morphism

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, the image of A under f is the subset of Y defined f(A) = {f(x) | x is in A} So, for example, the image of {-3,2,3} under the squaring function sqr is sqr({-3,2, 3}) = {4, 9}. This extension is consistent as long as no subset of the domain is also an element of the domain. A ...

See also:

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 - Image of a set

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 fact that |YX| = |Y||X|. See the article on cardinal numbers for more details. It is traditional to write f: X → Y to mean f ∈ [X → Y]; that is, "f is a function from X to Y". This statement is sometimes read "f ...

See also:

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 - Set of all functions

The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is ...

See also:

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 - Composing functions

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(x). More commonly, a function is defined by a formula, or more generally an algorithm — that is, a recipe that tells how to compute the value of f(x) given any x in the domain. See the squaring function sqr above. More generally, a function can also be defined by any mathematical condition relating the argument to the corresponding val ...

See also:

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 - Specifying a function

Function mathematics - Functions of two or more variables. The concept of function can be extended to an object that takes a combination of two (or more) argument values to a single result. This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets. For example, consider the multiplication function that associates two integers to their product: f(x, y) = x·y. This function can be defined formally as having domain Z ...

See also:

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 - Functions with multiple inputs and outputs

The set of automorphisms of an object X form a group under composition of morphisms. This group is called the automorphism group of X. That this is indeed a group is simple to see: Closure: composition of two endomorphisms is another endomorphism. Associativity: morphism composition is associative by definition. Identity: the identity is the identity morphism from an object to itself which exists by definition. Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphi ...

See also:

Automorphism, Automorphism - Definition, Automorphism - Automorphism group, Automorphism - Examples, Automorphism - Inner and outer automorphisms, Automorphism - Reference

Read more here: » Automorphism: Encyclopedia II - Automorphism - Automorphism group

Most Marxists missed the fact that Marx identified (though often not very clearly) three main types of production prices: the private or enterprise production price. This price equals the cost-price and profit applying to the new output of a specific enterprise when this output is sold by the enterprise. the sectoral production price. This price equals the average cost-price and average profit applying to the output of a specific sector or branch of production (at "producer's market price ...

See also:

Prices of production, Prices of production - Basic definition, Prices of production - Two interpretations of production prices, Prices of production - Three types of production prices, Prices of production - Production prices and the transformation problem, Prices of production - Value and price, Prices of production - Facts and logic

Read more here: » Prices of production: Encyclopedia II - Prices of production - Three types of production prices

Unfortunately, Marx never prepared the manuscript of the third volume of Das Kapital for publication. Therefore his draft text, which sketches complicated issues in a "shorthand" way, is sometimes ambiguous and incomplete. Some writers argue that Marx's production price is similar, or performs the same theoretical function, as the "natural prices" of classical political economy found e.g. in the writings of Adam Smith and David Ricardo. In that case, Marx's production price would be essentially a "centre of gravity" around whic ...

See also:

Prices of production, Prices of production - Basic definition, Prices of production - Two interpretations of production prices, Prices of production - Three types of production prices, Prices of production - Production prices and the transformation problem, Prices of production - Value and price, Prices of production - Facts and logic

Read more here: » Prices of production: Encyclopedia II - Prices of production - Two interpretations of production prices

The concept of production prices is one "building block" in Marx's theory of the "equalising tendency of the rate of profit", which aimed to tackle a problem left unsolved by David Ricardo. This problem concerned the question of explaining how an average or "normal" return on capital invested (e.g. 8-12%) could become established, so that capitals of equal size reaped equal profits, even although the enterprises diff ...

See also:

Prices of production, Prices of production - Basic definition, Prices of production - Two interpretations of production prices, Prices of production - Three types of production prices, Prices of production - Production prices and the transformation problem, Prices of production - Value and price, Prices of production - Facts and logic

Read more here: » Prices of production: Encyclopedia II - Prices of production - Production prices and the transformation problem

A lot of criticism of Marx's concept originates from the ambiguities referred to earlier. Consequently, many of the criticisms can be dispelled simply by a more exact definition of the cost, product and revenue aggregates used, and of the timing of transactions. In doing so, it must be admitted though that Marx's draft manuscript often shows sloppy use of terminology and concepts, and that the Marx's purpose was often not fully explicit. At a high level of abstraction, he moves very easily and cavalierly from values to prices and back again, and restricts his discussion of "capital invested" to ...

