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. The collection of subsets of X which are countable or whose complements are countable is a σ-algebra, which is distinct from the powerset of X iff X is uncountable. If {Σa} is a family of σ-algebras over X, then the intersection of all Σa ...

See also:

Sigma-algebra, Sigma-algebra - Notation, Sigma-algebra - Examples

Read more here: » Sigma-algebra: Encyclopedia II - Sigma-algebra - Examples

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: (0..+∞) → R is surjective. The function g: R → R defined by g(x) = x² is not surjective, because (for e ...

See also:

Surjection, Surjection - Examples and counterexamples, Surjection - Properties, Surjection - Category theory view

Read more here: » Surjection: Encyclopedia II - Surjection - Examples and counterexamples

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 all Borel functions are continuous. Random variables are by definition measurable functions defined on sample spaces. In the category of measurable spaces, the measurable functions are the morphisms. If X=Y and Σ=Τ, a measurable function f is called an endomorphism or a measure-preserving or stationary transformation of the measure space (X,Σ,μ) if and only if the measure μ is i ...

See also:

Measurable function, Measurable function - Special Measurable Functions, Measurable function - Properties of Measurable Functions

Read more here: » Measurable function: Encyclopedia II - Measurable function - Special Measurable Functions

Chart Cobordism Codimension. The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold. Connected sum Connection Cotangent bundle, the vector bundle of cotangent spaces on a manifold. Cotangent space ...

See also:

Glossary of differential geometry and topology, Glossary of differential geometry and topology - A, Glossary of differential geometry and topology - B, Glossary of differential geometry and topology - C, Glossary of differential geometry and topology - D, Glossary of differential geometry and topology - E, Glossary of differential geometry and topology - F, Glossary of differential geometry and topology - G, Glossary of differential geometry and topology - H, Glossary of differential geometry and topology - I, Glossary of differential geometry and topology - L, Glossary of differential geometry and topology - M, Glossary of differential geometry and topology - P, Glossary of differential geometry and topology - S, Glossary of differential geometry and topology - T, Glossary of differential geometry and topology - V, Glossary of differential geometry and topology - W

Read more here: » Glossary of differential geometry and topology: Encyclopedia II - Glossary of differential geometry and topology - C

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 group. Order of an element of a group. Suppose x∈G and there exists a positive integer m such that xm = e, then the smallest possible m is called the order of x. The order of a finite group is divisible by the order of every element. Subgroup. A subset H of a group (G,*) which remains a group when the operation * is r ...

See also:

Glossary of group theory, Glossary of group theory - Basic definitions, Glossary of group theory - Types of groups

Read more here: » Glossary of group theory: Encyclopedia II - Glossary of group theory - Basic definitions

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 vector of W, then the kernel of T is the preimage of the singleton set {0W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0W. The kernel is usually denoted "ker TSee also:

Kernel algebra, Kernel algebra - Survey of examples, Kernel algebra - Linear operators, Kernel algebra - Group homomorphisms, Kernel algebra - Ring homomorphisms, Kernel algebra - Monoid homomorphisms, Kernel algebra - Universal algebra, Kernel algebra - General case, Kernel algebra - Mal'cev algebras

Read more here: » Kernel algebra: Encyclopedia II - Kernel algebra - Survey of examples

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 are disjoint if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory; we include it here because it is the weakest in the sequence of different notions. For more ...

See also:

Separated sets, 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

Read more here: » Separated sets: Encyclopedia II - Separated sets - Definitions

Stanislaw Ulam, while working at the Los Alamos National Laboratory in the 1940s, studied the growth of crystals, using a simple lattice network as his model. At the same time, John von Neumann—Ulam's colleague at Los Alamos—was working on the problem of self-replicating systems. Von Neumann's initial design was founded upon the notion of one robot building another robot. This design is known as the kinematic model. As he developed this design, von Neumann came to realize the great difficulty of building a self-replicating robot, and of ...

See also:

Cellular automaton, 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

Read more here: » Cellular automaton: Encyclopedia II - Cellular automaton - History of cellular automata

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 surjection satisfying a local triviality condition outlined below. B is called the base space of the bundle, E the total space, and F the fiber. The map π is called the projection map. We shall assume in what ...

See also:

Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

Read more here: » Fiber bundle: Encyclopedia II - Fiber bundle - Formal 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 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 ...

See also:

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

Read more here: » Decision problem: Encyclopedia II - Decision problem - Definition

A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-one correspondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows. The function is bijective iff for all , there is a unique such that f(a) = b. A function f : A → B is bijective if and only if it ...

See also:

Bijection injection and surjection, 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

Read more here: » Bijection injection and surjection: Encyclopedia II - Bijection injection and surjection - Bijection

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 have different jectivity. Bijection, injection and surjection - Injective and surjective bijective. For every set A the identity function idA and thus specifically . and thus also its inverse . The exponential function and thus also its inverse the natural logarithm Bijection, injection and surjection - Injective and non- ...

See also:

Bijection, injection and surjection, 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

Read more here: » Bijection, injection and surjection: Encyclopedia II - Bijection, injection and surjection - Examples

A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-one correspondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows. The function is bijective iff for all , there is a unique such that f(a) = b. A function f : A → B is bijective if and only if it ...

See also:

Bijection, injection and surjection, 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

Read more here: » Bijection, injection and surjection: Encyclopedia II - Bijection, injection and surjection - Bijection

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 ...

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 - 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 - Mathematical definition of a function

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

A CA is said to be reversible if for every current configuration of the CA there is exactly one past configuration (preimage). If one thinks of a cellular automaton as a function mapping configurations to configurations, reversibility implies that this function is bijective. For one dimensional CA there are known algorithms for finding preimages, and any 1D rule can be proved either reversible or irreversible. For CA of two or more dimensions it has been proved that the reversibility is undecidable for arbitrary rules. The p ...

See also:

Cellular automaton, 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

Read more here: » Cellular automaton: Encyclopedia II - Cellular automaton - Reversible cellular automata

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. More generally, a function can be defined by any mathematical condition relating the argument to the corresponding value. There are many other ways of defining functio ...

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 - 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 - Specifying a function

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 - 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 - History of the concept

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 in X, there exists some y in Y such that x is related to y. f is single-valued: for each x in X, there is at most one y in Y such that x is related to y. In some contexts, a relation that satisfies condition (1), but not necessarily (2) ...

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 - Partial functions and multi-functions

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 - 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 - Functions in other fields

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), Ker(f,g), or Ker(f - g). The last notation shows where this terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f - g. Conversely, the kernel of a single function f can be reconstructed as the difference kernel Eq(f,0), w ...

See also:

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

Read more here: » Equaliser: Encyclopedia II - Equaliser - Difference kernels