1. For each of the following, is it an equivalence relation? Prove or provide a counterexample. (a) R² \ {(0,0)}, (a, b) ~ (c, d) if there exists k € R with k ‡ 0 such that (a, b) = (kc, kd). (b) R2, (a, b)~ (c,d) if a² + b² ≥ c² +d².

1. For each of the following, is it an equivalence relation? Prove or provide a counterexample. (a) R² \ {(0,0)}, (a, b) ~ (c, d) if there exists k € R with k ‡ 0 such that (a, b) = (kc, kd). (b) R2, (a, b)~ (c,d) if a² + b² ≥ c² +d².

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...

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Question

1. For each of the following, is it an equivalence relation? Prove or provide
a counterexample.
(a) R² \ {(0,0)}, (a, b) ~ (c,d) if there exists k € R with k ‡ 0 such that
(a, b) = (kc, kd).
(b) R², (a, b) ~ (c,d) if a² + b² ≥ c² + d².

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Follow-up Question

Then, there exists k1≠0, k2≠0 such that a,b=k1c,k1d and c,d=k2e,k2f.

a,b=k1c,d=k1k2e, k2f=k1k2e,f.

For k1=1k≠0 we have, c,d=k1a,k1b.

Then, there exists k1≠0, k2≠0 such that a,b=k1c,k1d and c,d=k2e,k2f.

a,b=k1c,d=k1k2e, k2f=k1k2e,f.

hi i am confused with " k1 k2 1ka k1c k1d........" I dont know how to understand the number "1 ,2". I dont know how to express them in paper. Can u tell me? Thank you!

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