Assignment four was pretty long, but I would hope that by now, working with other students in the class, you were able to get through most of the ones that you were tempted. I'm just going to focus on three of them, I think, I'll look at number 6, 10, and 12. Most of the other ones were just straightforward through table calculation and I'm pretty sure that by now, you should have done those. And if not then do go back and talk with friends and other students and try and sort it out. And you may not have had too much difficulty with 6, 10 and 12 as well. But let me just play it safe and work through these as there's at least one here that's a little bit tricky, it's not the first one. Incidentally the first one, it's a true statement, this number is a prime number but that's not what the question is about. The question is what's the negation or the denial and then the simplest way to to say, well 34,159 is not a prime number. Or else, you might say 34,159 is a composite number, either way I think it's fairly straight forward. This one is straightforward too but I can't think of any nice way of writing the answer that sounds like good English. You would have to say, Roses are not red or violets are not blue, which is the kind of sentence that you will only see or hear in a class of logic as we are looking at now. Mathematical uses of language, precision in language. So it's correct but it's just not an elegant sentence, you wouldn't normally say something like that, this one is tricky, actually. And it's tricky because there's a negation floating around in here. Remember when you got a conditional phi yield psi, when you deny it what you get is you get the antecedent conjoined with the negation of the consequence. In this case, the antecedent is there are hamburgers. Sorry, in this case, the antecedent is that there are no hamburgers. There we are, I almost slipped into that mistake myself. You've gotta be very careful with this. The antecedent is there are no hamburgers, so let's just write that down, there are no hamburgers. Conjoined with the fact that I won't have a hot dog and I think in this case it's more natural to write the conjunction using but rather than and. And say, but I won't have a hot dog. Okay, so we've got the antecedent that there are no hamburgers conjoined, in this case with a but because I think it's more natural to say that as a but. But I won't have a hot dog. Okay, well think about that one for a bit. In my experience, students often have trouble with this because of that negation, and it throws you off. So, you have to be a little bit careful. Fred will go, but he will not play, again the but is really the same as an, and so when we negate that, we'll get a, Fred won't go. The and becomes an or, we've got or, he'll play. In fact I think in English we've probably more likely to write it differently. We're probably more likely to say, Fred will play, Or, He won't go. I think we do that because we'd read into this some causality, some connection between the two. Now that's going beyond just doing the straight negation, but I think that's typically how we would understand it. So to me that seems a more natural way of saying it. Just a clause with the kind of thing people normally do. But in terms of negation, we just took the negation of the two parts together with the negation of the but which becomes a disjunction, which is a or. So I got, Fred won't go or he'll play. Or more naturally, Fred will play or he won't go. This one is actually messy if we do it in English but it is much easier to do it symbolically. So, let's do it symbolically first, the original statement is that x is negative or x is greater than ten. If we take that statement and negate it, then the x less than 0 becomes an x greater than or equal to 0. The or becomes an and, x greater than 10 becomes x less than or equal to 10, so that just becomes 0, less than or equal to x, less than or equal to 10. If you do it in English, you're going to have to say something like. The number x. Is non-negative, you can't say positive because the negation of negative is not positive. Because of zero, you've got to say a non-negative. The or becomes an and, and the negation of greater than ten is not less than ten, it's less than or equal to ten. You have to say less than or equal to 10. And I don't see any way of doing it in English, other than with an awkward sounding sentence like this. Symbols provide a much more efficient way of expressing this idea, and indeed, for dealing with the negation. Okay, but we've found an answer to the question, the last one, part (f), we will win the first game or the second. The negation is, we'll lose the first game and we'll lose the second game, and the simplest way to write that is to say we lose the first two games. Remember the question asked for a denial, now that's just not the strict negation, it's a natural, well not necessarily natural, but it's an equivalent version of the negation. And for some of these we were able to find pretty good answers. In others it was kind of messy, but there were a couple of tricky ones floating around in there. So I think you need to be a little bit careful about these kind of things. Dealing with language precisely is actually not easy because our mind jumps ahead. How many of you jumped from this one to something like If there are hamburgers, I won't have a hotdog. Now students often do that one, gotta be careful about the precision of language. Human beings are very smart with using language in everyday terms. But the cost is that we drop precision, we think in terms of meanings rather than what we think in terms of the way we understanding the meanings rather than what the literal meanings are. The whole point of this analysis of language that we're doing now, is to be very very precise. And then examples like this, remind us that precision is not easy. You have to work at it, okay? Well that was number six and what was the next one I was going to do? I was going to do number ten now. So, let's look at number ten. Well for number ten we have to refer back to question nine. I'm not going to do this one for you, this is just truth tables. By now you should be able to knock off truth table arguments easily. And so let me focus on the first one, which is b. And we want to show that these two things are equivalent. Now equivalence is a two-way street. We have to show that this implies that, and that that implies that, and so there's two parts. I'll first of all, go left to right, I'll show that this implies that. Well this tells us, That we can deduce psi and theta, That formula, from phi. But we know that we can deduce, Psi and theta, From psi and theta. Now it's just a conjunction, and if we know a conjunction, we know both of the conjuncts, okay? It follows, That we can deduce psi and theta, From phi. We can start with phi and deduce psi and theta, that