Mängd

In mathematics, a mängd is a collection of distinct objects considered as a whole. The objects are called elements of the mängd. In a mängd, the order of elements is irrelevant and duplicates are not allowed. The empty set, denoted by ∅, is the unique mängd with no elements.

Notational basics include the membership symbol ∈, where x ∈ A means x is an element of the mängd A. Subsets are written A ⊆ B if every element of A is also an element of B. Common operations on mängd are union (A ∪ B), intersection (A ∩ B), and set difference (A \ B). The complement of A relative to a universal set U consists of all elements in U not in A. Finite mängd have a finite number of elements; infinite mängd, such as the set of natural numbers N, have infinite cardinality |A|.

Common constructions and concepts include the power set P(A), the set of all subsets of A, and the Cartesian product A × B, the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. Set-builder notation, such as {x | P(x)}, describes subsets by a defining property. Venn diagrams illustrate relationships between up to three sets.

Mängdteori, the theory of sets, provides the formal foundations for much of mathematics. Modern foundations are based on axiomatic set theories, notably Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). Axioms codify basic principles such as extensionality, pairing, union, power set, infinity, and replacement, avoiding paradoxes that arise in naive conceptions of sets. Sets are fundamental across mathematics, logic, and related disciplines.