   Chapter 8.3, Problem 41ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Let R be an equivalence relation on s set A. Prove each of the statements in 36-41 directly from the definitions of equivalence relation and equivalence class without using the results of Lemma 8.3.2, Lemma 8.3.3, or Theorem 8.3.4. For every a and b in A, if a ∈ [ b ] then [ a ] = [ b ] .

To determine

To prove:

For all a and b in A, if a ∈ [ b ] then [ a ] = [ b].

Explanation

Given information:

Let R be an equivalence relation on a set A.

Proof:

R is an equivalence relation on a set A.

Let aA and bA such that a[b].

First part .

Let x[a].

By the definition of [a]={xA|x R a}:

x R a

By the definition of [b]={xA|x R b} and a[b]:

a R b

Since R is an equivalence relation, R is reflexive, symmetric and transitive.

Using the definition of transitive on x R a and a R b, we then obtain:

x R b

By the definition of [b]={xA|x R b}

x[b]

Since x[a] implies x[b], the definition of a subset tells us:

[a][b]

Second part

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