 Now in some cases, we can say that certain properties are inherited, but be prepared for a disputed inheritance. In other words, expect to prove your claim. So for example, we do get that the addition of integers is commutative, and so what do we want to show? We want to show that if we add two integers, we get the same thing if we add them in the reverse order. Well, we can always write down one side of inequality, so let's write down AB plus CD. Our definition of addition tells us what that is. And remember, ABC and D are natural numbers, and we know that the addition of natural numbers is commutative, so we can switch the order. Again, it's helpful to think about a proof as a bridge where we want to get to is commutativity. So let's go ahead and write that down as our last line. And again, definitions are the whole of mathematics, all else is commentary. The right-hand side has a very specific definition from our definition of addition. And so this second and third line are identical, and so we can join our bridge. And one way we might express this is that we could say that the commutativity of the addition of integers is inherited from the commutativity of the addition of natural numbers. But it still requires a proof. So mathematics falls from the sky. Well, mathematics is created by normal, I mean ordinary people. So let's talk a little bit about the creative process. So we have an additive identity for the addition of natural numbers. Let's try to create an additive identity for the addition of integers. Definitions are the whole of mathematics, all else is commentary. The additive identity for the integers would be some equivalence class x, y, where for any equivalence class a, b, we'd have the equivalence class plus x, y is the same thing we started with. Now, by definition, when we add two equivalence classes this way, we get an equivalence class where the class representative is the component y's sum. And if x, y is the additive inverse, we'd need the two equivalence classes to be the same. And again, remember if two equivalence classes overlap, they have to be identical. And so the way we can guarantee the two equivalence classes are the same is if the class representative a, b is equivalent to the class representative a plus x, b plus y. So let's find our identity. So again, we want this sum to be true, which would happen if our equivalence classes were equal, which would happen if the class representative were in the other equivalence class, which would happen if the class representatives were equivalent, which happens if the outer sum is equal to the inner sum. And remember at this point everything in sight is a natural number. So all the usual rules have already been proven and we can apply them without worrying about them. And so x equals y. And we have to make sure that we can drive forward. Suppose x is equal to y. Because they're natural numbers, we can add a and b to both sides. Disentangle our equivalence relation, which gives us an element of the equivalence class, which says that the two equivalence classes are the same, which gives us the additive identity. And so that proves the additive identity for the integers is any equivalence class where the two components of the class representative are the same. Now before we proceed a few more words about proof and mathematics, the theorems are great and useful and powerful, but they're actually the least important part of the process and you can forget the theorems. Wait, wait, I don't mean that literally. Come back. The important thing here is actually the proof and not the result. Because in order to prove this theorem, we had to review quite a bit of what we know or what we should know about these equivalence classes, such as the definition of when two things are equivalent, the fact that if one equivalence class contains another equivalence class representative, the two equivalence classes are the same, and the definition of the addition of the equivalence classes. And so it's important to keep in mind the proof is the pudding. The existence of an additive identity is very useful because it gives us a goal. If we have an additive identity, we can define the additive inverse. And we'll write it this way, let zero be the additive identity. The additive inverse of x, written this way, satisfies x plus the additive inverse is equal to zero. Now at this point, it's useful to keep in mind two important ideas. First of all, there are only so many symbols. And so the symbol here, when we write the additive inverse of x, should not have any significance read into it beyond the equation, x plus the additive inverse of x is equal to zero. The second important idea is how you speak influences how you think. You might look at this and say, oh, negative x, because that's how we're used to talking about it. But really, you should always refer to this as the additive inverse of x. Because you want to think about this as the additive inverse of x, namely that if you add it to x, you get the additive identity. So let's find the inverse. What's the additive inverse of the integer a, b? We want to find x, y, where if I add to a, b, I get the additive identity, which is the equivalence class generated by an ordered pair where both components are the same. So we'll need to come up with a very creative way of indicating that. How about let's call that ordered pair c, c. And so to find this ordered pair, we'll let you figure out that one for yourself.