Set (mathematics)

[5] Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences.

Together with other counterintuitive results, this led to the foundational crisis of mathematics, which was eventually resolved with the general adoption of Zermelo–Fraenkel set theory as a robust foundation of set theory and all mathematics.

In particular, algebraic structures and mathematical spaces are typically defined 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 which Cantor created for us.

In mathematical practice, sets can be manipulated independently of the logical framework of this theory.

The object of this article is to summarize the manipulation rules and properties of sets that are commonly used in mathematics, without reference to any logical framework.

[1][2][3][4] These things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, functions, or even other sets.

[8] Some definitions require the elements of a set to possess a distinctive common property.

The natural numbers form an infinite set, commonly denoted ⁠

[19] that specifies a set by listing its elements between braces, separated by commas.

When there is a clear pattern for generating all set elements, one can use ellipses for abbreviating the notation,[27][28] such as in

⁠ once for all and take the convention that every variable that appears on the left of the vertical bat of the notation represents an element of ⁠

For example, with the convention that a lower case Latin letter may represent a real number and nothing else, the expression

Likewise, the intersection is formally defined by the corresponding logical conjunction (

Satisfying associativity means that the order in which a series of operations is performed is trivial.

An expression that includes both unions and intersections, in addition, satisfies distributivity.

[8] Together the three operations satisfy De Morgan's laws, which are two identities that relate unions and intersection.

Both laws can be extended to hold true for series of either unions or intersections such as

These laws, formulated by Augustus De Morgan, have their roots in logic.

When one or both are infinite, multiplication of cardinal numbers is defined to make this true.

For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations.

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively.

The cardinality of a set S, denoted |S|, is the number of members of S.[39] For example, if B = {blue, white, red}, then |B| = 3.

[30] In fact, all the special sets of numbers mentioned in the section above are infinite.

[52] The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite.

[58] The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic.

According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.