(adsbygoogle = window.adsbygoogle || []).push({}); 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...

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# Equivalence relation proof

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