Question

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

For all sets s and t in (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

The distinct equivalence classes of the relation R on set A are {-1}, {0, -1, 1}, and {1, -1, 0}.

Step-by-step explanation:

To list the distinct equivalence classes of the relation R, we need to group the elements in A based on whether their sum is the same. Let's examine each possible sum for the elements in X:

If the sum is -1, there is only one set in the equivalence class: {-1}.

If the sum is 0, there are three sets in the equivalence class: {0}, {-1, 1}, and {1, -1}.

If the sum is 1, there are three sets in the equivalence class: {-1, 0, 1}, {0, 1, -1}, and {1, -1, 0}.

Therefore, the distinct equivalence classes of R are {-1}, {0, -1, 1}, and {1, -1, 0}.