sets

The word probability derives from the Latin probare (to prove, or to test). Informally, probable is one of several words applied to uncertain events or knowledge, being more or less interchangeable with likely, risky, hazardous, uncertain, and doubtful, depending on the context. Chance, odds, and bet are other words expressing similar notions. As with the theory of mechanics which assigns precise definitions to such everyday terms as work and force< ...

Including:

• Probability - Historical remarks
• Probability - Concepts
• Probability - Formalization of probability
• Probability - Representation and interpretation of probability values
• Probability - Distributions
• Probability - Probability in mathematics
• Probability - Remarks on probability calculations
• Probability - Applications of probability theory to everyday life
• Probability - Quotations

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Cartesian means of or relating to the French philosopher and mathematician René Descartes. See: René Descartes, after whom the term is named Cartesian dualism Cartesian theater Cartesian product, a direct product of two sets Cartesian coordinate system, modern rectangular coordinate system the Cartesian diver experiment Cartesian Meditations, a work by Edmund Husserl Carte

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Until its proof in 1995, the most famous of all conjectures was the mis-named Fermat's last theorem - this conjecture became a true theorem only after its proof. In the process, a special case of the Taniyama-Shimura conjecture, itself a longstanding open problem, was proven; this conjecture has since been completely proven. Other famous conjectures include: There are no odd perfect numbers Goldbach's conjecture The twin prime conjecture The Collatz conjecture The Riemann hypothesis P ≠ NP The Poinca ...

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Conjecture, Conjecture - Famous conjectures, Conjecture - Counterexamples, Conjecture - Use of conjectures in conditional proofs, Conjecture - Undecidable conjectures, Conjecture - Usage outside of mathematics

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A concrete category is formally defined as follows: a category C a faithful functor F : C → Set The faithful functor F is typically thought of as a forgetful functor, which assigns to every object of C its "underlying set", and to every morphism in C the corresponding function. Thus, a concrete category C consists not just of C itself, but of the category C and a corresponding forgetful functor F. In practice, the forgetful functor is usually clear, and we s ...

See also:

Concrete category, Concrete category - Definition, Concrete category - Not all categories are concrete, Concrete category - Alternate definition

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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 elements in B, but not in A. The relative complement of A in B is usually written B − A (also B \ A). Formally: Examples: {1,2,3} − {2,3,4}   =   {1} {2,3,4} −& ...

See also:

Complement set theory, Complement set theory - Relative complement, Complement set theory - Absolute complement

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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Ma ...

Including:

• Category theory - Background
• Category theory - Historical notes
• Category theory - Categories objects and morphisms
• Category theory - Some properties of morphisms
• Category theory - Functors
• Category theory - Natural transformations and isomorphisms
• Category theory - Universal constructions limits and colimits
• Category theory - Equivalent categories
• Category theory - Further concepts and results
• Category theory - Higher-dimensional categories

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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 C we are given a morphism 0XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram: Then the morphisms 0XY are c ...

Including:

• Zero morphism - Examples

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In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms ...

Including:

• Category of topological spaces - Top is a concrete category
• Category of topological spaces - Limits and colimits
• Category of topological spaces - Other properties
• Category of topological spaces - Relationships to other categories

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In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. The study of categories in their own right is known as category theory. For more extensive motivational background and historical notes, see category theory and the list of category theory topics. Category mathematics - Definition. A category C consists of Including:

• Category mathematics - Definition
• Category mathematics - Examples
• Category mathematics - Types of morphisms
• Category mathematics - Types of categories

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In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets). A class that is not a set is called a proper class. A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fr ...

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In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is ("aleph-null") and the cardinality of the real numbers is , the continuum hypothesis says: ...

Including:

• Continuum hypothesis - The size of a set
• Continuum hypothesis - Impossibility of proof and disproof
• Continuum hypothesis - Arguments pro and con
• Continuum hypothesis - The generalized continuum hypothesis

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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 whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes whose formulation of analytic geometry gave rise to this concept. For example, if set X is the 13-element set { A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, ...

