked through the streets of Athens crying out eureka, eureka! Which is Greek for I found it! I found it! now, I've known lots of mathematicians. I certainly haven't known Archimedes, but I doubt if even a mathematician deep in the throes of solving a problem would run naked through the streets. However, I can imagine him being extremely pleased with himself and having an adrenaline rush when he had that insight. Because that's a great example of thinking outside the box. He knew lots of techniques for calculating volumes, he invented many of them, but on this occasion he thought outside the box and found a really elegant solution that was different. And the puzzle I'm going to give you is very much along those lines and it actually is about taking a bath. And here it is if it takes half an hour for the cold water faucet to fill your bathtub and an hour for the hot water faucet to fill it, how long will it take to fill the tub if you run both faucets together? Now this looks like one of those, those frustrating little word problems you get in, in, in, in high school. Okay, where you, you end up, you sort of say that the, that the rate of flow of the cold water be f. And, and then the t. And you, you, you write down some equations and you, you you figure something out, okay. I'm sure you know how to do that. You can apply standard technique for doing this kind of thing involving rates of change, and you'll get the answer. But you don't need to know any of that, you don't need to do any calculations at all in order to solve this one. If you think outside the box you can answer it without doing any of those calculations based on the rates of flow or anything like that. You simply have to think of the problem a different way. it if helps if you're inspired by the Archimedes story, you might want to run a bath and get in the bath, and then and see if you can come up with the idea then. I, I must admit from personal experience that a lot of my best ideas in, in solving mathematical problems, including mathematical theorems throughout my career. Have actual been obtained. When I've been sitting in a bath or shortly after I've taken the bath, there's something so relaxing about getting in the bath that it freeze the mind to come up with these out-of-the-box solutions. So, this isn't, this isn't really about do you know the standard methods. You probably do. This is about can you think about this in a different way that allows you to solve it? Without really doing any arithmetic whatsoever. Okay, we'll give that hint. I'll leave you to it. Enjoy.
