3. The following relations are equivalence relations (you don't need to prove this). In each case determine how many equivalence classes there are, and describe them. If there are finitely many of them, list them explicitly. (i) On Z, with the equivalence relation a ~ b if a = b mod 3. (ii) On R {0}, with x ~y if xy > 0. (iii) On R², (x1, y₁) ~ (x2, Y2) if x² + y² = x² + y².
3. The following relations are equivalence relations (you don't need to prove this). In each case determine how many equivalence classes there are, and describe them. If there are finitely many of them, list them explicitly. (i) On Z, with the equivalence relation a ~ b if a = b mod 3. (ii) On R {0}, with x ~y if xy > 0. (iii) On R², (x1, y₁) ~ (x2, Y2) if x² + y² = x² + y².
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 28E
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Transcribed Image Text:3. The following relations are equivalence relations (you don't need to prove this). In
each case determine how many equivalence classes there are, and describe them. If
there are finitely many of them, list them explicitly.
(i) On Z, with the equivalence relation a~ b if a = b mod 3.
(ii) On R {0}, with xy if xy > 0.
(iii) On R², (x1, y1) ~ (x2, y2) if x² + y² = x² + y².
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