 We are now in a position where we can define the addition of two whole numbers, and it helps to keep in mind math ever generalizes. So remember that the cardinality expresses the notion of the number of objects in a set. If we combine two sets together, the new set has a new cardinality, and one way we might think of addition is as the following. Suppose A is the cardinality of some set, and B is the cardinality of some set. Then maybe A plus B is the cardinality of the union of the two sets. So let's prove that 2 plus 3 equals 5. So remember, definitions are the whole of mathematics, all else is commentary. So let's pull in our definition of addition. And so it looks like we need two sets. A set A, where the cardinality of A is 2, and a set B, where the cardinality of B is 3. Now here's a useful but somewhat risky thing to assume. If it's part of a definition, you can usually use it without proving it. What that means in this case is that because our definition of addition requires us to have the cardinality of two sets, what we can do is we can say, well, here are two sets, how about these? And we'll claim without further elaboration that the cardinality of A is 2 and the cardinality of B is 3. And again, the reason that we're allowed to do this is that they are part of our definition of addition, which is what we're using here. And so we can use the cardinality of the two sets without further comment. Well, the definition of addition requires us to find the union of A and B, so we'll find that. And again, because cardinality of the set is part of our definition of the addition, then we can find the cardinality of this set without further comment, the cardinality is 5. Now, so far so good, but here's a useful idea when doing anything creative, try to break things. And this is really another way of this idea that if you don't find the flaw in your work, someone else will. So apparently we have proven that 2 plus 3 equals 5, based on our definition of addition, but let's try and break this definition and see what happens. And in particular, we're concerned with a property that's sometimes referred to as being well-defined. We say that an operation is well-defined if the result doesn't depend on the representations we use. If the operation isn't well-defined, choosing a different representation may give us a different answer. For example, 2 plus 3 equals 5, where we use the representations this and this. But imagine that 2 plus 3 equals 4 if we use these representations. This would be a problem since changing our representations gives us a different result. And so the question you've got to ask yourself is, is addition well-defined? So let's try to break it. So let's use these two sets and our definition of addition. So again we have the cardinality of a is 2 and the cardinality of b is 3. And according to our definition, 2 plus 3 will be the cardinality of the union. So our union is going to be a, b, 1, 2. Because remember we don't list an element of a set more than once. And the cardinality of the union is 4. And that's a problem. When we use these representations for our sets of cardinality 2 and 3, we found that 2 plus 3 was equal to 5. But when we use these representations, we found that 2 plus 3 was equal to 4. And so 2 plus 3 depends on our representations. And our definition of addition is not well-defined. And this means we should fix it. So let's think about this. Why did this happen? And if we compare our two cases, we see that in the case that gave us what we think about as the correct results, a and b had nothing in common. Well, in the other case, a and b did have something in common. And since a and b had an element in common, and this element was not repeated in a union b. Now, since we're pretty sure the correct answer to 2 plus 3 is in fact 5, we're pretty sure that our first example is how we want to proceed. And so this means we have to modify our definition of addition, let a be the cardinality of some set, b the cardinality of some other set, and let the two sets have an empty intersection. Then our cardinality of the union is a plus b. Now, because we changed our definition of addition, we should reexamine our earlier proof. So again, we still want to prove 2 plus 3 equals 5, but this time we'll use our corrected definition of addition. So again, we still need to find two sets, a and b, with cardinalities 2 and 3. We might try these two, and this time we do have to make that additional check their intersection is the empty set. And again, because it's part of the definition, we can just claim it without comment. You should probably make sure that it's correct. Again, if you don't find the flaw in your work, someone else will. But looking at the two sets, a intersect b does appear to be the empty set, and so 2 plus 3 will be the cardinality of the union, which is 5. Now, it's also worth seeing why the other one didn't work. So we could have used a, cardinality of 2, b, cardinality of 3, but these two sets don't work. And so let's analyze that. While it is true they have the correct cardinalities, their intersection is not the empty set, and that means we can't use these two sets in our definition of addition. Another way we might look at this sets are irrelevant to the problem of finding 2 plus 3. Now, one advantage to this set theoretic definition of addition is that if we define addition in terms of sets, all of our results on sets are applicable, and this allows us to prove related properties of addition. For example, commutativity of the whole numbers. So definitions are the whole of mathematics. All else is commentary. So here's our definition of addition. And again, the definition requires certain things. We have two sets with the appropriate cardinalities and their intersection is empty. So let's claim we have two sets with the appropriate cardinality where the intersection is the empty set. Our definition says that a plus b is the cardinality of the union. Meanwhile, b plus a is the cardinality of the union, but in the reverse order because our add ends are in the reverse order. But because a and b are sets, we do know something about their union. Namely, we know that the union of a and b is the same as the union of b and a. So this union is the same as this union. And so b plus a is also the cardinality of a union b. And since they both have cardinality of a union b, it follows that b plus a is the same as a plus b. And there's our commutativity of addition.