Set (mathematics)

Core characteristics and notation

Several simple conventions capture what a set is and how it behaves:

• Unordered: The order in which members are listed does not matter. {1, 2, 3} is the same set as {3, 2, 1}.
• No repetition: A set cannot contain multiple copies of the same element; {1,1,2} denotes the same set as {1,2}.
• Notation: Finite sets are often written between curly braces, e.g. {a, b, c}. Set-builder notation describes members by a property, e.g. {x | x is even and 0 < x < 10}.
• Special sets: The empty set, written {} or Ø, contains no elements; the universal set (context-dependent) contains all objects under consideration.
• Cardinality: The size of a set is its cardinality; finite sets have a natural number cardinality, while infinite sets require more refined notions.

Basic operations and relations

Sets are combined and compared by standard operations that are central to many constructions:

• Subset: A is a subset of B if every element of A is also an element of B.
• Union: A ∪ B consists of elements that are in A or in B (or in both).
• Intersection: A ∩ B contains elements common to both A and B.
• Difference: A \ B contains elements of A that are not in B.
• Complement: Relative to a universal set, the complement of A contains all elements not in A.
• Cartesian product: A × B is the set of ordered pairs (a, b) with a in A and b in B; note that ordered pairs are distinct from sets because order matters.

History and foundations

The idea of collecting objects under a common description appears in ancient mathematics, but set theory emerged in the late 19th century as a systematic foundation for mathematics. Early, informal ("naive") set theory treated any definable collection as a set and led to paradoxes when applied without restriction. Famously, Russell's paradox showed that the unrestricted comprehension principle produces contradictions when one considers the set of all sets that are not members of themselves. These foundational issues motivated the development of axiomatic systems such as Zermelo–Fraenkel set theory, which provide rules designed to avoid such contradictions and to formalize which collections count as sets. See more on paradoxes in the literature and the specific example of paradoxes and Russell's paradox.

Uses, examples, and important distinctions

Sets are ubiquitous across mathematics and its applications. They underlie the definition of functions (as sets of ordered pairs), relations, spaces in topology and analysis, algebraic structures, and much of discrete mathematics. In computer science, sets model collections, support database queries, and inform type systems. It is important to distinguish sets from related concepts: multisets allow repeated members; sequences or lists preserve order and multiplicity; proper classes (in some frameworks) are too large to be sets. Also, the study of infinite sets leads to notions of different sizes of infinity and results such as Cantor's theorem, which shows that the power set of any set (the set of all its subsets) has strictly greater cardinality than the set itself.

Summary

In short, a set is an abstract, unordered collection of distinct elements. Its simple definition supports a rich structure of operations and relations and serves as a foundation for much of modern mathematics and theoretical computer science. For introductory materials, survey articles, and formal axioms, consult standard references in set theory and foundational mathematics.