single formula. And from that single formula, we can deduce those two formulas. And therefore, by chaining those two things together, it follows we can deduce psi and theta from phi, okay? That is, we know that we can deduce, Psi from theta, and we know that can deduce psi from phi, and we know that we can deduce theta from phi. Okay, we can start with phi and we can deduce psi. We can start with phi and we can deduce theta. So we can go from phi to psi, we can go from phi to theta. And if we can do those two, Then we have got their conjunct. If that's established, and that's established, then it means we've established phi yield psi. And phi, here's theta, okay? Well, that was left to right direction, what about going from right to left? Assume this guy, and deduce that one, okay? Well, let's see what that says. That says, We can deduce, If we can deduce a conjunction, we can deduce each conjunct. So we can deduce that, and we can deduce phi use theta. So we can deduce each of those two things are possible. Hence, We can deduce psi from theta and from phi, And we can deduce theta, From phi. But if we can deduce psi from phi, and we can deduce theta from phi, it means, We can deduce psi and theta from phi. And if we can deduce psi and theta from phi, that means we've got phi yields psi and theta. Well it's a little bit intricate. First of all, you have to keep track of what's been proved. And secondly, you have to remember to call phi phi, psi psi, and theta theta. And when you're in the middle of the logic as I was, you'll sometimes refer to things the wrong way. So if you rewind the tape and play it through a few times, made the obvious corrections when I meant to say phi and psi, and so forth. And then you should be able to unwrap that one. And I'm going to leave c for you to try it. So if you haven't got this one out already, having seen me get through part b, see if you can do part c. Whatever you do, don't take this literally and try to just change symbols. Get a handle on the kind of reasoning I was using, and then use that reasoning to do this. Throughout this course the worst thing you could do is look at any one answer and try to change the symbols and change the numbers. That might have got you through high school, but it ain't going to get you through much university mathematics, believe me. You've gotta go back and understand the method. So first of all, understand the reasoning. That already, I think, could be a bit of a challenge at first. And then when you've done that, have a shot at perhaps c, okay? Let's move on to the next one. Okay, and that next one I want to look at is question 12, and that is talking about the contrapositive. And these are the four statements that we have to write down the contrapositives of. So if two rectangles are congruent, they have the same area. So the contrapositive would be, If two rectangles, Do not have, The same area, They are not congruent. Number two, the contrapositive would be if in a triangle, With sides a, b, c, with c being the largest, It is the case that a squared + b squared is not equal to c squared, Then the triangle, Is not right angled. So if in a triangle with sides a, b, c, c the largest, a + b squared is not equal c squared, then the triangle is not right angled. Contrapositive to this one, if n, Is not prime, Then 2 to the n- 1 is not prime, or if you like, if n is composite then, 2 to the n-1 is composite. And then, for part d, If the dollar, Does not fall Then the yuan won't rise. Well, actually, writing these down was fairly straightforward. It's just a case of looking what the contrapositive does. It flips the order of things and puts negation in front of them, so actually writing them down was not what the challenge was. The point of this exercise was to give you an example to sort of help cement the fact that the contrapositive is actually logically equivalent. Because, is this one, two, yes, congruent rectangles have the same area. If rectangles do not have the same areas, they are not congruent. These are both true. If a triangle is right angled, so we got this Pythagoras Theorem. If a triangle with a side inside it, each one of these is clear that one is equivalent to the other. If this thing happened to be true, then other thing would happen to be true. That thing true then that will be true, so I think these four examples will illustrate the fact that contrapositives are logically equivalent to the original statements. Okay, I said I was just kind of those 6, 10, and 12, but I was waiting soon this one I thought it will be a good idea to look at the Question 14, which is very similar. So, let me do Question 14 as well. Okay, and number 14 is about converses. Where we're going as with a contrapositive you flip the order around but in this case you don't inject any negation sign, simply the implication in the opposite direction. Okay, which means actually these four are going to be very straight forward because we're just going to flip the order. So this one would be, If two triangles, Have the same area, Then they are congruent. Now already we've shown by an example now. The converse is not necessarily logically equivalent to the original statement. Because the original statement is true. But the converse is false Now this we know is the case, let's look at this one. The converse of this one is the following. If, in a triangle, with sides a, b, and c, with c the largest. It's the case that a squared plus b squared equals c squared. Then the triangle, Is right angled. Now this is true. It's Pythagoras' theorem. And the converse of Pythagoras' theorem is true. So these are both true. But what's going on is that we've actually started out with something that's not just an implication when it actually is an equivalent. Then when you've got an equivalence, and you take the converse, you've just got another part of it. The statement and the converse are both the two halves of an equivalence, so you can get the same truth values, but only when there's an equivalence already there. In the case of this one, The converse would be, if n is prime, And 2 to the n-1 is prime. And the converse of this one is, If the dollar falls, The yuan will rise. And in this case there's no reason to assume that one necessarily follows from the other, unless in some way the other two currencies are linked. Okay? Well. I hope you manage to get the four converters written down correctly. But as I indicated a moment ago that wasn't really the point of this. The point was to sort of give us examples of the fact that converses don't necessarily yield equivalent statements. When you take a converse you now get something that equivalent. Whereas with the contrapositive that we looked at in the previous example, in example 12. When you take the contrapositive you do get something that's equivalent. Okie dokie. Well, that should I think take care of assignment #4.
