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Here is an instance of a problem:

Instance: {U, S1, . . . Sn, k|U is a set of elements, the Si are different subsets of U, and k is a nonnegative integer}.

A YES instance is defined as follows: There exists C ⊆ U with |C| = k such that ∀i =6 j,(Si − C) = ( 6 Sj − C). That is, the sets Si − C remain different.

My question is to prove that this is NP-Complete. It is trivial to prove that this is in NP, but as yet I have not been able to find a reduction from an NP-Complete problem to prove NP-hardness.

My approach so far has been to reduce from 3-SAT as follows: consider an instance of 3-SAT, with clauses (a,b,c) and (a,~b,~c). Then, U = {a,~a,b,~b,c,~c}, S = { {a,~a}, {b,~b}, {c,~c}, {}, {a,b,c}, {a,~b,~c} }, and k = 3 (number of variable-complement pairs). If I have a NO instance of 3-sat, this correctly gives me a NO instance. However, a YES instance does not always return a YES instance, so I must modify my answer somehow.

I could not find any other online resource, so any help would be much appreciated!

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Something has gone wrong with the notation in your third paragraph. –  Gareth Rees Nov 21 '12 at 9:26

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