sets

Predicate grammar - Carlson classes. After the work of Greg Carlson, predicates have been divided into the following sub-classes, which roughly pertain to how a predicate relates to its subject: A stage-level predicate ("s-l predicate" for short) is true of a temporal stage of its subject. For example, if John is "hungry", that typically lasts a certain amount of time, and not his entire lifespan. S-l predicates can occur in a wide range of grammatical constructions an ...

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Predicate grammar, Predicate grammar - Predicate in traditional grammar, Predicate grammar - Secondary Predicates, Predicate grammar - Classes of predicates, Predicate grammar - Carlson classes, Predicate grammar - Collective vs. distributive predicates, Predicate grammar - Vendler classes, Predicate grammar - Predicate in logic and model-theoretic semantics

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

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

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PROPOSITION 1: The empty set is a subset of every set. Proof: Given any set A, we wish to prove that ø is a subset of A. This involves showing that all elements of ø are elements of A. But there are no elements of ø. For the experienced mathematician, the inference " ø has no elements, so all elements of ø are elements of A" is immediate, but it may be more troublesome for the beginner. Since ø has no members at all, how can "they" be members of anything else? It may help to think of it ...

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Subset, Subset - Notational variations, Subset - Examples, Subset - Properties, Subset - Other properties of inclusion

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Weyl was born in Elmshorn (a town near Hamburg), Germany. From 1904 to 1908 he studied in Göttingen and Munich, mainly mathematics and physics. His doctorate was awarded at Göttingen under the direction of Hilbert and Minkowski. In 1910, he obtained a teaching post of private lecturer at Göttingen. He took a professorship at the Technische Hochschule in Zürich, Switzerland in 1913, where he remained until 1949. ...

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Hermann Weyl, 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 2002, the station changed logos, and changed studio locations to a studio in downtown Sarasota. Until March 2004, the station's news department operated under the brand "News 40." The brand was changed to ABC 7, because most viewers watch the station on Brighthouse Networks or Comcast Cable, and both providers carry WWSB on channel 7. The conversion to ABC 7 involved a redress of the station's news sets, a new logo (to an unique version of the well known Circle 7 logo) and color scheme, a new slogan ("Local news. Every day. Every newscast."), and a major local promotion and advertising campaign. Logo until 2002 2002-M ...

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WWSB-TV, WWSB-TV - Changes, WWSB-TV - Current Personalities, WWSB-TV - Past Personalities

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The traditional approach to multi-threaded programming is to use locks to synchronize access to shared resources. Synchronization primitives such as mutexes, semaphores, and critical sections are all mechanisms by which a programmer can ensure that certain sections of code do not execute concurrently if doing so would corrupt shared memory structures. If one thread attempts to acquire a lock that is already held by an ...

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Non-blocking synchronization, Non-blocking synchronization - Motivation, Non-blocking synchronization - Implementation, Non-blocking synchronization - Wait-freedom, Non-blocking synchronization - Lock-freedom, Non-blocking synchronization - Obstruction-freedom, Non-blocking synchronization - Resources

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The above definition is usually all that's needed for the most common mathematical applications. However, it is possible to define the Cartesian product over an arbitrary (possibly infinite) collection of sets. If I is any index set, and {X i | i in I} is a collection of sets indexed by I, then we define that is, the set of all functions defined on the index set such that the value of the function at a particular index i is an element of Xi . For each i in I, the function ...

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Cartesian product, 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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The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, φX) of a functor F : J → C. Terminal objects. If J is the empty category, then the above definitions imply that every object of C is a cone of F. The limit of F is any object that has a unique factorization through any other object. This is just the definition of a terminal object.See also:

Limit category theory, 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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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 element commutes ...

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Commutative operation, Commutative operation - Mathematical meaning, Commutative operation - Neurophysiological meaning

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There are two definitions of k-place relations that are commonly encountered in mathematics. In order of simplicity, the first of these definitions is as follows: Definition 1. A relation L over the sets X1, …, Xk is a subset of their cartesian product, written L ⊆ X1 × … × Xk. Under this definition, then, ...

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Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography

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The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetric to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of GSee also:

Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry

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In informal use, a cardinal number is what is normally referred to as a counting number. They may be identified with the natural numbers beginning with 0 (i.e. 0, 1, 2, ...). The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic. More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is e ...

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Cardinal number, Cardinal number - History, Cardinal number - Motivation, Cardinal number - Formal definition, Cardinal number - Cardinal arithmetic, Cardinal number - The continuum hypothesis

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A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b where a and b are in ob(C). We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) (or homC(a, b)) to denote the hom-class of all morphisms from a to b. (Some a ...

