How did you get on with the assignment three? I kept it short since I wanted to give you a lot of time to come to terms with the tricky notion of implication. This lecture's going to be fairly short as well, though the assignment that goes with it will be the longest of the entire course. I'm not expecting you to complete it all by the next lecture, though. The next topic I want to look at is logical equivalence. Equivalence is closely related to implication. Two statements are said to be equivalent, or more fully, logically equivalent, if each implies the other. Equivalence is a central notion in mathematics. Many mathematical results are proofs that two statements are equivalent. In fact, equivalence is to logic as equations are to arithmetic and algebra. And you already know that equations play a central role in mathematics. Just as we had to introduce a formal version of implication that avoids the complex issue of causation, namely the conditional, we have to introduce an analogous version of equivalence. It's called the biconditional. Fortunately, we did all the difficult work with implication. Now we can reap the benefits of those efforts. Two statements phi and psi are said to be logically equivalent, or just equivalent, if each implies the other. Since we have a formal version of implication, named the conditional, it follows that there's a formal version of equivalence. We call it the biconditional. The biconditional of phi and psi is denoted by means of a double-headed arrow, like this. Formally, the biconditional is an abbreviation of phi conditional psi and psi conditional phi. Since the conditional is defined in terms of truth values, it follows that the biconditional is defined in terms of truth values. If you work out the truth table for phi conditional psi and for psi conditional phi and then you work out the conjunction, you'll get the truth table for phi biconditional psi. And if you do that, what you will find is that phi biconditional psi is true, if phi and psi are both true or if they're both false. One way to show that two statement phi and psi are equivalent is to show they have the same truth tables. Actually, to avoid confusion, let me change those to capital Phi and capital Psi, because I want to use the lower case phi and psi for something else. Because here's the example that I'm going to look at. I want to show that phi conjoined with psi or not phi- Is equivalent- To phi conditional psi. This is actually going to be my capital Phi, and this is going to be my capital Psi. If I was teaching this material at a high school level, I'd be very careful to choose different letters to denote everything but we're looking at college, university level mathematics now. And university mathematicians, professional mathematicians frequently use upper and lower case symbols in the same context. And part of being able to master university level mathematics is actually getting use to disambiguous notations. What we have to do is work up the truth table for this, the truth table for that, and show that they're the same. We'll start with phi, psi, and we're going to need to work out phi and psi. We're going to need to work out not phi. Then I'm going to need to work out phi and psi disjoined with not phi, and then I'm going to have to compare that with phi conditional psi. As usual, we start with the four combinations, TT, TF, FT, FF. By the way, it doesn't matter which order you right these in, so long as you get all four possible combinations. The way I've written it is the way that most mathematicians write it. Conjunction. Conjunction is true if the two conjuncts are true. TT gives T, here there's an F, so you're going to get an F. Here there's an F, so you're going to get an F. We have two Fs, so you're going to get an F. Negation simply flips T and F, so we have T T F F becomes F F T T. Now I'm going to disjoin this with that, and disjunction picks out at least one T. There's a T there, so I'm going to get a T here. There's no T, so I'm going to get an F. There's a T there, which gives me a T, and there's a T there, which gives me a T. I know the truth value is for Phi conditional Psi. They were T, F, T, T. If you look back at the earlier work, you'll see that that's what we worked out before and now I just compare that column with that column. T F T T is the same T F T T. Because these two columns are the same, I can conclude that is equivalent to that. I should mention that proving equivalence by means of truth tables is very unusual. It's only of a special case of equivalence. In general, proving equivalence is really quite hard. You have to look at what the two statements mean and develop a proof based on their meaning. Equivalence itself is not too difficult a notion to deal with. What is problematic is mastering the various nomenclatures that are associated with implication. We start out with the notion, phi implies psi. I'm really going to be thinking of genuine implication here, but you can interpret everything else in terms of the conditional. We'll look at that, I'll talk about that a little bit later. If mathematicians always describe this situation in this way, then life would be very simple. But we don't. There are many different expressions we use to describe phi implies psi. Some them are intuitively obvious and some of them are actually counter-intuitive when you first meet them. The following all mean phi implies psi. One, if phi, then psi. Okay, that's fairly obvious. Two, phi is sufficient for psi. In order to conclude that psi, it suffices to know phi. On the face of it, that's fairly straightforward but in complicated situations, people can