Encyclopedia - Probability: Encyclopedia - Probability
The word probability derives from the Latin probare (to prove, or to test). Informally, probable is one of several words applied to uncer...   » Read the article

Encyclopedia - Cartesianism: Encyclopedia - Cartesianism
Cartesian means of or relating to the French philosopher and mathematician René Descartes. See: René Descartes, after whom the term is...   » Read the article

Encyclopedia - Conjecture: Encyclopedia Ii - Conjecture - Famous Conjectures
Until its proof in 1995, the most famous of all conjectures was the mis-named Fermat's last theorem - this conjecture became a true theor...   » Read the article

Encyclopedia - Concrete Category: Encyclopedia Ii - Concrete Category - Definition
A concrete category is formally defined as follows: a category C a faithful functor F : C → Set The faithful functor F is typica...   » Read the article

Encyclopedia - Complement Set Theory: Encyclopedia Ii - Complement Set Theory - Relative Complement
If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of element...   » Read the article

Encyclopedia - Category Theory: Encyclopedia - Category Theory
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is...   » Read the article

Encyclopedia - Zero Morphism: Encyclopedia - Zero Morphism
In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in ...   » Read the article

Encyclopedia - Category Of Topological Spaces: Encyclopedia - Category Of Topological Spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose mor...   » Read the article

Encyclopedia - Category Mathematics: Encyclopedia - Category Mathematics
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories...   » Read the article

Encyclopedia - Class Set Theory: Encyclopedia - Class Set Theory
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia - Continuum Hypothesis
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept o...   » Read the article

Encyclopedia - Cartesian Product: Encyclopedia - Cartesian Product
In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs w...   » Read the article

Encyclopedia - Wallace And Gromit: Encyclopedia - Wallace And Gromit
Wallace and Gromit are the main characters in a series of three British animated short films and a feature-length film by Nick Park of Aa...   » Read the article

Encyclopedia - Universe Mathematics: Encyclopedia - Universe Mathematics
In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a...   » Read the article

Encyclopedia - Conjecture: Encyclopedia - Conjecture
In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able ...   » Read the article

Encyclopedia - Concrete Category: Encyclopedia - Concrete Category
In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional stru...   » Read the article

Encyclopedia - Commutative Operation: Encyclopedia - Commutative Operation
Commutative operation - Mathematical meaning. In mathematics, especially abstract algebra, a binary operation on a set S is commutativ...   » Read the article

Encyclopedia - Hermann Weyl: Encyclopedia - Hermann Weyl
Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. Although much of his working life was spent in Zürich and...   » Read the article

Encyclopedia - Complement Set Theory: Encyclopedia - Complement Set Theory
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complemen...   » Read the article

Encyclopedia - Limit Category Theory: Encyclopedia - Limit Category Theory
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions ...   » Read the article

Encyclopedia - Bracket: Encyclopedia - Bracket
apostrophe ( ' ) ( ’ ) brackets ( ( ) ) ( [ ] ) ( { } ) ( 〈 〉 ) colon ( : ) comma ( , ) dashes ( ‒ ) ( – ) ...   » Read the article

Encyclopedia - U: Encyclopedia - U
U is the twenty-first letter of the modern Latin alphabet. U was originally a positional variant of the letter V, as J was of I, used onl...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia - Adjoint Functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiqu...   » Read the article

Encyclopedia - Aleph: Encyclopedia - Aleph
Aleph or alef has several meanings: Aleph or Alef, first letter of many Semitic alphabets including Phoenician, Hebrew and Aramaic. ʾal...   » Read the article

Encyclopedia - Algebra Of Sets: Encyclopedia - Algebra Of Sets
The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, an...   » 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 - Information Theory: Encyclopedia - Information Theory
Information theory is the mathematical theory of data communication and storage, generally considered to have been founded in 1948 by Cla...   » Read the article

Encyclopedia - Definition: Encyclopedia - Definition
A definition may be a statement of the essential properties of a certain thing, or a statement of equivalence between one expression and ...   » Read the article

Encyclopedia - Distance: Encyclopedia - Distance
The distance between two points is the length of a straight line segment between them. In the case of two locations on Earth, usually the...   » Read the article

