Cartesian product of sets without repeated elements

In many applications, like database management systems, is very useful to have an expression to compute the cardinality of cartesian product of k sets without repeated elements; we designate this problem as T(k). The value of IT(k)I is upper-bounded by the multiplication of cardinalities of the sets. As long as we have searched, it has not been reported a general expression to compute T(k) using cardinalities of the intersections of sets, this is the main topic of this paper. Given three sets with indices {0,1, 2}, Ci is the cardinality of one set, C-i,C-j (i < j) and Ci,j,l (i < j < l) are respectively the cardinalities of the intersections of 2 and 3 sets, then the searched formulas for T(k) are: T(1) = C-0; T(2) = C0C1 -C-0,C-1; T(3) = C0C1C2 - (C0,1C2 +C0,2C1+C1,2C0) + 2C(0,1,2). In this paper, we prove formulas for computing T(k) and its specialization when a set is contained in the next sets. For this purpose, we will use concepts like partitions of the integer k in v parts, Bell numbers, Stirling numbers of the first kind and Stirling numbers of the second kind. Additionally, we present a complexity analysis for the computation of T(k). (C) 2021 Elsevier Inc. All rights reserved.