preimage - Article Index

Encyclopedia - Bijection Injection And Surjection: Encyclopedia - Bijection Injection And Surjection
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input exp...   » Read the article

Encyclopedia - Cellular Automaton: Encyclopedia - Cellular Automaton
A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory, mathematics, and theoretical biolog...   » Read the article

Encyclopedia - Null Vector Vector Space: Encyclopedia - Null Vector Vector Space
In linear algebra and related areas of mathematics, the null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all o...   » Read the article

Encyclopedia - Separated Sets: Encyclopedia - Separated Sets
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to eac...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia - Sheaf Mathematics
In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each ope...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Mathematical Definition Of A Function
A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an eleme...   » Read the article

Encyclopedia - Open And Closed Maps: Encyclopedia Ii - Open And Closed Maps - Examples
Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism iff it's open, or equivalentl...   » Read the article

Encyclopedia - Decision Problem: Encyclopedia Ii - Decision Problem - Definition
A decision problem is a countable set S and a function . Let A be the preimage of f for 1. The problem is called decidable if A is a...   » Read the article

Encyclopedia - Disjoint Union Topology: Encyclopedia Ii - Disjoint Union Topology - Definition
Let {Xi : i ∈ I} be a family of topological spaces indexed by I. Let be the disjoint union of the underlying sets. For each i i...   » Read the article

Encyclopedia - Quotient Space: Encyclopedia Ii - Quotient Space - Definition
Suppose X is a topological space and ~ is an equivalence relation on X. We define a topology on the quotient set X/~ (the set consisting ...   » Read the article

Encyclopedia - Sigma-algebra: Encyclopedia Ii - Sigma-algebra - Examples
If X is any set, then the family consisting only of the empty set and X is a σ-algebra over X, the so-called trivial σ-algebra. Another...   » Read the article

Encyclopedia - List Of Mathematical Functions: Encyclopedia Ii - List Of Mathematical Functions - Elementary Functions
List of mathematical functions - Polynomials. Polynomials: can be generated by addition and multiplication alone. Constant function: Z...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Introduction
The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler...   » Read the article

Encyclopedia - Open Set: Encyclopedia Ii - Open Set - Definitions
The concept of open sets can be formalized in various degrees of generality. Open set - Function-analytic. A point set in Rn is called ...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Introduction
Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary w...   » Read the article

Encyclopedia - Equaliser: Encyclopedia Ii - Equaliser - In Category Theory
Equalisers can be defined by a universal property, which allows the notion to be generalised from the category of sets to arbitrary categ...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - The Vocabulary Of Functions
An input to a function is called argument of the function. For each argument x, the corresponding unique y in the codomain is called the ...   » Read the article

Encyclopedia - Proper Map: Encyclopedia Ii - Proper Map - Algebraic Varieties And Schemes
Proper map - Definition. A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber pr...   » Read the article

Encyclopedia - Recursive Set: Encyclopedia Ii - Recursive Set - Definition
A subset S of the natural numbers is called recursive if there exists a total computable function with In other words the set S is ...   » Read the article

Encyclopedia - Recursively Enumerable Set: Encyclopedia Ii - Recursively Enumerable Set - Definition
A countable set S is called recursively enumerable if there exists a partial computable function such that S is the range of f : r...   » Read the article

Encyclopedia - Sigma-algebra: Encyclopedia Ii - Sigma-algebra - Examples
If X is any set, then the family consisting only of the empty set and X is a σ-algebra over X, the so-called trivial σ-algebra. Another...   » Read the article

Encyclopedia - Surjection: Encyclopedia Ii - Surjection - Examples And Counterexamples
f(x) = 2x + 1 is surjective, because for every real number y we have f(x) = y where x is (y - 1)/2. The natural logarithm function ln:&#...   » Read the article

Encyclopedia - Measurable Function: Encyclopedia Ii - Measurable Function - Special Measurable Functions
If (X,Σ) and (Y,Τ) are Borel spaces, a measurable function f is also called a Borel function. Continuous functions are Borel but not al...   » Read the article