sometimes get misled by the use of the word sufficient. The use of the word sufficient is by no means the most problematic In the context of implication. This one causes people a lot of problems at first. Phi only if psi. That's not the same as if psi then phi. The reason I'm emphasizing that is that the if here goes with the psi. Just as it does here. Let's put quotes on that one. The if goes with the psi, the if goes with the psi. But when it's expressed in this way with an only if, it flips around the order of the implication. For example, some of you may know that I'm a very keen cyclist. On the other hand, I've never qualified, nor have I ridden in the Tour de France. However, I could honestly say a person could ride in the Tour de France only if they have a bicycle. Well, that's true because if you don't have a bicycle you can't ride in the Tour de France. So a person can ride in the Tour de France only if they have a bicycle. Now that's true for me. But it doesn't mean the same as if you have a bicycle then you can ride in the Tour de France. I actually do have a bicycle, I have several, but I can't ride in the Tour de France. So it's true to say, Keith could ride in the Tour de France only if he had a bicycle, because that's true for everybody. You can only ride in the Tour de France if you have a bicycle. Otherwise you can't ride in it. But it's not the same as saying it this way around. That if you have a bicycle, then you can ride in the Tour de France. So, gotta be a little bit careful. These are often confused which is why I'm stressing the distinction. Okay? Let's look at this one because this one, that I'm going to write next, is one that I've already really encountered in my discussions about the Tour de France. Psi if phi, notice that we've flipped the order now. Phi is the antecedent, psi is the consequent. Let me write those down, just to remind us. Phi is the antecedent. And psi is a consequent. And those are the roles going to be played by phi and psi in all of these. So phi is the antecedent, psi is the consequent. Phi is the antecedent, psi is the consequent. Phi is the antecedent. Psi is the consequence. They've flipped. Phi is still the antecedent. Psi is still the consequence. This still means that Phi implies Psi. But now the consequent comes first. The antecedent comes second. Psi if phi. If like me you're a cyclist, you can think about that in terms of owning a bike and riding in the Tour de France or use your own favorite example. Think of something that you do that requires something, that requires some equipment or some preparation or something. And just interpret these things in terms of something that's meaningful in your life. Now that should help you to sort these out. Okay, there's a couple more I want to talk about. Psi Whenever phi. Again, the order is flipped from the order in implication. That's still the consequence. That's the antecedent, but they've been written the opposite way around, psi whenever phi. And finally, psi is necessary for phi. Again, the order is flipped around. That's still the consequent, that's still the antecedent. Psi is necessary for phi. In order to ride in the Tour de France it's necessary that you have a bicycle. It's not sufficient that you have a bicycle. To ride in the Tour de France it's necessary that you have a bicycle but it's not sufficient that you have a bicycle. That's the example that I would use for all of these but you can pick your own favorite example to understand these things. It's important to really master this terminology because it's used all the time, not just in mathematics, but in science and in analytic reasoning and thought in general. This is not just mathematical language, this is the language that people use in legal documents, in logical arguments, in analytic arguments, and in discussions and so forth. So understanding language as it's used is very important in many walks of life. And having introduced terminology commonly associated with implication. We have an associated terminology for equivalence. Phi is equivalent to psi is itself equivalent to, by the way this already shows how ubiquitous the equivalence is because the obvious word to describe this, let me put quotes around that just to ambiguate. The obvious way to describe that this is equivalent to something is to use the word equivalence as well. I mean, equivalence is just a very basic concept in mathematics. So this sentence, this statement, this claim is itself equivalent to, well, phi is necessary and sufficient for psi. This combination of necessary and sufficient is very common in mathematics. And b, phi, If and only if psi. That's also very common in mathematics, if and only if. Notice that we combining necessary and sufficient. With necessary we have psi before phi. With sufficient we have phi before psi and that get's us the implication in both directions, and equivalence means implication in both directions. So, the fact that it's in both directions is captured by the fact that here we have the psi before the phi, and here we have the phi before the psi. Similarly with b. If and only if combines only if, where phi comes before psi, with if, where psi comes before phi. So in both of these cases we have an implication from phi to psi and from psi to phi. Final remark. This expression is often abbreviated, iff. Iff is a standard mathematician's abbreviation. For if and only if. So, if and only if or if if means the two things are equivalent. Okay. Once you've mastered this terminology you should be able to read and make sense of Pretty well any mathematics that you come across. That doesn't mean to say you understand