- Key People:
- Georg Cantor
- Paul Erdős
- John von Neumann
- Saul Kripke
- Stanislaw Ulam
- On the Web:
- Mathematics LibreTexts - Set Theory (Oct. 24, 2026)
set theory
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What is a set in mathematics?
In mathematics and logic, a set is any collection of objects (elements) that may be mathematical (e.g., numbers and functions) or not. A set is commonly represented as a list of all its members enclosed in braces. The notion extends into the infinite. For example, the set of integers from 1 to 100 is finite, whereas the set of all integers is infinite.
To indicate that an object x is a member of a set A, one writes x ∈ A, while x ∉ A indicates that x is not a member of A. A set with no members is called an empty, or null, set and is denoted ∅. A set A is called a subset of a set B (symbolized by A ⊆ B) if all the members of A are also members of B.
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How do you show or write a set?
In set theory, a set is a collection of objects or elements. A set is represented by listing its members inside braces. For example, a set of prime numbers less than 10 can be written as {2, 3, 5, 7}.
To indicate that an object x is a member of a set A, the notation x ∈ A is used. Conversely, x ∉ A means that x is not a member of A.
Special sets include:
* Empty Set A set with no members, denoted by {} or Ø
* Universal Set The set of all elements under consideration, denoted by UOperations on sets include:
* Union The union of two sets A and B, denoted by A ∪ B, is the set of all elements in either A or B, or both.
* Intersection The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that belong to both A and B.
* Complement Given a universal set U, the complement of A, denoted by A′ or U − A, is the set of all elements in U that are not in A.
* Subset A set A is a subset of B, written A ⊆ B, if every member of A is also a member of B.
* Cartesian product The Cartesian product of two sets A and B, denoted by A × B, is defined as the set consisting of all ordered pairs (a, b) for which a ∈ A and b ∈ B.5 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
What does it mean for an object to be an element of a set?
In set theory, a set is a collection of objects, called elements or members, that is regarded as a single object. If an object x is a member of a set A, it is expressed as x ∈ A. Conversely, if x is not a member of A, it is expressed as x ∉ A.
A set can be defined by a rule, formula, or by listing its members within braces. For example, the set of prime numbers less than 10 can be given as {2, 3, 5, 7}. Georg Cantor, a German mathematician, created a theory of abstract sets, defining a set as a collection of definite and distinguishable objects conceived as a whole. Set theory is now a distinct branch of mathematics.
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What is the difference between a subset and a proper subset?
In set theory, a subset is a set contained within another set, allowing for the possibility that the two sets are identical. Meaning, all elements of set B are also elements of set A.
A proper subset, however, is more restrictive: it is a subset that is not identical to the original set. So, while every element of set B must be in set A, set A must contain at least one element that is not in set B.
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How are the union and intersection of sets defined?
In set theory, a branch of mathematics, a set is any collection of objects or elements. The union of two sets, symbolized by A ∪ B, combines all elements from both sets into a single set. For instance, if set A is {1, 2, 3, 4, 5} and set B is {1, 3, 5, 7, 9}, then A ∪ B is {1, 2, 3, 4, 5, 7, 9}.
The intersection of two sets, denoted by A ∩ B, includes only the elements that are common to both sets. Using the same example, the intersection of A and B (A ∩ B) would be {1, 3, 5}.
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How can set theory help us solve problems in mathematics?
Set theory, created by Georg Cantor between 1874 and 1897, is a branch of mathematics that studies collections of objects. It provides a precise terminology for defining complex mathematical concepts and can be used to derive virtually all mathematical concepts.
In topology, set theory helps define topological spaces by specifying collections of subsets that meet certain axioms. The principles of set theory, including the membership relation, also serve as a foundation for mathematics. The Zermelo-Fraenkel set theory, or ZFC, is an axiomatic system that helps avoid paradoxes and derive results in classical set theory.
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News •
set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts.
Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. A set, wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements or members of the set.
The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinity of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor’s work was the subject of much criticism to the effect that it dealt with fictions—indeed, that it encroached on the domain of philosophers and violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.
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At just that time, however, several contradictions in so-called naive set theory were discovered. In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to that developed for elementary geometry. The degree of success that has been achieved in this development, as well as the present stature of set theory, has been well expressed in the Nicolas Bourbaki Éléments de mathématique (begun 1939; “Elements of Mathematics”): “Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, The Theory of Sets.”
Introduction to naive set theory
Fundamental set concepts
In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes x ∊ A, while x ∉ A indicates that x is not a member of A. A set may be defined by a membership rule (formula) or by listing its members within braces. For example, the set given by the rule “prime numbers less than 10” can also be given by {2, 3, 5, 7}. In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers ℕ goes on forever. The empty (or void, or null) set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the status of being a set.
A set A is called a subset of a set B (symbolized by A ⊆ B) if all the members of A are also members of B. For example, any set is a subset of itself, and Ø is a subset of any set. If both A ⊆ B and B ⊆ A, then A and B have exactly the same members. Part of the set concept is that in this case A = B; that is, A and B are the same set.