# Question #6c402

Dec 31, 2016

The set is countable (countably infinite) and unbounded.

#### Explanation:

$S$ can be mapped one-to-one onto the ordered pairs of rationals.

The rationals are countable.

The set of ordered pairs of elements of a countable set is countable. (More generally, the Cartesian product of two countable sets is countable.) (Use a proof analogous to the proof that the rationals are countable.)

So, the ordered pairs of rationals are countable.

So, $S$ is countable.

$S$ contains a copy of the rationals in the form $x + i 0$, and the rationals are unbounded, so $S$ is unbounded (in the usual metric on $\mathbb{C}$.)