**Equivalence Relations and Equivalence Classes**Let $ R $ be an equivalence relation over set $ A $. Let $ a $ and $ b $ be any two elements in $ A $.a) Prove that $(a, b) \in R \Rightarrow [a] = [b]$. (Hint: show that $[a] \subseteq [b] \land [b] \subseteq [a]$.)b) Prove that $[a] = [b] \Rightarrow [a] \cap [b] \neq \emptyset$.c) Prove that $[a] \cap [b] \neq \emptyset \Rightarrow (a, b) \in R$.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.

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].)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Equivalence Relations and Equivalence Classes**

Let $ R $ be an equivalence relation over set $ A $. Let $ a $ and $ b $ be any two elements in $ A $.

a) Prove that $(a, b) \in R \Rightarrow [a] = [b]$.
(Hint: show that $[a] \subseteq [b] \land [b] \subseteq [a]$.)

b) Prove that $[a] = [b] \Rightarrow [a] \cap [b] \neq \emptyset$.

c) Prove that $[a] \cap [b] \neq \emptyset \Rightarrow (a, b) \in R$.

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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