If you were asked how many people are in your class, you would do the natural thing and count them. Thinking about this carefully, we see that what you are doing is assigning each person in the room one and only one number. If we asked whether or not there are more people in this class than were in your high school geometry class, you could certainly answer that question by comparing the two numbers. Now suppose we look at the positive integers and the natural numbers. Which set has more elements? What should that even mean? This is a more difficult question to answer correctly than you might think. The next few chapters are devoted to answering the questions. In this chapter, we begin our investigation by defining what it means for two sets to be equivalent and we give a rigorous definition for a set to be finite. As a consequence, we’ll have a rigorous definition of infinite set.