Encyclopedia - Infinity: Encyclopedia - Infinity
Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or c...   » Read the article

Encyclopedia - Algebraic Structure: Encyclopedia - Algebraic Structure
In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in t...   » Read the article

Encyclopedia - Areas Of Mathematics: Encyclopedia - Areas Of Mathematics
The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of t...   » Read the article

Encyclopedia - Union Set Theory: Encyclopedia - Union Set Theory
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to an...   » Read the article

Encyclopedia - Cardinal Number: Encyclopedia - Cardinal Number
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal nu...   » Read the article

Encyclopedia - Boolean Logic: Encyclopedia - Boolean Logic
Boolean logic, is a complete system for logical operations. It was named after George Boole, an English mathematician at University Colle...   » Read the article

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 - Associativity: Encyclopedia - Associativity
In mathematics, associativity is a property that a binary operation can have. It means that the order of evaluation is immaterial if the ...   » Read the article

Encyclopedia - Symmetry: Encyclopedia - Symmetry
Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to...   » Read the article

Encyclopedia - Beth Number: Encyclopedia - Beth Number
In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). The s...   » Read the article

Encyclopedia - Well-behaved: Encyclopedia - Well-behaved
Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object — a number, a function, a set, a ...   » Read the article

Encyclopedia - Intersection: Encyclopedia - Intersection
Intersection has various meanings in different contexts: In road transport, an intersection is a place where two roads meet. If there is...   » Read the article

Encyclopedia - ∈-induction: Encyclopedia - ∈-induction
In mathematics, ∈-induction is a variant of transfinite induction, which can be used in set theory to prove that all sets satisfy a giv...   » Read the article

Encyclopedia - Shattering: Encyclopedia - Shattering
The concept of shattering of a set of points plays an important role in Vapnik Chervonenkis theory, also known as VC-theory. Shattering a...   » Read the article

Encyclopedia - Continuum: Encyclopedia - Continuum
Continuum can refer to: The continuous black body radiation spectrum. continuum, a "fretless keyboard" continuum, or mathematical sets (...   » Read the article

Encyclopedia - Group Object: Encyclopedia Ii - Group Object - Definition
Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A gro...   » Read the article

Encyclopedia - Equivalence Relation: Encyclopedia Ii - Equivalence Relation - Generating Equivalence Relations
If two equivalence relations over the set X are given, then their intersection (viewed as subsets of X×X) is also an equivalence relatio...   » Read the article

Encyclopedia - Distributivity: Encyclopedia Ii - Distributivity - Examples
Distributivity is most commonly found in rings and distributive lattices. A ring has two binary operations (commonly called "+" and "*"),...   » Read the article

Encyclopedia - Groupoid: Encyclopedia Ii - Groupoid - Definitions
From one point of view, a groupoid is simply a category in which every morphism is an isomorphism (that is, invertible). To be explicit, ...   » Read the article

Encyclopedia - Distance Geometry: Encyclopedia Ii - Distance Geometry - Introduction
A straight line is the shortest path between two points. Therefore the distance from A to B is no bigger than the length of the straight-...   » Read the article

Encyclopedia - Vacuous Truth: Encyclopedia Ii - Vacuous Truth - Arguments Of The Semantic Truth Of Vacuously True Logical Statements
This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i...   » Read the article

Encyclopedia - Filming Location: Encyclopedia Ii - Filming Location - Pros And Cons
Location filming has several advantages over filming on a studio set: It can be cheaper than constructing large sets The illusion of rea...   » Read the article

Encyclopedia - Forbidden Planet: Encyclopedia Ii - Forbidden Planet - Plot
In 2257, the United Planets Cruiser C-57D is sent to planet Altair IV, of the star-system Alpha Aquilae, to search for survivors of the i...   » Read the article

Encyclopedia - Probability: Encyclopedia Ii - Probability - Historical Remarks
The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of p...   » Read the article

Encyclopedia - Georg Cantor: Encyclopedia Ii - Georg Cantor - Biography
Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm. ...   » Read the article

Encyclopedia - Goldeneye 007: Encyclopedia Ii - Goldeneye 007 - Storyline And Missions
Mission sequence Mission 1: Arkhangelsk Dam: Byelomorye Dam Facility: Arkhangelsk Runway: Runway Mission 2: Servernaya Surface: Se...   » Read the article

