In many problems concerning the natural numbers (1, 2,…), insight can be gained by representing the numbers by sets of objects. Since the particular choice of object is unimportant, we will usually use dots, squares, spheres, cubes, and other common easily drawn objects.
When one is faced with the task of verifying a statement concerning natural numbers (for example, showing that the sum of the first n odd numbers is n2), a common approach is to use mathematical induction. However, such an analytical or algebraic approach rarely sheds light on why the statement is true. A geometric approach, wherein one can visualize the number relationship as a relationship between sets of objects, can often provide some understanding.
In this chapter we will illustrate two simple counting principles, both of which involve the representation of natural numbers by sets of objects. The principles are:
if you count the objects in a set two different ways, you will get the same result; and
if two sets are in one-to-one correspondence, then they have the same number of elements.
The first principle has been called the Fubini principle [Stein, 1979], after the theorem in multivariate calculus concerning exchanging the order of integration in iterated integrals. We call the second the Cantor principle, after Georg Cantor (1845–1918), who used it extensively in his investigations into the cardinality of infinite sets. We now illustrate the two principles. [Note: The two principles are actually equivalent.]
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