Re: Rational Numbers/Irrational Numbers

In article <BZ6dnQn8QsXfuibYnZ2dnUVZ_t-mnZ2d@xxxxxxxxxxxx>,
"David T. Ashley" <dta@xxxxxxxx> wrote:

Again, the elements of my reasoning were:

a)There are two mutually exclusive (but perhaps not mutually exhaustive)
sets, A and B.

b)Every element of A maps to two different elements of B (through two
different functions f() and g()).

c)The mapping is unique in both directions, and if the domain of f() and g()
is A, the ranges of f() and g() are _disjoint_ subsets of B.

Note: (c) may not be immediately obvious, but given a real number such as
2/3 + PI, it can only be in the range of f(). 2/3 + PI + PI, on the other
hand, can only be in the range of g(). The range of f() and the range of
g() are disjoint.

That sort of partition of ranges merely requires that the common domain
of f() and g() not have any two members which differ by pi. But that
allows it to contain all sorts of irrationals.
.