Question

How can you define a function with domain the whole of `RR` and range the whole of `CC`?

(It is possible to define a bijection constructively)

Answer 1

See explanation...

Explanation:

Define `h(x): (0, 1) -> (-oo, oo)`:

`h(x) = (1-2x)/(x(x-1))`

graph{(sqrt(1/4-(x-1/2)^2))/(sqrt(1/4-(x-1/2)^2))(1-2x)/(x(x-1)) [-1, 2, -10, 10]}

Then:

`h^(-1)(y) = ((y-2)+sqrt(y^2+4))/(2y)`

So `h(x)` is a bijection between `(0, 1)` and `RR`, with inverse `h^(-1)(y)`

Let `S = { a+bi : a, b in (0, 1) }` i.e. the open unit square in Q1.

We can use `h(x)` on the Real and imaginary parts of a Complex number to define a bijection `k(x)` between `S` and `CC`:

`k(a+bi) = h(a)+h(b)i`

`k^(-1)(a+bi) = h^(-1)(a) + h^(-1)(b)i`

Having found these bijections, next consider decimal representations of numbers in `(0, 1)`.

Some numbers have two possible decimal representations; one with a tail of repeating `0`'s and the other with a tail of repeating `9`'s. For our purposes, we disallow representations with a tail of repeating `0`'s, choosing to use the representation with repeating `9`'s.

Then there is a bijection between `(0, 1)` and representations of the form `0.a_1 a_2 a_3 a_4 ...`, where `a_i in { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }` are decimal digits and no sequence has a tail of repeating `0`'s.

Given a number `0.a_1 a_2 a_3 a_4 ...`, split the sequence into a sequence of subsequences, each of which is as short as possible, but ends with a non-zero digit.

For example:

`0.0230040291...`

splits as:

`0." "02" "3" "004" "02" "9" "1" "...`

Then recombine alternate subsequences to form two Real numbers in `(0, 1)`:

`0.020049...`

`0.3021...`

These are the Real and imaginary parts of a corresponding Complex number in `S`

This process defines a function:

`s(x): (0, 1) -> S`

Conversely, given a Complex number in `S`, disassemble the Real and imaginary parts, then interlace them to form a Real number in `(0, 1)`:

So given:

`0.020049... + 0.3021... i`

Split into subsequences:

`0." "02" "004" "9" "...`

`0." "3" "02" "1" "...`

Interlace:

`0.0230040291...`

This defines the inverse function `s^(-1)(y): S -> (0, 1)`

Hence we can define a function with domain `RR` and range `CC` by:

`f(x) = k^(-1)(s(h(x)))`

This is a bijection, with inverse:

`f^(-1)(y) = h^(-1)(s^(-1)(k(y)))`