1. ## partition problem

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).

2. Originally Posted by crowned1
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 $A_1,A_2,A_3: \mathbb{R} \to \mathbb{N}$

$A_1 = \{ r \in \mathbb{R}: \cos \frac{r\pi}2 = 1\}$

$A_2 = \{ r \in \mathbb{R}: \cos \frac{r\pi}2 = 0\}$

$A_3 = \{ r \in \mathbb{R}: \cos \frac{r\pi}2 = -1\}$

Clearly $A_1 \cup A_2 \cup A_3 = \mathbb{N}$ and $\forall i,j \in \{1,2,3\}\text{ and } i\neq j, A_i \cap A_j = \phi$