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Equivalence relation proof

  1. Aug 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Suppose B ⊆ A and define a relation R on P(A) as follows:

    R = {(X,Y) ∈ P(A) x P(A) | (X∆Y) ⊆ B}

    a) Show that R is an equivalence relation on P(A).
    b) Prove that for every X ∈ P(A) there is exactly one Y ∈ [X]R such that Y ∩ B = { }.

    *P(A) is the powerset of A

    2. Relevant equations

    3. The attempt at a solution

    I have successfully completed part (a) of this exercise. I seem to be having a problem with part (b). My proof so far goes like this:

    Let X ∈ P(A) and suppose Y = X\B ∈ [X]R such that Y ∩ B = { }. Now let Z ∈ [X]R such that Y ∩ B = { }. We must show that Y = Z, so let w ∈ Y. Then w ∈ X and w ∉ B...
    Last edited: Aug 28, 2011
  2. jcsd
  3. Aug 29, 2011 #2
    First you must explicitly mention that Y = X\B ∈ [X]R, which I imagine you might already have done.

    Once this is done, consider this other Z ∈ [X]R disjoint from B. We know that X\Z ∪ Z\X ⊆ B. What does this say about Z relative to X?
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