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# Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. [1]

91 relations: Abstract algebra, Algebraic number, Algebraic structure, Alternative set theory, Augustus De Morgan, Axiom, Bernard Bolzano, Binary relation, Blackboard bold, Boolean ring, Bracket, Cambridge University Press, Cantor's paradox, Category of sets, Class (set theory), Closure (mathematics), Codomain, Colon (punctuation), Complex number, Countable set, De Morgan's laws, Dense set, Dimension, Domain of a function, Dover Publications, E (mathematical constant), Element (mathematics), Empty set, Euclidean space, Extension (semantics), Extensional and intensional definitions, Family of sets, Field (mathematics), First-order logic, Flag of France, Fraction (mathematics), Fuzzy set, Georg Cantor, Group (mathematics), Harvard University Press, If and only if, Infinite set, Integer, Internal set, Irrational number, Joseph Dauben, Letter case, Line (geometry), Line segment, Mathematical object, ... Expand index (41 more) »

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

## Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

## Alternative set theory

Generically, an alternative set theory is an alternative mathematical approach to the concept of set.

## Augustus De Morgan

Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician.

## Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

## Bernard Bolzano

Bernard Bolzano (born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Blackboard bold

Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled.

## Boolean ring

In mathematics, a Boolean ring R is a ring for which x2.

## Bracket

A bracket is a tall punctuation mark typically used in matched pairs within text, to set apart or interject other text.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Cantor's paradox

In set theory, Cantor's paradox is a statement derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite.

## Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets.

## Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

## Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

## Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall.

## Colon (punctuation)

The colon is a punctuation mark consisting of two equally sized dots centered on the same vertical line.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## De Morgan's laws

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference.

## Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A &mdash; for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

## E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

## Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Extension (semantics)

In any of several studies that treat the use of signs—for example, in linguistics, logic, mathematics, semantics, and semiotics—the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.

## Extensional and intensional definitions

Extensional and intensional definitions are two key ways in which the object(s) or concept(s) a term refers to can be defined.

## Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## First-order logic

First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

## Flag of France

The flag of France (Drapeau français) is a tricolour flag featuring three vertical bands coloured blue (hoist side), white, and red.

## Fraction (mathematics)

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

## Fuzzy set

In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership.

## Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Harvard University Press

Harvard University Press (HUP) is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Infinite set

In set theory, an infinite set is a set that is not a finite set.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Internal set

In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model.

## Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

## Joseph Dauben

Joseph Warren Dauben (born 29 December 1944, Santa Monica) is a Herbert H. Lehman Distinguished Professor of History at the Graduate Center of the City University of New York.

## Letter case

Letter case (or just case) is the distinction between the letters that are in larger upper case (also uppercase, capital letters, capitals, caps, large letters, or more formally majuscule) and smaller lower case (also lowercase, small letters, or more formally minuscule) in the written representation of certain languages.

## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

## Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

## Mathematical object

A mathematical object is an abstract object arising in mathematics.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mathematics education

In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research.

## Mereology

In philosophy and mathematical logic, mereology (from the Greek μέρος meros (root: μερε- mere-, "part") and the suffix -logy "study, discussion, science") is the study of parts and the wholes they form.

## Multiset

In mathematics, a multiset (aka bag or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements.

## Naive set theory

Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

## Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

## Paul Halmos

Paul Richard Halmos (Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-Jewish-born American mathematician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces).

## Pi

The number is a mathematical constant.

## Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Primitive notion

In mathematics, logic, and formal systems, a primitive notion is an undefined concept.

## Principia Mathematica

The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913.

## Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Rough set

In computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set.

## Russell's paradox

In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Set notation

Sets are fundamental objects in mathematics.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Set-builder notation

In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy.

## Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

## Square number

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Symmetric difference

In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection.

## Taxonomy (general)

Taxonomy is the practice and science of classification.

## The Paradoxes of the Infinite

The Paradoxes of the Infinite (German title: Paradoxien des Unendlichen) is a mathematical work by Bernard Bolzano on the theory of sets.

## Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

## Tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements.

## Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

## Universal set

In set theory, a universal set is a set which contains all objects, including itself.

## Universe (mathematics)

In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation.

## Venn diagram

A Venn diagram (also called primary diagram, set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets.

## Vertical bar

The vertical bar (|) is a computer character and glyph with various uses in mathematics, computing, and typography.

## Well-defined

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

## References

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