unionintersection

Unionintersection refers to the combined application of the union and intersection operations on a collection of sets. In set theory and computer science, the union of a family of sets collects all elements that appear in at least one member, while the intersection collects only those elements that appear in every member. A unionintersection operation can mean either computing both results separately or producing both results together, sometimes in a single pass or query plan to improve efficiency.

Formally, let F be a family of sets on a universe U. The union is U_F = ⋃{A : A ∈ F}, and the intersection is I_F = ⋂{A : A ∈ F}. The union gives all elements that appear in at least one member, and the intersection yields elements common to all members. For example, if F = {A, B} with A = {1, 2, 3} and B = {2, 3, 4}, then the union is {1, 2, 3, 4} and the intersection is {2, 3}.

Computationally, union and intersection can be computed in parallel or sequentially. The union typically requires aggregating all distinct elements, while the intersection may allow early termination if an element is missing from a member set. Properties include that both union and intersection are commutative and associative with respect to the order of the input sets. Note that the union of an empty family is the empty set, while the intersection of an empty family is often taken as the universal set in a given context.

In practice, the concept appears in database queries, data analysis, and algorithm design, where both results are of interest or where a joint evaluation improves performance. The term itself is not universally standardized and may be described as “union and intersection” operations or a combined unionintersection routine in different texts or libraries.