Including:

• Cartesian product - Cartesian square and n-ary product
• Cartesian product - Infinite products
• Cartesian product - Cartesian product of functions
• Cartesian product - Category theory

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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 Aardman Animations. All the characters were made from moulded Plasticine modelling clay on wire frames, and filmed with stop motion animation. This process is sometimes known as "claymation". Wallace is an absent-minded inventor, cheese enthusiast (especially for Wensleydale cheese), and companion to the dog Gromit who appears to be rather more intelligent than his master. Wallace is voiced by v ...

Including:

• Wallace and Gromit - Characters
• Wallace and Gromit - Wallace
• Wallace and Gromit - Gromit
• Wallace and Gromit - Studio Fire Incident
• Wallace and Gromit - Films
• Wallace and Gromit - Original 30-minute Shorts
• Wallace and Gromit - Feature Film
• Wallace and Gromit - New Shorts
• Wallace and Gromit - Video Games
• Wallace and Gromit - Stop-motion Technique

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In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. Universe mathematics - In a specific context. There are several precise versions of this general idea. Perhaps the simplest is that any set can be a universe, so long as you are studying that particular set. So if ...

Including:

• Universe mathematics - In a specific context
• Universe mathematics - In ordinary mathematics
• Universe mathematics - In set theory
• Universe mathematics - In category theory

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In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. Once a conjecture has been proven, it becomes known as a theorem, and it joins the realm of known mathematical facts. Until that point in time, mathematicians must be extremely careful about their use of a conjecture within logical structures. Conjecture - Famous conjectures. Until its proof in 1995, the most famous of all conjectures was the m ...

Including:

• Conjecture - Famous conjectures
• Conjecture - Counterexamples
• Conjecture - Use of conjectures in conditional proofs
• Conjecture - Undecidable conjectures
• Conjecture - Usage outside of mathematics

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In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. The prototypical concrete category is Set, the category of sets and functions. Most categories considered in everyday life are concrete; examples are Top, the category of topological spaces and continuous functions, and Grp the category of groups and gro ...

Including:

• Concrete category - Definition
• Concrete category - Not all categories are concrete
• Concrete category - Alternate definition

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Commutative operation - Mathematical meaning. In mathematics, especially abstract algebra, a binary operation on a set S is commutative if for all x and y in S. Otherwise, the operation is noncommutative. Additionally, if for a particular pair of elements x and y, then x and y are said to commute. Every element commutes with itself and, in a group, every elemen ...

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• Commutative operation - Mathematical meaning
• Commutative operation - Neurophysiological meaning

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Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. Although much of his working life was spent in Zürich and then Princeton, he is closely identified with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as pure disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and a key member of the Institute for Advanced Study in its early years, in term ...

Including:

• Hermann Weyl - Early life and interests
• Hermann Weyl - Geometric foundations of manifolds and physics
• Hermann Weyl - Foundations of mathematics
• Hermann Weyl - Mathematics of relativity
• Hermann Weyl - Topological groups Lie groups and representation theory
• Hermann Weyl - Harmonic analysis and analytic number theory
• Hermann Weyl - Later career
• Hermann Weyl - Personality
• Hermann Weyl - Quotes

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In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. 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 elements in B, but not in A. The relative complement of A in B is usually written Including:

• Complement set theory - Relative complement
• Complement set theory - Absolute complement

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In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. Accordingly, the dual notion of a colimit, generalizes disjoint unions and direct sums. Limits and colimits have strong relationships to the categorial concepts of universal morphisms and adjoint functors. Limit category theory - Definition. Before defining limits, it is useful to defin ...

Including:

• Limit category theory - Definition
• Limit category theory - Examples
• Limit category theory - Complete categories
• Limit category theory - Continuous functors
• Limit category theory - Colimits
• Limit category theory - Creation of Limits and Co-Limits

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