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1 Question 1: Let R be a relation on the set of real numbers defined as follows: For any two real numbers x and y, ((x, y), (u, v)) belongs to R if and only if x² + y² = u²+v². Prove that R is an equivalence relation. To do so, you need to demonstrate that R satisfies the three properties of an equivalence relation: reflexivity, sym- metry, and transitivity. Provide a rigorous proof for each of these properties and explain your rea- soning in detail.

1 Question 1: Let R be a relation on the set of real numbers defined as follows: For any two real numbers x and y, ((x, y), (u, v)) belongs to R if and only if x² + y² = u²+v². Prove that R is an equivalence relation. To do so, you need to demonstrate that R satisfies the three properties of an equivalence relation: reflexivity, sym- metry, and transitivity. Provide a rigorous proof for each of these properties and explain your rea- soning in detail.

Elements Of Modern Algebra
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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1 Question 1: Let R be a relation on the set of real numbers defined as follows: For any two real numbers and y, ((x, y), (u, v)) belongs to R if and only if x² + y² = u²+v². Prove that R is an equivalence relation. To do so, you need to demonstrate that R satisfies the three properties of an equivalence relation: reflexivity, sym- metry, and transitivity. Provide a rigorous proof for each of these properties and explain your rea- soning in detail.
Transcribed Image Text:1 Question 1: Let R be a relation on the set of real numbers defined as follows: For any two real numbers and y, ((x, y), (u, v)) belongs to R if and only if x² + y² = u²+v². Prove that R is an equivalence relation. To do so, you need to demonstrate that R satisfies the three properties of an equivalence relation: reflexivity, sym- metry, and transitivity. Provide a rigorous proof for each of these properties and explain your rea- soning in detail.
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