Paper 2, Section I, 2D2 \mathrm{D}

Numbers and Sets | Part IA, 2020

Define an equivalence relation. Which of the following is an equivalence relation on the set of non-zero complex numbers? Justify your answers. (i) xyx \sim y if xy2<x2+y2|x-y|^{2}<|x|^{2}+|y|^{2}. (ii) xyx \sim y if x+y=x+y|x+y|=|x|+|y|. (iii) xyx \sim y if xyn\left|\frac{x}{y^{n}}\right| is rational for some integer n1n \geqslant 1. (iv) xyx \sim y if x3x=y3y\left|x^{3}-x\right|=\left|y^{3}-y\right|.

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