 Definitions are some of the most important parts of mathematics. In fact, we might go so far to say that definitions are the whole of mathematics. All else is commentary. Now, this doesn't mean you just have to know the definitions. The commentary is actually very important, and it is the useful part of mathematics, but it is vitally important to keep in mind if you don't know the definitions, you can't do mathematics. The most important thing to remember about a definition is it's an equivalence. It says that two things are completely and totally interchangeable. And one of the things that this means is that if I've defined an object or a relationship by saying an object relationship is this thing with all of these properties, then all of the properties listed in the definition are possessed by the object or the relationship. We can also go the other way. Because definitions are equivalences, because they say things are completely and totally interchangeable, this also means that if I have all of the properties listed in the definition, I have to be the object or the relationship that's defined. And what this means is we can use definitions in two ways. First of all, an object will have all of the properties in its definitions, but also any object with all of the properties will be an example of the defined object. This is not true for many of the things we call definitions in the so-called real world. For example, suppose I throw down the following definition for a dog. A dog is an animal with four legs. Well, okay, if we accept this as our definition of a dog, then we can use it in two ways. First of all, I have a dog. Because the definition is supposed to be an equivalence, then if I have a dog, then I know that what I have is an animal and it has four legs. Okay, so far so good. But I have a cat. Well, let's see, what do I know about a cat? Well, it's an animal and it has four legs, which means that because it has all of the listed properties, it is an animal, it has four legs, because it has all of the listed properties, it has to be a dog. So I have a cat, it must be a dog. Well, that's what happens if I accept this as a definition of a dog. If we accept this as a definition of a dog, then cats are dogs. Well, we don't want that to happen, so we have to reject this as a definition of what a dog is. So let's think about that a little bit further. If we use a definition, there's two primary ways we can use it. First of all, we can use a definition as a checklist. If you have all of the listed properties, then you are the object defined or you have the relationship that's defined. For example, here is a relationship definition that we have. A is a subset of B, if every element of A is an element of B. And so there's a definition of a subset. Well, let's suppose A is the set A, B, C, and B is the set C, B, A. And let's see if A is a subset of B. And so we're going to check those requirements. Every element of A is an element of B. So let's check those requirements. So A has element A. And let's see. So does B. A has element B. And so does B. A has element C. So does B. And so every element of A is an element of B. And so we've met all of the requirements of being a subset. So we can say that A is a subset of B. And there's our use of a definition as a checklist. The other thing we can use definitions for is called instantiation. And this is just a fancy way of saying we're going to make a substitution. Once we've defined an object, then we know that we have all of the listed properties of that object. So for example, here is one of the ways we can define whole numbers. If N is a whole number, N plus 1 is the whole number after N. And so, for example, we might try to find 5 plus 1 from the definition. And that means we have to use this actual definition that we have of whole numbers. So in order to do that, we need to instantiate our definition. Now remember paper is cheap, so let's write stuff down and maybe the first thing we'll write down is the actual definition. So if N is a whole number, N plus 1 is the whole number after N. Now instantiation means we're going to make some substitutions here so that we can actually prove the thing we're trying to find. So here we want to find 5 plus 1. So we're going to look for the part of the definition that's most similar to 5 plus 1. And so I look through my definition, look through the definition, look through the definition, look through the definition, oh, here we go. Okay, so here, this part here, this N plus 1 looks an awful lot like the 5 plus 1 we're supposed to be working with. So that means that I want to make my N plus 1 the same as 5 plus 1. And if I compare these two, it seems that all I've done is I've replaced N with 5. And so if we are going to replace N with 5, our instantiation, we're going to replace N with 5 every place I see N. Well, every place I see N as a variable. I'm not going to replace this N with the 5. That would be a little bit silly. But every place I see N, this part of my definition, I'm going to replace it with 5. So here it goes. If, well, here's N, so I'm going to replace it with a 5, is a whole number. Well, that stays the same. However, it's a good idea to check our statement as we go. If 5 is a whole number, well, is that true? If this isn't true, there's no point in writing the rest of the definition because we can't use it. Oh, yeah, yeah, 5 is a whole number. I buy that. 5 is a whole number. So it's worth proceeding at this point. So 5 is a whole number. I'm replacing N with 5. So my next statement, N plus 1, is going to become 5 plus 1. And is the whole number after? Well, again, that doesn't change. And N, I'm replacing N with 5. So there's my definition so far instantiated. If 5 is a whole number, this one is a whole number after 5. And that puts us about 90% of the way through the proof. But we really ought to complete the thought. And so the whole number after 5, it would be kind of nice to make an observation of what that actually is. So I might say that's going to be 6. Now, here, it's very important to understand that the answer to this question find 5 plus 1 from the definition. If you want to answer this question, what you need to be sure to do is to include all of the parts in green. The complete answer is if 5 is a whole number, 5 plus 1 is a whole number after 5, which is 6. This is the only acceptable complete answer to the question. Find 5 plus 1 from the definition. So this is the question. There's the answer.