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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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In Exercises 6 10 , a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each.

x R y if and only if x + 3 y is a multiple of 4 .

To determine

The proof of the statement,” The relation defined by xRy if and only if x+3y is a multiple of 4 on the set of all integers Z is an equivalence relation” and the distinct equivalence classes of R with least four members of each.

Explanation

Formula Used:

A relation R on a nonempty set A is an equivalence relation if the following conditions are satisfied for arbitrary x, y, and z in A:

1. xRx for all xA. (Reflexive Property)

2. If xRy, then yRx. (Symmetric Property)

3. If xRy and yRz, then xRz. (Transitive Property)

If R is an equivalence relation on the nonempty set A, then for each aA, the set [a]={xA|xRa} is the equivalence class containing a.

Explanation:

Consider the relation xRy if and only if x+3y is a multiple of 4 defined on Z.

1. xRx, since x+3x=4x.

2. xRyx+3y=4kforsomekZx=4k3yforsomekZy+3x=y+3(4k3y)=y+12k9y=8y+12kforsomekZyRx

3. xRyandyRzx+3y=4kandy+3z=4mforsomek,mZx=4k3yandy=4m3zforsomek,mZ

x=4k3(4m3z)=4k12m+9z

x+3z=4k12m+9z+3z=4k12m+12z

So, x+3z is a multiple of 4

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