PROBLEM 30. For (x, y) and (u, v) in R2, define (x, y) ~ (u, v) if x² + y² =u? + v?. Prove thatequivalence classes geometrically.defines an equivalence relation on R2 and interpret the

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Answered: PROBLEM 30. For (x, y) and (u, v) in…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 12E: Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and...

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Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions. if and only ifand are parallel. if and only ifand are perpendicular.
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For a,y E R, let z~y if and only if (x - y) € Q. Show that ~ defined as such is an equivalence relation on R and describe the equivalence class containing 0.
Problem 4Let A be a set, and let R be a relation on A.1. Suppose R is reflexive. Prove that ∪x∈A[x] = A.2. Suppose R is symmetric. Prove that x ∈[y] if and only if y ∈[x], for all x, y ∈A.3. Suppose R is transitive. Prove that if xRy, then [y] ⊆[x] for all x, y ∈A.
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PROBLEM 30. For (x, y) and (u, v) in R2, define (x, y) ~ (u, v) if x² + y² = u? + v?. Prove that equivalence classes geometrically. defines an equivalence relation on R2 and interpret the