In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function is injective (one-to-one) if or, equivalently, if . One could also say that every element of the codomain (sometimes called range) is mapped to by at most one element (argument) of the domain; not every element of t ...

Including:

• Bijection injection and surjection - Injection
• Bijection injection and surjection - Surjection
• Bijection injection and surjection - Bijection
• Bijection injection and surjection - Cardinality
• Bijection injection and surjection - Examples
• Bijection injection and surjection - Injective and surjective bijective
• Bijection injection and surjection - Injective and non-surjective
• Bijection injection and surjection - Non-injective and surjective
• Bijection injection and surjection - Non-injective and non-surjective
• Bijection injection and surjection - Properties
• Bijection injection and surjection - Category theory
• Bijection injection and surjection - History

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A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory, mathematics, and theoretical biology. It consists of an infinite, regular grid of cells, each in one of a finite number of states. The grid can be in any finite number of dimensions. Time is also discrete, and the state of a cell at time t is a function of the states of a finite number of cells (called its neighborhood) at time t-1. These neighbors are a selection of cells relative to the spec ...

Including:

• Cellular automaton - History of cellular automata
• Cellular automaton - The simplest cellular automata
• Cellular automaton - Reversible cellular automata
• Cellular automaton - Totalistic cellular automata
• Cellular automaton - Uses in cryptography
• Cellular automaton - Related automata
• Cellular automaton - Cellular automata in nature
• Cellular automaton - Cellular automata in the chemistry lab
• Cellular automaton - Articles on specific cellular automata

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In linear algebra and related areas of mathematics, the null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero. It is usually written 0 or simply 0. For a general vector space, the null vector is the uniquely determined vector that is the identity element for vector addition. The zero vector is unique; if a and b are zero vectors, then a = a + b = ...

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In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but aren't the same thing. And separable spaces are a completely differe ...

Including:

• Separated sets - Definitions
• Separated sets - Relation to separation axioms and separated spaces
• Separated sets - Relation to connected spaces
• Separated sets - Relation to topologically distinguishable points

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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 open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

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A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x). Because two definitions of binary relation are in use, there are actually two definitions of function, in ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Mathematical definition of a function

Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism iff it's open, or equivalently, iff it's closed. If Y has the discrete topology (i.e. all subsets are open and closed) then every function f : X → Y is both open and closed (but not necessarily continuous). Whenever we have a product of topological spaces X=ΠXi, then the natural projections pi : X → Xi ...

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Open and closed maps, Open and closed maps - Examples, Open and closed maps - Facts and theorems

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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 recursive set. It is called partially decidable, solvable or provable if A is a recursively enumerable set. Otherwise, the problem is called undecidable. We can give an alternati ...

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Decision problem, Decision problem - Definition, Decision problem - Notes, Decision problem - Examples, Decision problem - History

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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 in I, let be the canonical injection. The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e. the final topology ...

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Disjoint union topology, Disjoint union topology - Definition, Disjoint union topology - Properties, Disjoint union topology - Examples, Disjoint union topology - Preservation of topological properties

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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 of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open if and only if their union is open in X. This is the quotient topology on the quotient set X/~. Equivalently, the quotient topology can be characterized in the following manner: Let q : X → X/~ be the projection map which sends each element of X to its equivalence class. Then the quotient topology on X/~ is the fine ...

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Quotient space, Quotient space - Definition, Quotient space - Examples, Quotient space - Properties, Quotient space - Compatibility with other topological notions

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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 σ-algebra over X is given by the full power set of X. If X is uncountable then the family of all E contained in X where E or the complement X-E is countable forms a σ-algebra. If {Σa} is a family of σ-algebras over X, then the intersection of all Σa ...

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Sigma-algebra, Sigma-algebra - Examples

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List of mathematical functions - Polynomials. Polynomials: can be generated by addition and multiplication alone. Constant function: Zero degree polynomial, fixed value regardless of arguments. Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial.See also:

List of mathematical functions, List of mathematical functions - Classes of functions, List of mathematical functions - Elementary functions, List of mathematical functions - Polynomials, List of mathematical functions - Elementary periodic functions, List of mathematical functions - Elementary transcendental functions, List of mathematical functions - Special functions, List of mathematical functions - Antiderivatives of elementary functions, List of mathematical functions - Gamma and related functions, List of mathematical functions - Elliptic and related functions, List of mathematical functions - Bessel and related functions, List of mathematical functions - Riemann zeta and related functions, List of mathematical functions - Hypergeometric and related functions, List of mathematical functions - Other standard special functions, List of mathematical functions - Number theoretic functions, List of mathematical functions - Miscellaneous

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The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler, in the 18th century. But the intuitive idea of a function as any rule or procedure that assigns an output to each given input proved to be naive. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. The concept of a function was not put on a rigorous basis u ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Introduction

The concept of open sets can be formalized in various degrees of generality. Open set - Function-analytic. A point set in Rn is called open when every point P of the set is an inner point. Open set - Euclidean space. A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in < ...

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Open set, Open set - Definitions, Open set - Function-analytic, Open set - Euclidean space, Open set - Metric spaces, Open set - Topological spaces, Open set - Uses, Open set - Manifolds

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Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (that is, depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as op ...

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Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia II - Sheaf mathematics - Introduction

Equalisers can be defined by a universal property, which allows the notion to be generalised from the category of sets to arbitrary categories. In the general context, X and Y are objects, while f and g are morphisms from X to Y. These objects and morphisms form a diagram in the category in question, and the equaliser is simply the limit of that diagram. In more explicit terms, the equaliser consists of an object E and a morphism eq : E → X satisfying ...

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Equaliser, Equaliser - Definitions, Equaliser - Difference kernels, Equaliser - In category theory

Read more here: » Equaliser: Encyclopedia II - Equaliser - In category theory

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 function value at x, or the image of x under f. The image of x can be written as f(x) or as y. Written mathematics sometimes omits the parentheses around the argument, thus: sin x, but calculators and computers require parentheses around the argument. In some branches of mathematics, such as automata theory, th ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - The vocabulary of functions

Proper map - Definition. A morphism f : X → Y of algebraic varieties or schemes is called universally closed if all its fiber products f × Id: X × Z → Y × Z are closed maps of the underlying topological spaces. A morphism f : X → Y of algebraic varieties or is called proper if it is separated and universally closed. A morphism of schemes is called proper if it is separated, of finite type and unive ...

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Proper map, Proper map - Topological spaces, Proper map - Definition, Proper map - Properties, Proper map - Examples, Proper map - Algebraic varieties and schemes, Proper map - Definition, Proper map - Examples, Proper map - Valuative criterion of properness, Proper map - Stein factorization

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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 recursive iff the indicator function 1S is computable. ...

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Recursive set, Recursive set - Definition, Recursive set - Examples, Recursive set - Properties

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A countable set S is called recursively enumerable if there exists a partial computable function such that S is the range of f : rng(f) = S f is called an enumerative function because it associates a rank in the enumeration t ...

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Recursively enumerable set, Recursively enumerable set - Definition, Recursively enumerable set - Remarks, Recursively enumerable set - Examples, Recursively enumerable set - Properties

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