thumb|A set of polygons in an [[Euler diagram]]
thumb|This set equals the one above since they have the same elements.
In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets.
Mathematics typically does not define precisely what constitutes a "set" or "collection", because such a definition would have to be in terms of something else previously defined. Instead, sets serve as foundational objects whose behavior is described by axioms modeled on intuition about collections, and then essentially all other mathematical objects are rigorously defined in terms of sets.
Set theory studies possible axiom systems and their consequences.
Since the first half of the 20th century, ZFC (Zermelo–Fraenkel set theory with the axiom of choice) has been the axiom system most commonly used.
Context
Before the end of the 19th century, sets were not studied specifically, and they were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets. For example, a line was considered not as a set of points, but as a locus where a point may be located.
The mathematical study of infinite sets began with Georg Cantor (1845–1918). This provided some counterintuitive statements and paradoxes. For example, the number line has an infinite number of elements that is strictly larger than the infinite number of natural numbers, and any line segment has the same number of elements as the whole line. Assuming the existence of a set of all sets led to a contradiction, Russell's paradox. This led to the foundational crisis of mathematics, and to proposed resolutions. One of these, Zermelo–Fraenkel set theory, has been generally adopted as a foundation of set theory and all mathematics, though much of mathematics does not require its full power.
Meanwhile, sets started to be widely used in all mathematics. In particular, algebraic structures and mathematical spaces are typically defined in terms of sets. Also, many older mathematical results are restated in terms of sets. For example, Euclid's theorem is often stated as "the set of the prime numbers is infinite". This wide use of sets in mathematics was prophesied by David Hilbert when saying: "No one will drive us from the paradise that Cantor created for us."
The object of this article is to summarize the manipulation rules and properties of sets that are commonly used in mathematics, without reference to a specific logical framework. For the branch of mathematics that studies sets, see Set theory; for an informal presentation of the corresponding logical framework, see Naive set theory; for a more formal presentation, see Axiomatic set theory and Zermelo–Fraenkel set theory.
Basic notions
In mathematics, a set is a collection of different things, called elements or members of the set. A set may also be called a collection or family, especially when its elements are themselves sets; this may avoid confusion between the set and its members. A set may be specified either by listing its elements or by giving a property that characterizes its elements, such as for the set of the prime numbers or the set of all students in a given class.
If is an element of a set , one says that belongs to or is in , and one writes . The statement " is not in " is written as . For example, if is the set of all integers, then and . The axiom of extensionality states that two sets are equal if and only if they have the same elements.
There exists a set with no elements, and extensionality implies that there is only one such set. It is called the empty set (or null set) and is denoted , , or .
A singleton is a set with exactly one element.