Encyclopedia - Glossary Of Differential Geometry And Topology: Encyclopedia Ii - Glossary Of Differential Geometry And Topology - C
Chart Cobordism Codimension. The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifol...   » Read the article

Encyclopedia - Glossary Of Group Theory: Encyclopedia Ii - Glossary Of Group Theory - Basic Definitions
Order of a group. Order of a group (G,*) is the cardinality (i.e. number of elements) of G. A group with finite order is called a finite ...   » Read the article

Encyclopedia - Kernel Algebra: Encyclopedia Ii - Kernel Algebra - Survey Of Examples
Kernel algebra - Linear operators. Let V and W be vector spaces and let T be a linear transformation from V to W. If 0W is the zero vec...   » Read the article

Encyclopedia - Separated Sets: Encyclopedia Ii - Separated Sets - Definitions
There are various versions of the concept. The terms are defined below, where X is a topological space. First, two subsets A and B of X a...   » Read the article

Encyclopedia - Cellular Automaton: Encyclopedia Ii - Cellular Automaton - History Of Cellular Automata
Stanislaw Ulam, while working at the Los Alamos National Laboratory in the 1940s, studied the growth of crystals, using a simple lattice ...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Formal Definition
A fiber bundle consists of the data (E, B, π, F), where E, B, and F are topological spaces and π : E → B is a continuous surject...   » Read the article

Encyclopedia - Decision Problem: Encyclopedia Ii - Decision Problem - Definition
A decision problem is a countable set S and a function . Let A be the preimage of f for 1. The problem is called decidable if A is a...   » Read the article

Encyclopedia - Bijection Injection And Surjection: Encyclopedia Ii - Bijection Injection And Surjection - Bijection
A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-one correspondence). A function ...   » Read the article

Encyclopedia - Bijection, Injection And Surjection: Encyclopedia Ii - Bijection, Injection And Surjection - Examples
It is important to specify the domain and codomain of each function since by changing these, functions which we think of as the same may ...   » Read the article

Encyclopedia - Bijection, Injection And Surjection: Encyclopedia Ii - Bijection, Injection And Surjection - Bijection
A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-one correspondence). A function ...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Mathematical Definition Of A Function
A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an eleme...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Classes Of Functions
Function mathematics - Injective surjective bijective. Three important properties that a function may have are: injective (or one-to-o...   » Read the article

Encyclopedia - Cellular Automaton: Encyclopedia Ii - Cellular Automaton - Reversible Cellular Automata
A CA is said to be reversible if for every current configuration of the CA there is exactly one past configuration (preimage). If one thi...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Specifying A Function
If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - History Of The Concept
As a mathematical term, "function" was coined by Gottfried Leibniz in 1694, to describe a quantity related to a curve, such as a curve's ...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Partial Functions And Multi-functions
The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x i...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions In Other Fields
Functions are used in every quantitative science, to model relationships between all kinds of physical quantities — especially when one...   » Read the article

Encyclopedia - Equaliser: Encyclopedia Ii - Equaliser - Difference Kernels
A binary equaliser (that is, an equaliser of just two functions) is also called a difference kernel. This may also be denoted DiffKer(f,g...   » Read the article

Encyclopedia - Cellular Automaton: Encyclopedia Ii - Cellular Automaton - The Simplest Cellular Automata
The simplest nontrivial CA would be one-dimensional, with two possible states per cell, and a cell's neighbors defined to be the adjacent...   » Read the article

Encyclopedia - Equaliser: Encyclopedia Ii - Equaliser - Definitions
Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equaliser of f and g is the set of elements x of X such that f(...   » Read the article

Encyclopedia - Cellular Automaton: Encyclopedia Ii - Cellular Automaton - Related Automata
There are many possible generalizations of the CA concept. One way is by using something other than a rectangular (cubic, etc.) grid. For...   » Read the article

