# Partitioning number systems into sets

1. Oct 19, 2009

### MathDude

I've been having trouble partitioning number systems into sets. The complex and rational number systems blow me away, so I'll stick with all reals, integers, and naturals for now.

1. The problem statement, all variables and given/known data

1a) With five sets of infinitely many positive integers, partition the set of all real numbers.
1b) With five infinite sets, partition the set of all real numbers.

2a) With infinitely many infinite sets, partition the set of all real numbers/
2b) With infinitely many infinite sets, partition the set of integers.

2. Relevant equations

(see above)

3. The attempt at a solution

1a)

A1 = { k $$\in$$ R>0 : k $$\equiv$$ 0(mod 5) }
A2 = { k $$\in$$ R>0 : k $$\equiv$$ 1(mod 5) }
A3 = { k $$\in$$ R>0 : k $$\equiv$$ 2(mod 5) }
A4 = { k $$\in$$ R>0 : k $$\equiv$$ 3(mod 5) }
A5 = { k $$\in$$ R>0 : k $$\equiv$$ 4(mod 5) }

1b) Same as above except k $$\in$$ R.

2a) Using Fundamental Thm of Arithmetic,

Sp = {pn : n $$\in$$ N, p is prime)

Let L, Sp, ... partition R.

where L = R - $$\bigcup$$Sp

2b) Same as above except instead of L = R -..., R is instead Z (set of integers).

2. Oct 19, 2009

### MathDude

Am I better off splitting these up into separate threads?