Encyclopedia - Disjoint Sets: Encyclopedia Ii - Disjoint Sets - Explanation
Formally, two sets A and B are disjoint if their intersection is the empty set, i.e. if This definition extends to any collection of s...   » Read the article

Encyclopedia - Hash Table: Encyclopedia Ii - Hash Table - Collision Resolution
If two keys hash to the same index, the corresponding records cannot be stored in the same location. So, if it's already occupied, we mus...   » Read the article

Encyclopedia - Hausdorff Dimension: Encyclopedia Ii - Hausdorff Dimension - Informal Discussion
Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe...   » Read the article

Encyclopedia - Relation Mathematics: Encyclopedia Ii - Relation Mathematics - Definition
A relation over the sets X1, ..., Xn is an (n + 1)-tuple R=(X1, ..., Xn, G(R)) where G(R) is a subset of X1 × ... × Xn (the Cartesian p...   » Read the article

Encyclopedia - The Movies: Encyclopedia Ii - The Movies - The Movies
There is no limit on movie length. The average movie length is between 30 seconds and 3 minutes, but user created films can be as long as...   » 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 - Infinity: Encyclopedia Ii - Infinity - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which sta...   » Read the article

Encyclopedia - Initial Object: Encyclopedia Ii - Initial Object - Properties
Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an...   » Read the article

Encyclopedia - Internal Set Theory: Encyclopedia Ii - Internal Set Theory - Formal Axioms For Ist
There are three axioms of IST to add to the established ZFC set theoretic axioms (note that use of the ZFC axiom schemas is restricted: t...   » Read the article

Encyclopedia - Java Programming Language: Encyclopedia Ii - Java Programming Language - History
Java programming language - Early history. The Java platform and language began as an internal project at Sun Microsystems in December ...   » Read the article

Encyclopedia - Inverse Limit: Encyclopedia Ii - Inverse Limit - Formal Definition
Inverse limit - Algebraic objects. We start with the definition of an inverse system of groups and homomorphisms. Let (I, ≤) be a dir...   » Read the article

Encyclopedia - Equivariant: Encyclopedia Ii - Equivariant - Intertwiners
A completely analogous definition holds for the case of linear representations of G. Specifically, if X and Y are two linear representati...   » Read the article

Encyclopedia - Category Theory: Encyclopedia Ii - Category Theory - Background
The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. Consider t...   » Read the article

Encyclopedia - Recursion: Encyclopedia Ii - Recursion - Recursion In Computing
Recursion in computer programming defines a function in terms of itself. One example application of recursion is in parsers for programmi...   » Read the article

Encyclopedia - Direct Limit: Encyclopedia Ii - Direct Limit - Formal Definition
Direct limit - Algebraic objects. In this section we will understand objects to be sets with a given algebraic structure such as groups...   » Read the article

Encyclopedia - Graph Mathematics: Encyclopedia Ii - Graph Mathematics - Definitions
Definitions in graph theory vary in the literature. Here are the conventions used in this encyclopedia. Graph mathematics - Undirected g...   » 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 - Ham Sandwich Theorem: Encyclopedia Ii - Ham Sandwich Theorem - Reduction To The Borsuk-ulam Theorem
The ham sandwich theorem can be proved as follows using the Borsuk-Ulam theorem. This proof follows the one described by Steinhaus and ot...   » Read the article

Encyclopedia - Areas Of Mathematics: Encyclopedia Ii - Areas Of Mathematics - Algebra
The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which ar...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Informal Discussion
Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting ...   » Read the article

Encyclopedia - Hom Functor: Encyclopedia Ii - Hom Functor - Formal Definition
Let C be a locally small category (i.e. a category for which Hom-classes are actually sets and not proper classes). For all objects A in ...   » Read the article

Encyclopedia - Georg Cantor: Encyclopedia Ii - Georg Cantor - Biography
Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a musician of German descent, Maria An...   » Read the article

Encyclopedia - Functional Predicate: Encyclopedia Ii - Functional Predicate - Introducing New Function Symbols
In a treatment of predicate logic that allows one to introduce new predicate symbols, one will also want to be able to introduce new func...   » 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 - Philosophy Of Language: Encyclopedia Ii - Philosophy Of Language - Major Problems And Sub-fields
Philosophy of language - Composition and parts. A major question in the field - perhaps the single most important question for formalis...   » Read the article

