restmenge

Restmenge is a term used in German mathematical texts to denote the remainder or residual set obtained when a specified subset is removed from a universal set. In English-language contexts the corresponding concept is usually called the complement of a set, or the set difference relative to a universal set.

Etymology and scope: The word combines Rest (remainder) and Menge (set). The Restmenge is defined with respect to a fixed universal set U. If A is a subset of U, then the Restmenge of A in U is U \ A, the elements of U that are not in A. This is distinct from the relative complement A \ B, which is defined within A rather than within U.

Notation and basic properties: If U is the universal set and A ⊆ U, the Restmenge of A in U is denoted U \ A or A^c when the universal context is clear. It satisfies U = A ∪ (Restmenge) and A ∩ (Restmenge) = ∅, assuming A ⊆ U. If A is not a subset of U, the precise definition of the Restmenge may depend on the chosen universal reference and requires adjustment to maintain consistency.

Example: Let U = {1, 2, 3, 4, 5} and A = {2, 4}. The Restmenge is {1, 3, 5}. In probability terms, the rest of the sample space after event A is the complement of A with respect to U, having probability 1 − P(A) when probabilities are defined on U.

See also: Complement (set theory), Set difference, Universal set, Relative complement.