Restmengen

Restmengen is a German mathematical term used to describe the remaining part of a set after removing a specified subset or collection of subsets. The phrase is descriptive rather than a single canonical construction, and its precise meaning depends on the context within a text.

In its simplest form, if U is a universal (or ambient) set and B is a subset of U, the restmenge (singular) or restmengen (plural) is the set difference U \ B, i.e., the elements of U not in B. If several subsets Bi are specified, the restmengen can refer to U \ (⋃i Bi), the complement of the union of those subsets. This makes restmengen a general way to denote the “complement” or “leftover” part relative to a chosen reference.

Applications and contexts include:

- Probability and statistics: with a sample space Ω and an event B ⊆ Ω, the restmengen of B in Ω is Ω \ B, the set of outcomes not in B.

- Measure and probability theory: the restmengen of a measurable set A is the complement A^c within a fixed universal space.

- Combinatorics, data analysis, and computer science: the remaining elements after applying a filter, exclusion, or partition.

Example: Ω = {1, 2, 3, 4, 5} and B = {2, 4}. The restmengen of B in Ω is {1, 3, 5}.

Notes: The term is widespread in German-language texts but does not denote a single universal object; its formal definition is given by context, typically as a set difference or complement relative to a chosen universal set. See also set difference, complement, and universal set.