Set Theory
Introduction
Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.
History
Set theory was developed in the 19th century by mathematicians such as Georg Cantor and Richard Dedekind. Cantor's work in particular was instrumental in the development of the theory, as he introduced the concept of a "set" and used it to develop a theory of infinite numbers.
Basic Concepts
The basic concepts of set theory are surprisingly simple. A set is a collection of distinct objects, considered as an object in its own right. Sets are usually denoted by uppercase letters, and their elements are denoted by lowercase letters. The objects in a set are called its elements or members. If an object 'a' is a member of a set 'A', we write 'a ∈ A'. If 'a' is not a member of 'A', we write 'a ∉ A'.
Empty Set
The empty set, denoted by '∅', is the set with no elements. It is a subset of every set, including itself.
Singleton Set
A singleton set is a set with exactly one element. If 'a' is an object, the singleton set containing 'a' is denoted by '{a}'.
Subset
If every element of a set 'A' is also an element of a set 'B', then 'A' is a subset of 'B', denoted 'A ⊆ B'. Every set is a subset of itself, and the empty set is a subset of every set.
Power Set
The power set of a set 'A', denoted by 'ℙ(A)', is the set of all subsets of 'A'.
Cardinality
The cardinality of a set 'A', denoted '|A|', is the number of elements in 'A'. The cardinality of the empty set is zero.
Axiomatic Set Theory
Axiomatic set theory is a rigorous axiomatic theory developed in response to the paradoxes of naive set theory. The most widely studied systems of axiomatic set theory are Zermelo-Fraenkel set theory (ZF) and Von Neumann–Bernays–Gödel set theory (NBG).
Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel set theory (ZF) is the most commonly used system of axiomatic set theory. It is named after its creators, Ernst Zermelo and Abraham Fraenkel. ZF includes the axiom of choice, which states that for any set of nonempty sets, there exists a choice function that selects exactly one element from each set.
Von Neumann–Bernays–Gödel Set Theory
Von Neumann–Bernays–Gödel set theory (NBG) is a conservative extension of ZF that includes proper classes in addition to sets. It was developed by John von Neumann, Paul Bernays, and Kurt Gödel.
Applications of Set Theory
Set theory is used throughout mathematics. It is used as a foundational system, in which all mathematical objects (numbers, functions, geometric figures, etc.) are defined as sets. Set theory is also used in many areas of computer science, such as in the design of data structures and algorithms.