Let R be an equivalence relation over set A. Let a and b be any twoelements in A.a) Prove that (a, b) ER =→ [a]b) Prove that [a] = [b] = [a] n [b] # Øc) Prove that [a] n [b] ± Ø = (a, b) E R.= [b]. (Hint: show that [a] C [b] ^ [b] C [a].)This establishes that every equivalence relation partitions the set it is defined over into a set of pairwisedisjoint equivalence classes. Every element of the set belongs to exactly one equivalence class.

Given that R is an equivalence relation over the set A and a and b are any two elements in A. Since…

Let R be an equivalence relation over set A. Let a and b be any two elements in A. a) Prove that (a, b) E R = b) Prove that [a] = [b] = [a] n [b] # Ø c) Prove that [a]n [b] # Ø = (a, b) E R. [a] = [b]. (Hint: show that [a] C [b] ^ [b] C [a].)

Let R be an equivalence relation over set A. Let a and b be any two elements in A. a) Prove that (a, b) E R = b) Prove that [a] = [b] = [a] n [b] # Ø c) Prove that [a]n [b] # Ø = (a, b) E R. [a] = [b]. (Hint: show that [a] C [b] ^ [b] C [a].)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 7E: In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R...

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Question

Let R be an equivalence relation over set A. Let a and b be any two
elements in A.
a) Prove that (a, b) ER =→ [a]
b) Prove that [a] = [b] = [a] n [b] # Ø
c) Prove that [a] n [b] ± Ø = (a, b) E R.
= [b]. (Hint: show that [a] C [b] ^ [b] C [a].)
This establishes that every equivalence relation partitions the set it is defined over into a set of pairwise
disjoint equivalence classes. Every element of the set belongs to exactly one equivalence class.

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