komplements

Komplements, in mathematics, refers to the collection of elements not in a given subset relative to a universal set. Given a universal set U and a subset A ⊆ U, the complement of A is denoted A^c (or A′) and defined as U \ A. Elements of A^c are precisely those in U that are not in A. The concept is used across set theory, logic, and related disciplines.

Basic properties include: A ∪ A^c = U and A ∩ A^c = ∅. Taking the complement twice yields the original set, (A^c)^c = A. The relative complement (or set difference) of A from B is B \ A, which can be expressed in terms of complements when B = U.

De Morgan's laws describe how complements interact with unions and intersections: (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c. These laws underpin many logical and algebraic structures, such as Boolean algebra, where the complement operation corresponds to logical negation.

Applications span several areas. In propositional logic, a complement corresponds to negation of a statement. In probability, if P(U) = 1, then P(A^c) = 1 − P(A). In computer science and information theory, the bitwise complement operator flips all bits, effectively producing the complement of a data pattern relative to a fixed width.

Example: with U = {1, 2, 3, 4} and A = {1, 2}, the complement A^c is {3, 4}. Komplements provide a simple and fundamental tool for describing what remains outside a specified subset within a universal context.