   Chapter 8.3, Problem 12ES ### 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 − { − 4 , − 3 , − 2 , − 1 , 2 , 3 , 4 } . R is defined on A as follows: For all ( m , n ) ∈ A . m R n         ⇔     5 | ( 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. Find the distinct equivalence classes of R.

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

R n       5|(m2n2).

Calculation:

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

Let us first determine which values are in the same equivalence class as − 4. a is in the same equivalence class as − 4, when (4,a)R and thus when (4)2a2=16a2 is divisible by 5 (( 4)2a2=16a2 is a multiply of 5).

Element (4)2a2=16a2 in same equivalence class as − 5

− 3     16 − (- 3)2 = 16 − 9 = 7     No

− 2     16 − (- 2)2 = 16 − 4 = 12     No

− 1     16 − (- 1)2 = 16 − 1 = 15     Yes

0     16 − 02 = 16 − 0 = 16     No

1     16 − 12 = 16 − 1 = 15     Yes

2     16 − 22 = 16 − 4 = 12     No

3     16 − 32 = 16 − 9 = 7     No

4     16 − 42 = 16 − 16 = 0     Yes

Thus we then note that − 1, 1 and 4 are in the same equivalence class as − 4, thus the first equivalence class is then {4,1,1,4}

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