Encyclopedia - Cellular Automaton: Encyclopedia Ii - Cellular Automaton - Cellular Automata In Nature
Patterns of certain seashells, like the ones in Conus and Cymbiola genus, are generated by natural cellular automata. The pigment cells r...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions In Category Theory
The notion of function is generalizes to the notion of morphism in the context of category theory. A category is a collection of objects ...   » Read the article

Encyclopedia - Decision Problem: Encyclopedia Ii - Decision Problem - Notes
It should be noted that a decision problem is always a set of related problems which is in some sense large enough. A single problem P is...   » Read the article

Encyclopedia - Cellular Automaton: Encyclopedia Ii - Cellular Automaton - Uses In Cryptography
Rule 30 was originally suggested as a possible stream cipher for use in cryptography. Cellular automata have been proposed for public key...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Pointwise Operations
If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can defi...   » Read the article

Encyclopedia - Disjoint Union Topology: Encyclopedia Ii - Disjoint Union Topology - Properties
The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a t...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions With Multiple Inputs And Outputs
Function mathematics - Functions of two or more variables. The concept of function can be extended to an object that takes a combinatio...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Restrictions And Extensions
Informally, a restriction of a function f is the result of trimming its graph to a smaller domain. More precisely, if f is a function fro...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Is A Function More Than Its Graph?
Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codo...   » Read the article

Encyclopedia - Proper Map: Encyclopedia Ii - Proper Map - Topological Spaces
Proper map - Definition. A function f : X → Y between two topological spaces is proper if and only if the preimage of every comp...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Generalizations
It is possible to define a cohomology theory for sheaves of abelian groups (sheaf cohomology) that can give much useful, more concrete in...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - History
The first origins of sheaf theory are hard to pin down — they may be co-extensive with the idea of analytic continuation. It took about...   » Read the article

Encyclopedia - Glossary Of Group Theory: Encyclopedia Ii - Glossary Of Group Theory - Types Of Groups
Abelian group. A group (G,*) is abelian if * is commutative, i.e. g*h=h*g for all g,h ∈ G. Likewise, a group is nonabelian if this rela...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - The étale Space Of A Sheaf
In early developments of sheaf theory, it was shown that giving a sheaf F on X is as good as giving a certain topological space E togethe...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Stalks Of A Sheaf At A Point And Germs Of Functions
Fix a point x of X. We would like to study the behavior of F near the point x. In analytical terms, we would like to somehow take the lim...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Examples
In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sect...   » Read the article

Encyclopedia - Recursively Enumerable Set: Encyclopedia Ii - Recursively Enumerable Set - Remarks
Since the Church-Turing thesis states that computable functions are defined equivalently by Turing machines and other models of computati...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - Morphisms Of Sheaves
Let F and G be two sheaves on X both with values in the category C. We define a morphism from G to F to be a family of morphisms φU ...   » Read the article

Encyclopedia - Glossary Of Differential Geometry And Topology: Encyclopedia Ii - Glossary Of Differential Geometry And Topology - V
Vector bundle, a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps. Vector field, a section of a...   » Read the article

Encyclopedia - Glossary Of Differential Geometry And Topology: Encyclopedia Ii - Glossary Of Differential Geometry And Topology - T
Tangent bundle, the vector bundle of tangent spaces on a differentiable manifold. Tangent field, a section of the tangent bundle. Also ca...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Pointwise Operations
If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can defi...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Restrictions And Extensions
Informally, a restriction of a function f is the result of trimming its graph to a smaller domain. More precisely, if f is a function fro...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Partial Functions And Multi-functions
The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x i...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Is A Function More Than Its Graph?
Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codo...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions In Category Theory
The notion of function is generalizes to the notion of morphism in the context of category theory. A category is a collection of objects ...   » Read the article

Encyclopedia - Glossary Of Differential Geometry And Topology: Encyclopedia Ii - Glossary Of Differential Geometry And Topology - D
Diffeomorphism. Given two differentiable manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both and its inv...   » Read the article

Encyclopedia - Glossary Of Differential Geometry And Topology: Encyclopedia Ii - Glossary Of Differential Geometry And Topology - S
Section Submanifold. A submanifold is the image of a smooth embedding of a manifold. Submersion Surface, a two-dimensional manifold or su...   » Read the article

