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

Question

Let A = {−5, −4, −3, −2, −1, 0, 1, 2, 3} and define a relation R on A as follows:

For all m, n ∈ A, m R n  ⇔ 5|(m² − n²).

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.)

Answer 1

mRn ⇔ 5|(m²−n²) means (m² − n²) is divisible by 5

Therefore, looking at the set A, we can clearly tell that, m² − n² can take only 2 values i.e. m² − n² ∈ {0, 5, -5}

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 and the fact that m² − n² ∈ {0, 5, -5}, we can list all the equivalent classes as follows:

[-5] = [0] = {0, -5}

[-4] = {-4}

[-3] = [-2] = [2] = [3] = {-3, 2, -2, 3}

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

Formally, for any element a ∈ A, [a] = {b∈ A : |a² − b²| ∈ {0, 5}}

You should note that, the equivalence classes partition the set A in a unique way.