dintersection

Dintersection is a generalization of the standard set intersection used in set theory and computer science. Given a finite collection of sets A1, A2, ..., An and an integer d with 1 ≤ d ≤ n, the d-intersection of these sets is the collection of elements that belong to at least d of the input sets. It can be denoted as D_d(A1,…,An) and can be expressed equivalently as the union of all d-wise intersections: D_d = ⋃_{I ⊆ {1..n}, |I|=d} (⋂_{i∈I} A_i). When d = 1, this reduces to the union of all A_i; when d = n, it is the ordinary intersection A1 ∩ A2 ∩ … ∩ An.

Properties and remarks: Dintersection is commutative and symmetric in the input sets. It is monotone decreasing in d (as d increases, the d-intersection becomes smaller). The operation is often computed by counting, for each element, how many A_i contain it, and selecting those with a count of at least d. Computationally, naive calculation is combinatorially intensive for large n, but efficient implementations exist in contexts such as frequent itemset mining and thresholded data analysis.

Examples: Let A1 = {1,2,3,4}, A2 = {3,4,5}, and A3 = {4,5,6}. D_2(A1,A2,A3) = {3,4,5}, since these elements appear in at least two of the three sets. D_3(A1,A2,A3) = {4}.

Applications and variants: The concept appears in data mining, fault-tolerant design, and threshold-based querying. Variants include weighted or probabilistic versions and extensions to multisets or fuzzy sets. See also threshold intersection, k-wise intersection, and set operations.