   Chapter 8.3, Problem 9ES ### 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, the relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. X = { − 1 , 01 } and A = P ( X ) ,R is defined on P ( X ) as follows: For all sets s r in P ( X ) , s R r     ⇔ the   sum   of the elements in  s   equals the sum of the elements in  r .

To determine

To find the distinct equivalence classes of R.

Explanation

Given information:

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

X = {-1, 0, 1} and A = P(X). R is defined on P(X) as follows: For all sets s and r in P(X),

S R T       the sum of the elements in S equals the sum                        of the elements in T

Calculation:

X={a,b,c}A=P(X)R={(S,T)A×A|SumofelementsinS is the same as sum of elements in T}

Let us first determine all elements in A, which are thus all subsets of X :

A={0,{1},{0},{1},{1,0},{1,1},{0,1},{1,0,1}}

Two elements are then related if the sum of the elements of the sets is the same:

R={(ϕ,ϕ),(ϕ,{0}),(ϕ,{ 1,1}),(ϕ,{ 1,0,1}),      ({0},ϕ),({0},{0}),({0},{1,1}),({0},{1,0,1}),      ({1,1},ϕ),({1,1},{0}),({1,1},{1,1}),({1,1},{1,0,1}),       ({1,0,1},ϕ),({1,0,1},{0}),({1,0,1},{1,1}),({1,0,1},{1,0,1}),       ({1},{1}),({1},{1,0}),({1,0},{1}),({1,0},{1,0}),       ({1},{1}),({1},{ 1,0}),({ 1,0},{1}),({ 1,0},{ 1,0})}

Let us next group the ordered pairs in R that have at least one common element (with at least one of the other elements in the group)

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