Looking for some help with this problem, Thanks ahead of time
Describe the partition of N(set of natural numbers) induced by R(set of real numbers).
Define $\displaystyle A_1,A_2,A_3: \mathbb{R} \to \mathbb{N}$
$\displaystyle A_1 = \{ r \in \mathbb{R}: \cos \frac{r\pi}2 = 1\}$
$\displaystyle A_2 = \{ r \in \mathbb{R}: \cos \frac{r\pi}2 = 0\}$
$\displaystyle A_3 = \{ r \in \mathbb{R}: \cos \frac{r\pi}2 = -1\}$
Clearly $\displaystyle A_1 \cup A_2 \cup A_3 = \mathbb{N}$ and $\displaystyle \forall i,j \in \{1,2,3\}\text{ and } i\neq j, A_i \cap A_j = \phi $