bijection

The above diagram represents a partial function that is not a total function since the element 1 in X is not associated with anything. Until the second half of the 20th century, only total functions were considered "well-defined". Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the ...

See also:

Partial function, Partial function - Domain of a partial function, Partial function - Discussion and examples

Read more here: » Partial function: Encyclopedia II - Partial function - Discussion and examples

The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and * on R (called addition and multiplication, resp.), a total order ≤ on R, satisfying the following properties. 1. (R, +, *) forms a field. In other words, For all x, y, and z in RSee also:

Construction of real numbers, Construction of real numbers - Synthetic approach, Construction of real numbers - Explicit constructions of models, Construction of real numbers - Construction from Cauchy sequences, Construction of real numbers - Construction by Dedekind cuts, Construction of real numbers - Construction by decimal expansions, Construction of real numbers - Construction from ultrafilters, Construction of real numbers - Construction from surreal numbers, Construction of real numbers - Construction from the group of integers

Read more here: » Construction of real numbers: Encyclopedia II - Construction of real numbers - Synthetic approach

In Zermelo-Fränkel set theory, Cantor's theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantor's theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. In particular, the power set of a countably infinite set is un-countably infinite. To illustrate the validity of Cantor's theorem for infinite sets, just test an infinite set in the proof below. Cantor's theorem - The proof. Including:

• Cantor's theorem - The proof
• Cantor's theorem - A detailed explanation of the proof when X is countably infinite
• Cantor's theorem - History

Read more here: » Cantor's theorem: Encyclopedia - Cantor's theorem

Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years after his first proof. His original argument did not mention decimal expansions, nor any other numeral system. Since this technique was first used, si ...

Including:

• Cantor's diagonal argument - Real numbers
• Cantor's diagonal argument - Why this does not work on integers
• Cantor's diagonal argument - General sets

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A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, di ...

Including:

• Manifold - Introduction
• Manifold - Motivational example: the circle
• Manifold - Charts atlases and transition maps
• Manifold - Construction
• Manifold - Charts
• Manifold - Patchwork
• Manifold - Zeros of a function
• Manifold - Identifying points of a manifold
• Manifold - Cartesian products
• Manifold - Gluing along boundaries
• Manifold - Topological manifolds
• Manifold - Differentiable manifolds
• Manifold - Orientability
• Manifold - Möbius strip
• Manifold - Klein bottle
• Manifold - Real projective plane
• Manifold - Other types and generalizations of manifolds
• Manifold - History

Read more here: » Manifold: Encyclopedia - Manifold

In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism that is a bijection from an object to itself. In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism, φ(xy) = φ(y)φ(x) for all x,y in X. The map that sends x to x ...

Including:

• Antihomomorphism - Examples

Read more here: » Antihomomorphism: Encyclopedia - Antihomomorphism

In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). See names of numbers in English. In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. While for finite sets the size is given by a natural number, the number of elements, cardinal numbers (cardinality ...

Including:

• Cardinal number - History
• Cardinal number - Motivation
• Cardinal number - Formal definition
• Cardinal number - Cardinal arithmetic
• Cardinal number - The continuum hypothesis

Read more here: » Cardinal number: Encyclopedia - Cardinal number

In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. This means that if we denote one such relation by ≤ then the following statements hold for all a, b and c in X: if a ≤ b and b ≤ a then a = b (antisymmetry) if a ≤ b and b ≤ c then a ≤ c (transitivity) a ≤ < ...

Including:

• Total order - Examples
• Total order - Mathematical errata
• Total order - Order topology
• Total order - Chains
• Total order - Finite total orders

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In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is ("aleph-null") and the cardinality of the real numbers is , the continuum hypothesis says: ...

Including:

• Continuum hypothesis - The size of a set
• Continuum hypothesis - Impossibility of proof and disproof
• Continuum hypothesis - Arguments pro and con
• Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia - Continuum hypothesis

See Cartesian coordinate system or Coordinates (mathematics) for a more elementary introduction to this topic. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. If the space or manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called charts, are ...

Including:

• Coordinate system - Examples
• Coordinate system - Transformations
• Coordinate system - Singularities
• Coordinate system - Systems commonly used
• Coordinate system - Astronomical systems

Read more here: » Coordinate system: Encyclopedia - Coordinate system

In mathematics the term countable is used to describe the size of a set, i.e. the number of elements it contains. The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision. A set is called countable if the number of elements is finite or if it has the same number of elements as the natural numbers. (Cantor defined a countable set as a set which can be put into one-to-one correspondence with a subset of the natural numbers). The term countable stems from the fac ...

