restmengder

Restmengder, in mathematics, are the complements of a subset within a fixed universal set. If U is the universal set and A is a subset of U, then the restmengde of A relative to U is U \ A. It is often denoted A^c or A^∁_U. The restmengde consists of all elements of U that do not belong to A.

Example: Let U = {1, 2, 3, 4, 5} and A = {2, 4}. Then the restmengde is U \ A = {1, 3, 5}.

Key properties include:

- A ∪ A^c = U and A ∩ A^c = ∅.

- (A^c)^c = A.

- If A = ∅, then A^c = U; if A = U, then A^c = ∅.

- For any B with A ⊆ B ⊆ U, the rest of B after removing A is B \ A = B ∩ A^c (the relative complement of A in B).

In probability and measure theory, the restmengde corresponds to the complement of an event or measurable set within the sample space Ω. If A is an event in a probability space, A^c represents the event that A does not occur, and P(A^c) = 1 − P(A) when a total probability of 1 is assumed.

The concept is related to, and often interchangeable with, the ideas of set complement and set difference, depending on whether the universal set is made explicit. See also: complement, set difference, universal set, probability.