Addition

Consider the natural, or counting, numbers 1, 2, 3, 4,... Each natural number can be defined in terms of sets. The number 1 is the name of the collection containing every conceivable set with one element, such as the set containing 0 or the set containing the Washington Monument. The number 2 is the name of the collection containing every conceivable set with two elements, and so on. The sum of two natural numbers is determined by counting the number of elements in the union of two sets chosen to represent them. For example, let the set {A, B, C} represent 3 and the set {W, X, Y, Z} represent 4. Then 3 + 4 is determined by counting the elements in {A, B, C, W, X, Y, Z}, which is the union of {A, B, C} and {W, X, Y, Z}. The result is seven, and we write 3 + 4 = 7. In this way, the operation of addition is carried out by counting.