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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Label each of the following statements as either true or false.

Let R be an equivalence relation on a nonempty set A and let a and b be in A . If b [ a ] , then [ b ] = [ a ] .

To determine

Whether the statement, “If R is an equivalence relation on the nonempty set A and let a and b be in A. if b[a] then [b]=[a] ” is true or false.

Explanation

Consider the statement, “If R is an equivalence relation on the nonempty set A and let a and b be in A. if b[a] then [b]=[a]

Let R be an equivalence relation on nonempty set A. For each aA, the set [a]={xA|xRa} is called equivalence class containing a.

Given that b[a].

By using definition of equivalence class,

bRa ………… (1)

Let x[a]

By using definition of equivalence class,

xRa

From (1),

bRa

By using R is symmetric,

aRb

Thus xRa and aRb

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