Set theory

Venn A intersect B

Georg Cantor 1894

Von Neumann Hierarchy

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can theoretically be collected into a set, set theory is applied most commonly to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.

The modern study of set theory was formalized in the late 19th century by Georg Cantor and Richard Dedekind. Set theory is now a major area of research in mathematics, with many applications in the foundations of mathematics, abstract algebra, and mathematical logic.

Basic Concepts and Notations

Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, the notation o ∈ A is used. A set is described by listing its elements between braces, for example, {1, 2, 3} is the set containing the elements 1, 2, and 3. Sets can also contain other sets as elements.

Sets and Membership

The concept of a set is one of the most fundamental in mathematics. A set can be defined informally as a collection of distinct objects, considered as an object in its own right. For example, the numbers 1, 2, and 3 are distinct objects when considered separately, but when they are considered collectively as the set {1, 2, 3}, they form a single object.

Subsets

A set A is considered a subset of a set B if every element of A is also an element of B. The notation A ⊆ B denotes that A is a subset of B. A subset A of B is called a proper subset if A is not equal to B.

Union, Intersection, and Difference

The union of two sets A and B is the set of elements that are in A, in B, or in both A and B. The intersection of two sets is the set of elements that are in both A and B. The difference between two sets A and B (denoted A \ B) is the set of elements that are in A but not in B.

Important Concepts in Set Theory

Set theory introduces several important operations and concepts that are widely used in mathematics:

- Cardinality: The cardinality of a set is a measure of the "number of elements" in the set. - Power set: The power set of a set A is the set of all possible subsets of A. - Ordered pair and Cartesian product: An ordered pair is a collection of two elements where order matters. The Cartesian product of two sets is the set of all possible ordered pairs that can be formed by the elements of the two sets. - Function: A function is defined as a special type of relation between sets. - Infinite sets and finite sets: Sets can be classified based on whether they have a finite number of elements or an infinite number.

Axiomatic Set Theory

In the early 20th century, several paradoxes were discovered in naive set theory, leading to the development of axiomatic set theories. The most widely studied form of axiomatic set theory is Zermelo–Fraenkel set theory (ZF), which includes the Axiom of Choice (AC). Together, ZF and AC form the foundation for much of modern mathematics.

Applications of Set Theory

Set theory is not only a foundation for mathematics but also has applications in various fields such as computer science, where it is used in the study of algorithms and data structures, and in linguistics, where it helps in the analysis of language structure.