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Garey and Johnson describe in their book many NP-complete problems which are based on sets, for example Hitting Set, Minimum Test Set, Set Packing, Set Splitting, and many more.

The traditional mathematical definition of a set does not allow duplicates; when multiplicities count, the object is called a multiset.

Are the sets in the NP-complete problems described by Garey and Johnson allowed to contain duplicates?

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It doesn't matter. If a certain problem has one version in which the encoding of sets allows for repeated elements (which are ignored semantically), and another in which repeated elements are forbidden, then there is a simple polynomial time reduction from the former to the latter showing that if the first version is NP-complete, so is the second one.

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