 Hello and welcome to the screencast on working with definitions. So I want you to think back to the very first screencast we saw in this series where we were discussing statements and it gave you a couple of examples of statements. January is the first month of the year and July is the first month of the year. Now these were definitely statements in the logical sense so they have a definite truth value. The first one is true while the second one is false, right? Well it depends on what we mean by year it turns out. If we mean the calendar year then yeah, of course January is first and July is not. But suppose what I really meant was the year in which I've been alive. You see my birthday is in July and so for me I could say that what I meant by a year quote unquote is a full year of being born. In which case July is the first month of that quote unquote year and January is not. So it might be a little annoying to play these kinds of semantic games with our statements but it's an important idea here and that is the definitions matter. Before we can work with any kind of logical system like mathematics we need to learn how to read definitions and in keeping with the theme from earlier videos how to play with definitions. And that's what we're going to learn how to do in this video. So let's start with a very simple definition namely that of an even number. You have an innate sense of what even means and you can certainly pick out even numbers in a line up but forming a rigorous mathematical definition does take some work. You might start by saying that an even number is defined to be one of the numbers 2, 4, 6, 8 and so on. Now this definition is partially true. Those four numbers you see there 2, 4, 6, 8 are indeed even but there's two big problems with this definition. First of all it leaves a lot of stuff out. For example 0, negative 2, negative 4 and so on aren't specified in the list and it's a little hard to see just from looking at the list that those 0 and negative even numbers ought to be in the list of even numbers. And second of all this definition doesn't scale up well. That means it doesn't provide an easy way to tell if a new number is or isn't even. For example let's look at the number 1, 2, 3, 4, 8, 0, 2. Now is that number even or not? Now the only way to tell from this definition is to try to extend the list out until we get into the neighborhood of this number and see if 1, 2, 3, 4, 8, 0, 2 is in it. Now that's really inefficient and takes a long time. And in fact that's not how you determine whether that number is even or not anyway. The way you did it is to do it in some way that doesn't involve listing things. What you did probably was look at the 1's digit of that number and realize hey that's a 2 and anything that ends in a 2 must be even. So based on that reasoning we can upgrade our definition to something that looks more like this. An even number is a number that ends in 0, 2, 4, 6, or 8. And in fact this is the definition my kids get in elementary school and maybe some of you are in the same boat. But there's still a problem with this definition. It's ambiguous. And that means that there are some ways to gain this definition that allow numbers that are clearly not even to be misidentified as being even. For example take the number 1.008. Now you could say that this number quote unquote ends in 8. So by the definition above that would make it an even number. But that's crazy. That's not even a whole number. So the problem here is that we haven't been precise enough about our terms and our definition. We haven't described what we mean by a number precisely. And we haven't really talked about what it means to quote something. 1.008 is not what we want to be identified as an even number. But our definition does it. So we have some work to do. Now as an aside before we go on from here on out we're going to use the term integer to mean whole number. That's a shorter word and more common in mathematical practice. So how can we repair the definition we have? Well we could do something like this. An even number is an integer that ends in 0, 2, 4, 6, or 8. So you see how everything matters in a definition? We can't remove the word integer from this definition and just use number without allowing some non-examples to creep in and be accidentally counted. But if you look closely we still have included up the ambiguity. We still don't know what it means to end in something. What does that even mean? A more precise way to state this definition would be the following. An even number is an integer whose 1's digit is 2, 0, 2, 4, 6, or 8. Now this is better because the notion of 1's digit is a mathematically well-defined and well-understood term. But ends in is not. We don't have another definition for ends in but we do have a definition for 1's digit that we learned in arithmetic. So here at last I think we have a workable definition. Note that it scales up well. That means if I give you a new number like negative 2, 2,382 then I can immediately look at the definition and tell that that number is even very efficiently. And I can also make up non-examples like 11 and 4 million and 1. And those are clearly not even numbers by the same definition. So for a concept check we're going to look at another definition of even that once we have gotten it fully up to specs will become our official definition of even number. Here's a draft version. An integer is said to be even if it is equal to 2 times another number. Now the question this time is under this definition which of the following numbers is or are even. Now remember that this definition could have flaws in it. And you must take it literally. That's something we always do with mathematical definitions. We're very, very strict about our definitions. So the definition might identify some numbers as even when in reality we know they aren't. So here are the choices. 12, 13. The set of all integers would be counted even. Or the set of all real numbers period would be counted even. So think about this for a minute while you pause the video and then play it again when you come back and you're ready. So the best answer here is C. That under this definition every integer, every whole number period would be considered even. Now let's see why that is. Let's take the number 13 for example from the list. 13 is an integer of course. So the definition does actually apply. And does, is the number 13 equal to 2 times another number? Well, actually yes it is. 13 is 2 times another number, namely 13 halves. And so since 13 is 2 times another number, we'd have to call 13 even under this definition and that's clearly ridiculous. So our definition is busted and we have some fixing to do. In fact we could play this game for any integer at all. If n is any integer whatsoever, then n is always 2 times n over 2. So all integers would be included as even. Not all real numbers would be included because we do have this filter on the definition that only allows integers to be considered. So we can fix this definition in one little way here. And this will be our official definition of even. An integer is said to be even if it is equal to 2 times another integer. So if we require the number we multiply by 2 to get our integer to itself be an integer, instead of something like 13 halves, we're eliminating the possibility of 13 being even because 13 is not 2 times an integer, it's 2 times a fraction. And we can take this statement one step farther and make it more precise still. And this is going to be our real official definition here that we'll use many, many times in the future. An integer n is said to be even if there exists another integer k such that n equals 2 times k. Now we introduced some notation here, so let's unpack that for a moment. What this means is that n, the integer in question, is even if there's some other integer k such that n is 2 times k. I can get n by taking some other integer and multiplying that by 2. For example, 18 is even because there is another integer, namely 9, such that 18 is 2 times 9. And in that example, the n is 18 and the k is 9. And 17 is not even because 17 is not equal to 2 times k for any integer k, any whole number k. Again, the term integer is all important. If you lose that, the definition is busted. So what we just did here is what we call instantiating the definition. We're making instances out of it. We took the definition and built an example and a non-example out of it. If you're having trouble still with this definition, just instantiate it some more. For example, is 66 even? Well, can you phrase 66 as 2 times k where k is an integer? Well, of course you can. In that case, k is equal to 33. So let's wrap up. And here's what we've learned. First of all, definitions matter. Without clear definitions that are free of ambiguity and error, we can't really talk about anything, especially math. Every part of a mathematical definition has to be clearly defined and mean what you want to mean. And every part of a mathematical definition has a role to play. And finally, instantiating a definition means building examples and non-examples of it to test the boundaries of that definition to see what it really means. And instantiating a definition is the first thing you should do when you encounter a new definition after reading it. Thanks for watching.