delmängd

Delmängd (Swedish for subset) is a fundamental concept in set theory. A delmängd of a set B is a set A such that every element of A is also an element of B. The standard notation is A ⊆ B, read “A is a delmängd of B.” If A ⊆ B and A ≠ B, then A is a proper delmängd of B; some authors use ⊂ to denote this strict relation.

Key properties: A is always a delmängd of itself, and the empty set ∅ is a delmängd of every set. The collection of all delmängder of a set B is called its power set, denoted P(B).

The subset relation ⊆ is a partial order on sets: it is reflexive (A ⊆ A), antisymmetric (if A ⊆ B and B ⊆ A then A = B), and transitive (if A ⊆ B and B ⊆ C then A ⊆ C). Subset relations are closed under union and intersection: if A ⊆ B then A ∪ B = B and A ∩ B = A.

Examples illustrate: with B = {1, 2, 3}, the set A = {1, 3} is a delmängd of B, while {1, 4} is not. For finite sets of size n, the number of delmängder is 2^n.

Applications appear across mathematics, including probability, logic, and combinatorics, where the idea of inclusion and containment underpins definitions of events, subspaces, and various indexings.