Set theory

Basic concepts and notation

Key notions include element membership (a \in A means a is an element of A), subset (B \subseteq A), equality (A = B when they have the same members) and the empty set denoted \varnothing. Sets are unordered and duplicates are ignored: {1,1,2} is the same as {1,2}. Common constructions are the power set P(A) (all subsets of A) and the Cartesian product A × B (ordered pairs from A and B).

Representation and methods

Three standard ways to describe sets are:

• Roster (listing) method: {a, b, c}.
• Rule (set‑builder) method: {x : property of x}, e.g. {x in N : x is even}.
• Descriptive names or definitions: A = "the set of prime numbers".

History and axiomatic development

Set theory emerged in the 19th century with work on real numbers and infinite processes, and was formalized by Georg Cantor. Early, unrestricted collections led to paradoxes such as Russell's paradox, which motivated the development of axiomatic systems like Zermelo–Fraenkel set theory (ZF) and ZF with the Axiom of Choice (ZFC). Axiomatic set theory supplies rules that avoid contradictions while allowing the construction of numbers, functions and most mathematical objects.

Operations, types and examples

Basic set operations include union (A ∪ B), intersection (A ∩ B), difference (A \ B) and complement. Sets are classified as finite or infinite, and infinite sets may be countable (like the integers) or uncountable (like the real numbers). Cardinality measures size: two sets have the same cardinality if there is a one‑to‑one correspondence between them.

Applications and notable facts

Set theory underpins nearly every part of modern mathematics: it provides definitions of functions, relations, sequences and structures used in algebra, analysis, topology and logic. It also appears in computer science (data structures and databases), probability (events as sets), and linguistics. Notable principles include the Axiom of Choice, independence results (some statements neither provable nor refutable from ZF), and Cantor's theorem that the power set of any set has strictly greater cardinality than the set itself.