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Math Help - partition problem

  1. #1
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    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).
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  2. #2
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    Quote Originally Posted by crowned1 View Post
    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
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