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 find the distinct equivalence classes of R.

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|Sum of elements in S 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)