What is the correct answer to the distinct equivalence classes of R?

Question

Let X = {−1, 0, 1} and A = 𝒫(x) and define a relation R on A as follows:

For all sets s and t in 𝒫(x), s R t  ⇔  the sum of the elements in s equals the sum of the elements in t.

It is a fact that R is an equivalence relation on A. Use set-roster notation to list the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets. Enter EMPTY or ∅ for the empty set.)

Answer 1

Let X = {−1, 0, 1} and A = 𝒫(X) and define a relation R on A as follows:

For all sets s and t in 𝒫(X), sRt  ⇔  the sum of the elements in s equals the sum of the elements in t.

That is, aRb ⇔ |a| = |b|

A = 𝒫(X) = {∅, {-1}, {0}, {1}, {-1,0}, {0,1}, {-1,1}, {-1,0,1}}

For a given set A and an equivalence relation R on A, the equivalence class of an element a in A, denoted by [a], is the set {x∈A | aRx} i.e. [a] = {x∈A | aRx}

Following, this definition we can list all the equivalent classes as follows:

[∅] = [{-1,0,1}] = {∅, {-1,0,1}}

[{-1}] = [{0}] = [{1}] = {{-1}, {0}, {1}}

[{-1,0}] = [{0,1}] = [{-1,1}] = {{-1,0}, {0,1}, {-1,1}}

Formally, for any element a ∈ 𝒫(X), [a] = {b∈ 𝒫(X) : |a| = |b|}