The correct answer is: D. superset of both closed frequent item sets and maximal frequent item sets.
A frequent item set is a set of items that occurs in a transaction database with a frequency greater than or equal to a user-specified minimum support threshold. A closed frequent item set is a frequent item set that cannot be extended to a larger frequent item set by adding any additional items. A maximal frequent item set is a frequent item set that cannot be extended to a larger frequent item set by adding any additional items, and that is not a subset of any other frequent item set.
Therefore, all frequent item sets are supersets of both closed frequent item sets and maximal frequent item sets.
Here is a more detailed explanation of each option:
- Option A: Frequent item sets are a superset of only closed frequent item sets. This is not true, because frequent item sets also include sets of items that are not closed. For example, the set of items {A, B} is a frequent item set, but it is not a closed frequent item set, because it can be extended to the larger frequent item set {A, B, C}.
- Option B: Frequent item sets are a superset of only maximal frequent item sets. This is also not true, because frequent item sets also include sets of items that are not maximal. For example, the set of items {A, B} is a frequent item set, but it is not a maximal frequent item set, because it is a subset of the larger frequent item set {A, B, C}.
- Option C: Frequent item sets are a subset of maximal frequent item sets. This is not true, because frequent item sets include sets of items that are not maximal. For example, the set of items {A, B} is a frequent item set, but it is not a maximal frequent item set, because it is a subset of the larger frequent item set {A, B, C}.
- Option D: Frequent item sets are a superset of both closed frequent item sets and maximal frequent item sets. This is the correct answer, because frequent item sets include all closed frequent item sets and all maximal frequent item sets.