[Math] Show that R is an equivalence relation and determine all distinct classes

discrete mathematicselementary-set-theoryequivalence-relationsrelations

Let R be a relation on Z define as follows:

m R n <–> 3|($m^2$-$n^2$)

show that R is an equivalence relation and determine all distinct equivalence classes.

EDIT:
I looked several places and found R to be defined on A as follows: A={-5,-4,-3,-2,-1,0,1,2,3,4,5), How would I solve this type of problem?

I tried solving this problem this way:

  1. Equivalence relations are relations that are reflexive, symmetric, and transitive. Therefore we need a combination of sets that give these results.
  2. Knowing that I came up with R to be (2,2), (2,4), (2, 5), (4,4), (4, 2), (4,5), (5,5), (5,2) and (5,4).

  3. With my logic, I came up with the equivalence class of every element.

     [2] = {2,4}
 [4] = {2,4}
 [5] = {4,5}

  4. I concluded that the distinct equivalence classes in the relation are: {2,4}, {4,5}.

I feel like this problem is incomplete though. This is what my professor gave us, but I just feel like he's supposed to give us R. Looking past this, is my logic correct in answering this problem?

Best Answer

As it is given here, the solution is incomplete.

Item 1: You must prove that the relation is reflexive, symmetric and transitive. I will give a few hints.

  • Is n R n?
  • Is n R -n?
  • Is it true that (3 | x) <=> (3 | -x)?
  • Is it true that (a - b) mod 3 + (b - c) mod 3 = (a - c) mod 3? (mod is remainder of division)

Item 2: It was not required to explicitly list R as a subset of A^2. Omit it from the answer.

Item 3: What is [0] = { x such that 0 R x }? Find [n] for all n in A, then remove the duplicate sets (there are several). From each set, choose one element to be its representative.

Finally, a reference: Equivalence Relation (Wikipedia)

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