complements

In mathematics and related fields, a complement is what remains when something is removed from a larger whole or what is needed to complete meaning. The notion appears in set theory, probability, logic, linguistics, and computer science, where a complement is often defined relative to a fixed universe or context.

In set theory, the complement of a subset A within a universal set U is the set of elements in U that are not in A. It is denoted A^c or U\A. Complementation interacts with unions and intersections: the complement of a union equals the intersection of the complements, and the complement of an intersection equals the union of the complements (De Morgan's laws).

In probability, the complement of an event A is the event that A does not occur. If P(A) is the probability of A, then P(A) + P(A^c) = 1, and P(A^c) = 1 − P(A).

In linguistics and grammar, a complement is a word, phrase, or clause that completes the meaning of a predicate or linking verb. For example, in "She believes that the test will be easy," the clause "that the test will be easy" is a complement; in "They are tired," "tired" serves as a predicative complement.

In logic and computer science, the complement often refers to negation or the NOT operation, which inverts a truth value or a bit pattern. The logical complement of true is false, and the bitwise complement flips every bit in a binary word.