Equivalence and Cardinality

We have seen that the rational numbers are densely distributed on the real line in the sense that there is always a rational between any two distinct real numbers. But even more is true. In fact, it follows that there must be infinitely many rational numbers between any two distinct reals. (Why?) In sharp contrast to this picture of the rationals as a “dense” set, we will show in this section that the rational numbers are actually rather sparsely represented among the real numbers. We will do so by “counting” the rationals!

We say that two sets A and B are equivalent if there is a one-to-one correspondence between them. That is, A and B are equivalent if there exists some function f : A → B that is both one-to-one and onto. As a quick example, you might recall from calculus that the map x ↦ arctan x is a strictly increasing (hence one-to-one) function from ℝ onto the open interval (−π/2, π/2). Thus, ℝ is equivalent to (−π/2, π/2). For convenience we may occasionally write A ~ B in place of the phrase “A is equivalent to B.” Please note that the relation “is equivalent to” is an equivalence relation.

The notion of equivalence is supposed to lead us to a notion of the relative sizes of sets. Equivalent sets should, by rights, have the same “number” of elements.