Fullcomplement

Fullcomplement is a term used in set theory to denote the complement of a subset within a fixed universal set. While not universally standardized, the phrase often emphasizes that the complement is taken with respect to the entire universe, rather than a restricted context.

Definition and notation: Let U be a universal set and A a subset of U (A ⊆ U). The full complement of A is A^c = { x in U | x ∉ A }. It is also written A′. The full complement operation is an involution: (A^c)^c = A, and it satisfies A ∪ A^c = U and A ∩ A^c = ∅.

Relation to probability and logic: In probability, P(A^c) = 1 − P(A) when a universal set U underlies the sample space. In logic, the complement corresponds to negation. De Morgan's laws apply: (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c.

Variants and caveats: In contexts with a restricted universe V ⊆ U, one may consider the relative complement A^c_V = V \ A. If the universal set is not fixed, the term "fullcomplement" can be ambiguous; many texts prefer simply "complement" with an explicit universal set.

Examples: U = {1,2,3,4,5}, A = {1,3,5}. Then A^c = {2,4}. If V = {1,2,3,4}, the relative complement V \ A = {2,4}; note that this can differ from A^c when the universe changes.

Usage: The term is encountered in expository material and some historical texts; in modern practice, authors typically use "complement" and specify the universal set. See also complement (set theory), universal set, relative complement, and De Morgan's laws.