See also:

Prices of production, Prices of production - Basic definition, Prices of production - Two interpretations of production prices, Prices of production - Three types of production prices, Prices of production - Production prices and the transformation problem, Prices of production - Value and price, Prices of production - Facts and logic

Read more here: » Prices of production: Encyclopedia II - Prices of production - Value and price

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 slope at a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the ...

See also:

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 - History of the concept

Proper map - Definition. A function f : X → Y between two topological spaces is proper if and only if the preimage of every compact set in Y is compact in X. An equivalent, possibly more intuitive definition is as follows: We say an infinite sequence of points {pi} in a topological space X escapes to infinity if, for every compact set S ⊂ X, only finitely many points pi are in See also:

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

Read more here: » Proper map: Encyclopedia II - Proper map - Topological spaces

For most political economists, this kind of price corresponds roughly to Adam Smith's concept of "natural prices" and the modern neoclassical concept of long-term competitive equilibrium prices under constant returns to scale. However, the function of prices of production within Marxian theory is different from that of these other concepts in their own theories. Simply put, for a single commodity, Marx's "price of production" (P) is a price which applies to sales of new output produced, and it equals cost price (< ...

See also:

Prices of production, Prices of production - Basic definition, Prices of production - Two interpretations of production prices, Prices of production - Three types of production prices, Prices of production - Production prices and the transformation problem, Prices of production - Value and price, Prices of production - Facts and logic

Read more here: » Prices of production: Encyclopedia II - Prices of production - Basic definition

Let U : D → C be a functor from a category D to a category C, and let X be an object of C. A universal morphism from X to U consists of a pair (A, φ) where A is an object of D and φ : X → U(A) is a morphism in C, such that the following universal property is satisfied: Whenever Y is an object of D and f : X → U(Y) is a morphism in C, then there exists a unique morphism g : A → < ...

See also:

Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

Read more here: » Universal property: Encyclopedia II - Universal property - Formal definition

Functions are used in every quantitative science, to model relationships between all kinds of physical quantities — especially when one quantity is completely determined by another quantity. Thus, for example, one may use a function to describe how the temperature of water affects its density. Functions are also used in computer science to model data structures and the effects of algorithms. However, the word is also used in computing in the very different sense of pro ...

See also:

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 - Functions in other fields

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 an 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 by (or under) the function. The value of a function f at an argument x is traditionally written f(xSee also:

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 - Domain codomain argument image

In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two classes. In the case of groups: The inner automorphisms are the conjugations by the elements of the group itself. For each element a of a group G, conjugation by a is the operation φa : G → G given by φa(g) = aga−1. One can easily check that conjugation by a is actually a group automorphism. They form ...

See also:

Automorphism, Automorphism - Definition, Automorphism - Automorphism group, Automorphism - Examples, Automorphism - Inner and outer automorphisms, Automorphism - Reference

Read more here: » Automorphism: Encyclopedia II - Automorphism - Inner and outer automorphisms

We give a few worked examples to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction. Universal property - Tensor algebras. Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K (assumed to be unital and associative). Let U be the forgetful functor which assig ...

See also:

Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

Read more here: » Universal property: Encyclopedia II - Universal property - Examples

Function mathematics - Injective surjective bijective. Three important properties that a function may have are: injective (or one-to-one, or an injection) if it associates different arguments to different values; i.e., if f(a) = f(b) implies a = b, for any arguments a and b; surjective (or onto, or a surjection) if its range is equal to its codomain; in other words, if for every y in the ...

See also:

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 - Classes of functions

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 (or subsets thereof), then this definition coincides with the familiar sense of "graph" as a picture or plot of the function, with the ordered pairs being the Cartesian coordinates of the plot's points There are theorems formulated or proved most eas ...

See also:

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 - Graph of a function

In cartesian closed categories, a "function of two variables" (a morphism f:X×Y → Z) can always be represented as a "function of one variable" (the morphism λf:X → ZY). In computer science applications, this is known as currying; it has led to the realization that simply-typed lambda calculus can be interpreted in any cartesian closed category. Certain cartesian closed categories, the topoi, have been proposed as a general s ...

See 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 - Applications

An associative unitary algebra over K is based on a morphism A×A→A having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K→A identifying the scalar multiples of the multiplicative identity. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra. < ...

See also:

Associative algebra, Associative algebra - Definition, Associative algebra - Examples, Associative algebra - Algebra homomorphisms, Associative algebra - Index-free notation, Associative algebra - Generalizations, Associative algebra - Coalgebras, Associative algebra - Representations, Associative algebra - Motivation for a Hopf algebra, Associative algebra - Motivation for a Lie algebra

Read more here: » Associative algebra: Encyclopedia II - Associative algebra - Coalgebras