Including:

• Countable set - Definition
• Countable set - Gentle introduction
• Countable set - A more formal introduction
• Countable set - Further theorems about uncountable sets

Read more here: » Countable set: Encyclopedia - Countable set

In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. Intuitively, a function is continuous if it maps nearby points to nearby points. For metric spaces, nearness is measured in terms of distance, leading to the ε-δ definition used in real analysis. For more general topological spaces, nearness is measured less directly in terms of open sets, leading to the definition below. If a top ...

Including:

• Continuous function topology - Definitions
• Continuous function topology - Open and closed set definition
• Continuous function topology - Neighborhood definition
• Continuous function topology - Closure and interior operator definition
• Continuous function topology - Closeness relation definition
• Continuous function topology - Useful properties of continuous maps
• Continuous function topology - Other notes

Read more here: » Continuous function topology: Encyclopedia - Continuous function topology

In mathematics, there are a number of ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered ...

Including:

• Construction of real numbers - Synthetic approach
• Construction of real numbers - Explicit constructions of models
• Construction of real numbers - Construction from Cauchy sequences
• Construction of real numbers - Construction by Dedekind cuts
• Construction of real numbers - Construction by decimal expansions
• Construction of real numbers - Construction from ultrafilters
• Construction of real numbers - Construction from surreal numbers
• Construction of real numbers - Construction from the group of integers

Read more here: » Construction of real numbers: Encyclopedia - Construction of real numbers

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. Cardinality - Comparing sets. We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positi ...

Including:

• Cardinality - Comparing sets
• Cardinality - Countable and uncountable sets
• Cardinality - Cardinal numbers
• Cardinality - Examples and other properties

Read more here: » Cardinality: Encyclopedia - Cardinality

In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: where t is a real number and E denotes the expected value. If FX is the cumulative distribution function, then the characteristic function is given by the Riemann-Stieltjes integral In cases in which there is a probabili ...

Including:

• Characteristic function probability theory - The inversion theorem
• Characteristic function probability theory - The continuity theorem
• Characteristic function probability theory - Uses of characteristic functions
• Characteristic function probability theory - Related concepts

Read more here: » Characteristic function probability theory: Encyclopedia - Characteristic function probability theory

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivists. See constructive proof. Constructivism is often confused with intuitionism, but in fact, intuitionism is only one kind of constructivism. Intuitionism maintains that the foundations ...

Including:

• Constructivism mathematics - Constructivist mathematics
• Constructivism mathematics - Example from real analysis
• Constructivism mathematics - Cardinality
• Constructivism mathematics - Attitude of mathematicians
• Constructivism mathematics - Mathematicians who have contributed to constructivism
• Constructivism mathematics - Branches

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In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. Involution - General properties. The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other exa ...

Including:

• Involution - General properties
• Involution - Involutions in Euclidean geometry
• Involution - Involutions in differential geometry
• Involution - Involutions in ring theory
• Involution - Involutions in group theory

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Examples of cartesian closed categories include: The category Set of all sets, with functions as morphisms, is cartesian closed. The product X×Y is the cartesian product of X and Y, and ZY is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X×Y → Z is naturally identified with the function g : X → ZY defined by gSee also:

Cartesian closed category, Cartesian closed category - Definition, Cartesian closed category - Examples, Cartesian closed category - Applications, Cartesian closed category - Equational theory

Read more here: » Cartesian closed category: Encyclopedia II - Cartesian closed category - Examples

Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite; i.e. c is strictly greater than the cardinality of the natural numbers, ℵ0 (aleph-null) In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in a couple of different ways. See Cantor's ...

See also:

Cardinality of the continuum, Cardinality of the continuum - Properties, Cardinality of the continuum - The continuum hypothesis, Cardinality of the continuum - Sets with cardinality c

Read more here: » Cardinality of the continuum: Encyclopedia II - Cardinality of the continuum - Properties

We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positive even numbers has the same cardinality as the set N = {1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. We say that a set A has cardinality greater than or equal to the cardinality of B (and B has cardinality l ...

See also:

Cardinality, Cardinality - Comparing sets, Cardinality - Countable and uncountable sets, Cardinality - Cardinal numbers, Cardinality - Examples and other properties

Read more here: » Cardinality: Encyclopedia II - Cardinality - Comparing sets