   Chapter 8.3, Problem 10ES ### 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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# In each of 7-14, relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. A = { − 5 , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , 5 } R is defined on A as follows: For all m , n ∈ Z . m R n           ⇔     3 ( m 2 − n 2 ) .

To determine

To find the distinct equivalence classes of R.

Explanation

Given information:

The relation R is an equivalence relation on the set A.

A = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}. R defined on A as follows: For all m,nZ,

R n       3|(m2n2).

Calculation:

A={5,4,3,2,1,0,1,2,3,4,5}R={(a,b)A×A|3|( a 2 b 2 )}

Let us first determine the values are in the same equivalence class as − 5. A is in the same equivalence class as − 5, when (5,a)R and thus when (5)2a2=25a2 is divisible by 3 (( 5)2a2=25a2 is a multiply of 3).

Element (5)2a2=25a2 in same equivalence class as − 5

− 4     25 − (- 4)2 = 25 − 16 = 9      Yes

− 3     25 − (- 3)2 = 25 − 9 = 16      No

− 2     25 − (- 2)2 = 25 − 4 = 21      Yes

− 1     25 − (- 1)2 = 25 − 1 = 24      Yes

0     25 − 02 = 25 − 0 = 25 &e

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