Encyclopedia - Bracket: Encyclopedia Ii - Bracket - In Writing
Brackets are punctuation marks, used in pairs to set apart or interject text within other text. Types of brackets include parentheses...   » Read the article

Encyclopedia - Infinity: Encyclopedia Ii - Infinity - History
Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800...   » Read the article

Encyclopedia - Cantor–bernstein–schroeder Theorem: Encyclopedia Ii - Cantor–bernstein–schroeder Theorem - Visualization
The definition of h can be visualized with the following diagram. Displayed are parts of the (disjoint) sets A and B together with parts...   » Read the article

Encyclopedia - Cartesian Closed Category: Encyclopedia Ii - Cartesian Closed Category - Examples
Examples of cartesian closed categories include: The category Set of all sets, with functions as morphisms, is cartesian closed. The pro...   » Read the article

Encyclopedia - Tennis: Encyclopedia Ii - Tennis - Manner Of Play
Tennis - The court. Main article: Tennis court Tennis is played on a rectangular flat surface, usually of grass, clay, or concrete (h...   » 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 - Relation Mathematics: Encyclopedia Ii - Relation Mathematics - Formal Definitions
There are two definitions of k-place relations that are commonly encountered in mathematics. In order of simplicity, the first of these d...   » Read the article

Encyclopedia - Ontological Commitment: Encyclopedia Ii - Ontological Commitment - Background
The sentence “Napoleon is one of my ancestors” asserts only the existence of two individuals (i.e., Napoleon and the speaker) and a l...   » Read the article

Encyclopedia - Naive Set Theory: Encyclopedia Ii - Naive Set Theory - Unions Intersections And Relative Complements
Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of bot...   » Read the article

Encyclopedia - Monotonic Function: Encyclopedia Ii - Monotonic Function - Monotonicity In Calculus And Analysis
In calculus, there is often no need to call upon the abstract methods of order theory. As already noted, functions are usually mappings b...   » Read the article

Encyclopedia - Order Theory: Encyclopedia Ii - Order Theory - Introduction To The Basic Definitions
This section aims at giving a first guide to the realm of ordered sets. It addresses readers who have basic knowledge of set theory and a...   » Read the article

Encyclopedia - Bijective Proof: Encyclopedia Ii - Bijective Proof - Example
For instance, consider the number of ways in which a committee can be formed from a total of n people: Set B: Each committee K can be rep...   » Read the article

Encyclopedia - Bijection: Encyclopedia Ii - Bijection - Composition And Inverses
A function f is bijective if and only if its inverse relation f-1 is a function. In that case, f-1 is a bijection. The composition (mathe...   » Read the article

Encyclopedia - Interval Mathematics: Encyclopedia Ii - Interval Mathematics - Higher Mathematics
In higher mathematics, a formal definition is the following: An interval is a subset S of a totally ordered set T with the property that ...   » Read the article

Encyclopedia - Constructive Solid Geometry: Encyclopedia Ii - Constructive Solid Geometry - Operations
In modeling packages, basic geometric objects such as the cube or 'box', sphere or ellipse, torus, and a number of other shapes that can ...   » 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 - Equivariant: Encyclopedia Ii - Equivariant - Intertwiners
A completely analogous definition holds for the case of linear representations of G. Specifically, if X and Y are two linear representati...   » Read the article

Encyclopedia - Equation Solving: Encyclopedia Ii - Equation Solving - Methods Of Solution
In simple cases, it is rather easy to solve an equation provided certain conditions are met. However, in more complicated cases, exact sy...   » Read the article

Encyclopedia - Information Theory: Encyclopedia Ii - Information Theory - Basic Concepts For Discrete Channels
Information theory - Self-information. Shannon defined a measure of information content called the self-information or surprisal of a m...   » Read the article

Encyclopedia - Homomorphism: Encyclopedia Ii - Homomorphism - Homomorphism For Beginners
Homomorphism is one of the fundamental concepts in abstract algebra. Because abstract algebra studies sets with operations that generate ...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia Ii - Continuum Hypothesis - The Size Of A Set
To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if ther...   » Read the article