Let $X = \{−1,0,1\}$ and $A =\mathcal{P}(X)$, and $R$ is defined on $A$ as for all sets $S,T \in A$, $\ldots$

Question

Let $X = \{−1,0,1\}$ and $A = \mathcal{P}(X)$, and $R$ is defined on $A$ as for all sets $S,T \in A$, $$ SRT \Longleftrightarrow \text{the sum of the elements in $S$ equals the sum of the elements in $T$}. $$ Find the distinct equivalence classes of $R$.

Answer 1

Subsets should be $\{\},\{-1\},\{0\},\{1\},\{-1,0\},\{-1,1\},\{0,1\},\{-1,0,1\}$

One set will be: $\{\},\{0\},\{-1,1\}$, and $\{-1,0,1\}$

Second set: $\{-1,0\}$ and $\{-1\}$

Last set: $\{1\},\{0,1\}$

The equivalence classes of $RR$ are:

1. $\{\{\},\{0\},\{-1,1\},\{-1,0,1\}\}$
2. $\{\{1\},\{0,1\}\}$
3. $\{\{1\},\{0,1\}\}$