komplementaarsest

Komplementaarsest is a term found in some Scandinavian mathematical texts to denote the complement of a subset with respect to a fixed universal set. The concept is equivalent to the standard set-theoretic complement; the difference lies mainly in terminology used by certain authors or traditions.

Definition: Let U be a universal set and A a subset of U (A ⊆ U). The komplementaarsest of A is the set A^c = { x ∈ U : x ∉ A }. It is the unique subset of U that together with A yields the whole universe, and whose intersection with A is empty.

Properties: The komplementaarsest operation is involutive, meaning (A^c)^c = A. It satisfies A ∪ A^c = U and A ∩ A^c = ∅. De Morgan’s laws also apply: (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c for any A, B ⊆ U.

Example: If U = {1, 2, 3, 4} and A = {1, 2}, then the komplementaarsest of A is A^c = {3, 4}.

Etymology and usage: The term komplementaarsest appears as a compound formed from komplementær (complementary) and est in some Nordic languages. In English-language texts, the standard term is typically “complement” or “complement set.” The concept remains the same: it is defined relative to a chosen universal set, and the chosen universe determines the resulting complement.

See also: Complement (set theory); Set theory; Boolean algebra.