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Question Number 66103 by Rio Michael last updated on 09/Aug/19

$$AbinaryrelationRisdefinedon\mathbb{N},thesetofnaturalnumbersby$$$${}_x{R}_y\iff \mathrm{\exists }n\in \mathbb{Z}:x=2^{n}y,x,y\in \mathbb{N}$$$$showthatRisanequivalencerelation$$

Commented by Prithwish sen last updated on 09/Aug/19

$$\mathrm{for}\mathrm{\forall }\mathrm{x}\in \mathbb{N}\mathrm{x}=2^{\mathrm{n}}\mathrm{x}\Rightarrow \mathrm{n}=0$$$${\therefore }_{\mathrm{x}}{\mathrm{R}}_{\mathrm{x}}\mathrm{holds}\mathrm{i}.\mathrm{e}\mathrm{R}\mathrm{is}\mathrm{reflexive}$$$$\mathrm{for}\mathrm{\forall }\mathrm{x},\mathrm{y}\in \mathbb{N}\mathrm{x}=2^{\mathrm{n}}\mathrm{y}\Rightarrow \mathrm{y}=2^{\mathbf{m}}\mathrm{x}\Rightarrow \mathrm{m}=-\mathrm{n}\mathrm{as}\mathrm{m},\mathrm{n}\in \mathbb{Z}$$$$\mathrm{i}.\mathrm{e}{}_{\mathrm{x}}{\mathbb{R}}_{\mathrm{y}}{\iff }_{\mathrm{y}}{\mathbb{R}}_{\mathrm{x}}\because \mathrm{the}\mathrm{R}\mathrm{is}\mathrm{symmetric}$$$$\mathrm{for}\mathrm{\forall }\mathrm{x},\mathrm{y},\mathrm{z}\in \mathbb{N}\mathrm{x}=2^{\mathrm{n}}\mathrm{y}\mathrm{and}\mathrm{y}=2^{\mathrm{n}}\mathrm{z}$$$$\Rightarrow \mathrm{x}=2^{\mathrm{n}}.2^{\mathrm{n}}\mathrm{z}=2^{2\mathrm{n}}\mathrm{z}\because \mathrm{n}\in \mathbb{Z}\Rightarrow 2\mathrm{n}\in \mathbb{Z}$$$$\mathrm{i}.\mathrm{e}{}_{\mathrm{x}}{\mathrm{R}}_{\mathrm{y}}\mathrm{and}{}_{\mathrm{y}}{\mathrm{R}}_{\mathrm{z}}{\Rightarrow }_{\mathrm{x}}{\mathrm{R}}_{\mathrm{z}}$$$$\therefore \mathrm{R}\mathrm{is}\mathrm{transitive}$$$$\therefore \mathrm{R}\mathrm{is}\mathrm{an}\mathrm{equivalance}\mathrm{relation}.$$$$\mathrm{please}\mathrm{check}.$$

Commented by Rio Michael last updated on 09/Aug/19

$$thankyousir$$

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