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I have the following problem and I don’t know if I’m even setting it up right. Any help is much appreciated!

I’m given: Let U be any set and let P(U) be the power set of U. Prove that for every A ∈ P(U) there is a unique B ∈ P(U) such that for every C ∈ P(U), C\A = C ∩ B.

I’m setting it up as follows but again, I don’t know if this is right: ∀A ∈ P(U) ∃!B ∈ P(U) (∀A ∈ P(U)(C\A = C ∩ B).

Now…I don’t know if this is saying that there is one B set in total that will solve C\A = C ∩ B for every A and C (which I am thinking this isn’t possible) or is it saying that for every iteration of A and C we use, there is a unique B. In other words, for instance A1 and C1 we have a unique B1 and then for instance A2 and C2 we have a unique B2.

Thanks again guys!

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    B is the complement of A. Why do you want to first formalize the information? – 2017-02-13
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    Thanks Casper. Formalizing it is part of the exercise. – 2017-02-13
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    Ok. Your formalization is correct except for what I think is a typo: $\forall A\in P(U)\exists ! B\in P(U)\forall C\in P(U)(C\setminus A=C\cap B)$ – 2017-02-13

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