Numbers and Sets | Part IA, 2003

What is an equivalence relation? For each of the following pairs (X,)(X, \sim), determine whether or not \sim is an equivalence relation on XX :

(i) X=R,xyX=\mathbb{R}, x \sim y iff xyx-y is an even integer;

(ii) X=C\{0},xyX=\mathbb{C} \backslash\{0\}, x \sim y iff xyˉRx \bar{y} \in \mathbb{R};

(iii) X=C\{0},xyX=\mathbb{C} \backslash\{0\}, x \sim y iff xyˉZx \bar{y} \in \mathbb{Z};

(iv) X=Z\{0},xyX=\mathbb{Z} \backslash\{0\}, x \sim y iff x2y2x^{2}-y^{2} is ±1\pm 1 times a perfect square.

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