Question

Question #6c402

Answer 1

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+i0`, and the rationals are unbounded, so `S` is unbounded (in the usual metric on `CC`.)