the mathematics itself, but at least you should be able to understand what it's talking about. And that's a first step towards understanding the mathematics itself. And that's all there is to say about this. The rest is really up to you to spend some time mastering the concepts and the associated terminology. Time for a quiz. This quiz comes in four parts. So how did you do? Well, let's see what's going on here. Which of the following conditions is necessary for the natural number n to be a multiple of 10? So the question we have to ask ourselves is. Does n being a multiple of 10 Imply the statement. To be necessary, n being a multiple of 10 has to imply the statement. Well let's see. Does n being a multiple of 10 imply that it's a multiple of 5? Yes it does, so that one's necessary. Does n being a multiple of 10 imply this is a multiple of 20? 10 is itself a multiple of 10. But 10 is certainly not a multiple of 20. Does n being multiple of 10 imply that n is even and that it's a multiple of 5? Yes, does n being a multiple of 10 imply that it's a multiple of 100? No, does n being a multiple of 10 imply that n squared is a multiple of 100? Yes, so the three conditions that are necessary for the number n to be a multiple of 10. Condition one, condition three, condition five. Okay, let's move on to Part 2. Well this time, we have to ask the question, does the statement imply n is a multiple of 10. Okay well, does this statement imply that n is a multiple of 10? No it doesn't. 5 is itself a multiple of 5. 5 is not a multiple of 10. Look at number two, is it the case that if n is a multiple of 20 then it has to be a multiple of 10? Does this imply that? The answer is yes. Does n being even and a multiple of 5 imply that it's a multiple 10? Yes. If n is 100, does that imply that it's a multiple of 10? Yes, if n squared is a multiple of 100, does that imply that it's a multiple of 10? Yes, so in this case, 2, 3, 4, and 5 are the correct answers. They're all sufficient for n being a multiple of 10. Okay, let's move on to part three. For this one, we have to compare our answers for the two previous questions. The first question was necessity and the second question is sufficiency. For necessity, we had 1, we had 3, and we had 5. For sufficiency we had 2, 3, 4, and 5. This asks for necessity and sufficiency. Here we've just got necessity, there we've just sufficiency, there we have both of them. There we just have sufficiency, there we have both of them. So the one's that are necessary and sufficient, this one and that one. Okay, let's move on to part four. For this one, the question we have to ask ourselves is, what does the implying? Well, in number one, this does the implying. So that's the antecedent. In question two of statement two. Well, even though it's written the opposite way around, it's essentially the same statement. It's the alarm ringing that does the implying. What about number three? Keith cycles only if the sun shines, what's doing the implying? Keith cycling, if you see me cycling, you can conclude that the Sun's shining because I cycle only if the sun shines. Incidentally, I was brought up in England, so that's not the case. I'm quite happy to, well, I'm not happy to ride in the rain, but I do ride in the rain. But it's a good example. Number four, what does the implying? Well, Amy arrives does the implying. So far, I've distinguished between genuine implication and equivalents, and they're far more counterparts. The conditional and the bi-conditional. In the daily work however, Mathematicians are very not particular. For instance, we often use the arrow symbol as an abbreviation for implies. On the double headed arrow is an abbreviation for is equivalent to. Although this is a very confusing to beginners, it's simply the way a mathematical practice is evolved and there's no getting around it. In fact, once you get used to the notions, it's not all this confusing as it might seem at first and here is why. The conditional and bi-conditional only differ from implication and equivalents in situations that do not adrise in the cause of normal mathematical practice. In any real mathematical context, the conditional effectively is implication, and the bi-conditional effectively is equivalent. So having made note of where the formal notions differ from the everyday ones, mathematicians simply move on and turn their attention to other things. The very act of formulating formal definitions creates an understanding of implication and equivalence that allows us to use the everyday notion safely. Of course, computer programmers and people who develop aircraft control systems don't have such freedom. They have to make sure all the notions in their programs are defined and give answers in all circumstances. Okay, that's the end of lecture four. As I said at the start, it's been a fairly short lecture. My reason for keeping the lecture short is that the upcoming assignment is much longer than the others, it has to be. Implication and equivalence are at the heart of mathematics. Mastery of those concepts and of the terminology associated with them is fundamental to mathematical thinking. You simply have to master implication and equivalents before you can go much further. And there's only one way to achieve mastery, right? Remember the story of the elderly lady who approached a New York City policeman and asked, officer, how do I get to Carnegie Hall? The officer smiled and said, lady, there's only one way, practice, practice, practice. So, I suggest you carve out some time, grab some food and drink, and head off somewhere quiet to complete as much of assignment four as you possibly can.
