   Chapter 8.3, Problem 16ES ### 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 the relation of congruence modulo 3. Which of the following equivalence classes are equal? [ 7 ] , [ − 4 ] , [ − 6 , ] [ 4 ] , [ 27 ] , [ 19 ] Let R be the relation of congruence modulo 7. Which of the following equivalence are equal? [ 35 ] , [ − 7 ] , [ 12 ] , [ 0 ] , [ − 2 ] , [ 17 ]

To determine

(a)

The equal equivalence classes.

Explanation

Given information:

Let R be the relation of congruence modulo 3.

Equivalence classes are [7 ], [-4 ], [-6 ], [17 ], [4 ], [27 ], .

Concept used:

Let p be any integer, by definition, of an equivalence class

[p]={xZ|xp(modn)}={xZ|xp=knforsomekZ}

Calculation:

R=Relation of congruence modulo 3.

Let us first determine which equivalence classes are equal to the equivalence class of 7. The equivalence class of a is equal to the equivalence class of 7 if and only if a7 mod 3 that is if and only if 3 divides a − 7.

Equivalence class a − 7     3 divides a − 7     Equal to equivalence class 

 − 4 − 7 = − 11     No     No

 − 6 − 7 = − 13     No     No

 17 − 7 = 10     No     No

 4 − 7 = − 3     Yes     Yes

 27 − 7 = 20     No     No

 19 − 7 = 12     Yes     Yes

This implies: ==.

Next determine which equivalence classes are equal to the equivalence class of − 4. The equivalence class of a is equal to the equivalence class of − 4 if and only if a4 mod 3 if and only if 3 divides a − (- 4). Note: you only need to check the remaining equivalence classes that were unequal to the equivalence class 

To determine

(b)

The equal equivalence classes.

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