Encyclopedia - Glossary Of Differential Geometry And Topology: Encyclopedia Ii - Glossary Of Differential Geometry And Topology - P
Parallelizable. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being tr...   » Read the article

Encyclopedia - Glossary Of Differential Geometry And Topology: Encyclopedia Ii - Glossary Of Differential Geometry And Topology - F
Fiber. In a fiber bundle, π: E → B the preimage π−1(x) of a point x in the base B is called the fiber over x, often denoted Ex. Fib...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Set Of All Functions
The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is justified by the f...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Composing Functions
The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then apply...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Examples
Let E = B × F and let π : E → B be the projection onto the first factor. Then E is a fiber bundle over B. Here E is not just loc...   » Read the article

Encyclopedia - List Of Mathematical Functions: Encyclopedia Ii - List Of Mathematical Functions - Special Functions
List of mathematical functions - Antiderivatives of elementary functions. Logarithmic integral function: Integral of the reciprocal of...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Sections
A section (or cross section) of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all x in B. Since bundles do...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions Of More Than One Variable
A straightforward generalization is to allow functions depending on several arguments. For instance, g(x,y) = xy is a function which ta...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - History
As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Images And Preimages
The image of an element x in X under f is the output f(x). The image of a subset A of X under f is the subset of Y formally defined by f...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Domains Codomains And Ranges
X, the set of input values, is called the domain of f, sometimes denoted dom(f), and Y. The distinction between "codomain" and "range" is...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Formal Definition
Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is ...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Structure Groups And Transition Functions
Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization chart...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Graph Of A Function
The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X. There are theorems formulated or proved mo...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Examples Of Functions
(More can be found at list of functions.) The relation wght between persons in the United States and their weights at a particular time....   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Domain Codomain Argument Image
If f is a function from X to Y, the set X is called the domain of f, and Y is called its codomain. Each element of the domain is called a...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions In Other Fields
Functions are used in every quantitative science, to model relationships between all kinds of physical quantities — especially when one...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - History Of The Concept
As a mathematical term, "function" was coined by Gottfried Leibniz in 1694, to describe a quantity related to a curve, such as a curve's ...   » Read the article

Encyclopedia - Quotient Space: Encyclopedia Ii - Quotient Space - Properties
Quotient maps q : X → Y are characterized by the following property: if Z is any topological space and f : Y → Z is any fun...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Graph Of A Function
The graph of a function f is the set of all ordered pairs (x, f(x)), for all x in the domain X. If X and Y are the set of real numbers (o...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Specifying A Function
If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Symbols And Language
The terms function, mapping, map and transformation are usually used synonymously. It is not necessary that we be able to display the exp...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Image Of A Set
One often extends the concept (and notation) of image of an argument to sets of arguments. Namely, if A is any subset of the domain X, th...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Functions With Multiple Inputs And Outputs
Function mathematics - Functions of two or more variables. The concept of function can be extended to an object that takes a combinatio...   » Read the article

Encyclopedia - Bijection Injection And Surjection: Encyclopedia Ii - Bijection Injection And Surjection - Examples
It is important to specify the domain and codomain of each function since by changing these, functions which we think of as the same may ...   » Read the article

Encyclopedia - Sheaf Mathematics: Encyclopedia Ii - Sheaf Mathematics - The Formal Definition
To define sheaves we will proceed in two steps. The first step is to introduce the concept of a presheaf, which captures the idea of asso...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Pointwise Operations
If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can defi...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Composing Functions
The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then apply...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - N-ary Function: Function Of Several Variables
Functions in applications are often functions of several variables, or multivariate functions: the values they take depend on a number of...   » Read the article

Encyclopedia - Function Mathematics: Encyclopedia Ii - Function Mathematics - Properties Of Functions
Functions can be odd or even continuous or discontinuous real or complex scalar or vectorial. Function mathematics - Ambiguous functio...   » Read the article