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Let S=\{n+(-1)^n: n\in \mathbb{Z}\} (\forall s\in S)(s\in \mathbb{Z}) \implies S\subseteq \mathbb{Z} (\forall x\in \mathbb{Z})(x\in \mathbb{Z}_{even}\lor x\in \mathbb{Z}_{odd}) \[(\forall k\in \mathbb{Z}_{even})((-1)^k=1) \[(\forall k\in \mathbb{Z}_{odd})((-1)^k=-1) \[(\forall n\in \mathbb{Z})(\exists k\in S)(k=n\pm 1 )\implies \mathbb{Z}\subseteq S\implies \mathbb{Z}=S \\ \blacksquare \]
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Fri, 25 Jun 2021 21:22 GMT