Let’s define two real numbers to be *equivalent* (denoted by )
if their difference is a rational number:

Now consider the resulting *equivalence class*, which, for any real number ,
is the set of real numbers that are equivalent to under :

For example, all the rational numbers trivially fall into .

*How many such equivalence classes are there in ?*
The answer to this question is a key (if small) step along the way to a proof
about the nonexistence of a

*universal measure*(

*i.e.*, a measure defined on all subsets) on the real numbers, which I came across recently while watching this intro video about Measure Theory. The answer, which may seem obvious to some, took me a while to figure out, so I figured I’d share my thought process:

First, note that each equivalence class is *countable*. In fact, each class is
exactly as large as the set of rational numbers, which is of course countable.
This becomes obvious if we rewrite the class like so:

Next, notice that the *union* of all these equivalence classes needs to cover
the real numbers. After all, every real number belongs to (exactly) one of these
classes.

It follows that there must be an **uncountable** number of these classes, since
otherwise, we would have the union of a countable number of countable sets, which
cannot possibly cover (which is uncountable). *Neat!*