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Number Theory

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Moderate

What Sets Are Equal to $\mathbb{Z}$?

NUMTH-LWFS4$

Determine which of the following sets are equal to $\mathbb{Z}$.

A

$\cup_{m \in \mathbb{Z}} \{ m \}$

B

$3\mathbb{Z} \cup 3\mathbb{Z} + 1 \cup 3\mathbb{Z} + 2$ $\left(\text{where}~3\mathbb{Z} = \{3j~|~ j \in \mathbb{Z} \}, 3\mathbb{Z} + 1 = \{3k + 1~|~k \in \mathbb{Z} \},~\text{and}~3\mathbb{Z} + 2 = \{ 3m + 2~|~m \in \mathbb{Z}\} \right)$

C

$-\mathbb{N} \cup \{ 0 \} \cup \mathbb{N}$ (where $-\mathbb{N} = \{ -n~ | ~n \in \mathbb{N} \}$)

D

$\frac{1}{\pi}\sin^{-1}(0)$ (where $\sin^{-1}(0)$ is the preimage of the sine function at $0$)