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

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The most common choice is to take "small" to mean finite. In this case, two sets are almost disjoint if their intersection is finite, i.e. if (Here, '|X|' denotes the cardinality of X, and '< ∞' means 'finite'.) For example, the closed intervals [0, 1] and [1, 2] are almost disjoint, because their intersection is the finite set {1}. However, the unit interval [0, 1] and the set of rational numbers Q are not almos ...

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Almost disjoint sets, Almost disjoint sets - Definition, Almost disjoint sets - Other meanings

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Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor U : Top → Set to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function. ...

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Category of topological spaces, 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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Principal photography for the film began on June 29, 2005 at the Fitzgerald Theater in Saint Paul, Minnesota (the usual venue for the radio show). Filming ended on July 28, 2005. Because the Fitzgerald is a rather small building, other stage theaters in the Minneapolis-St. Paul region had been considered as stand-ins. With some effort, the necessary film equipment was crammed into the structure. The basement is also being used for sets due to lack of space. Set design also had to make the show more visually interesting, and fake dress ...

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A Prairie Home Companion film, A Prairie Home Companion film - Production notes, A Prairie Home Companion film - Trivia

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The intuitive meaning of a sequent such as the one given above is that under the assumption of Γ the conclusion of Σ is provable. In a classical setting, the formulae on the left of the turnstile are interpreted conjunctively while the formulae on the right are considered as a disjunction. This means that, when all formulae in Γ hold, then at least one formula in Σ also has to be true. If the succedent is empty, this is interpreted as falsity, i.e. means that Γ proves falsity and is thus inconsistent. On the other hand an empty anteced ...

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Sequent, Sequent - Explanation, Sequent - Intuitive meaning, Sequent - Example, Sequent - Property, Sequent - Rules, Sequent - Variations, Sequent - History

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According to Yoneda's lemma, natural transformations from Hom(A,–) to F are in one-to-one correspondence with the elements of F(A). Given a natural transformation Φ : Hom(A,–) → F the corresponding element of u ∈ F(A) is given by Conversely, given any element u ∈ F(A) we may define a natural transformation Φ
...

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Representable functor, Representable functor - Definition, Representable functor - Universal elements, Representable functor - Uniqueness, Representable functor - Examples, Representable functor - Relation to universal morphisms and adjoints

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Boolean logic - Digital electronic circuit design. Boolean logic is also used for circuit design in electrical engineering; here 0 and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if, and only if, the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior ca ...

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Boolean logic, Boolean logic - Terms, Boolean logic - Example, Boolean logic - Chaining operations together, Boolean logic - Use of parentheses, Boolean logic - Properties, Boolean logic - Truth tables, Boolean logic - Other notation, Boolean logic - Basic mathematics use of Boolean terms, Boolean logic - English language use of Boolean terms, Boolean logic - Applications, Boolean logic - Digital electronic circuit design, Boolean logic - Database applications, Boolean logic - Search engine queries

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Brackets are punctuation marks, used in pairs to set apart or interject text within other text. Types of brackets include parentheses ( ) (the singular is parenthesis), box brackets or square brackets [ ], curly brackets or braces { }, and angle brackets 〈 〉. All these forms may be used according to typographical conventions that may vary from publication to publication and may vary even more from language to language. Some typical uses in English texts follow. Brack ...

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Bracket, Bracket - In writing, Bracket - Types of brackets, Bracket - In computing, Bracket - Layout rules, Bracket - In mathematics, Bracket - In sports, Bracket - In mechanics and structures, Bracket - In sociology, Bracket - Reference

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Let be a cumulative probability distribution on the states of nature. From a Bayesian point of view, we would regard it as a prior distribution, that is, it is our believed probability distribution on the states of nature, prior to observing data. For a frequentist, it is merely a function on with no such special interpretation. The Bayes risk of the decision rule with respect to is the expectation If the Bayes risk is finite, we can minimize with respect to to obtain , a Bayes rule with respect to . There may be more than one Bayes rule. If th ...

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Admissible decision rule, Admissible decision rule - Bayes rules, Admissible decision rule - Admissible rules and Bayes rules

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In universal algebra, one studies algebraic structures consisting of a set and a collection of operations defined on the set which are required to satisfy certain identities. Simple structures Set: a set is a degenerate algebraic structure, one that has zero operations defined on it Pointed set: a set S with a distinguished element s of S Unary system: a set S with a unary operation, i.e. a function S → S Pointed unary system: a unary system with a distinguish ...

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Algebraic structure, Algebraic structure - In the sense of universal algebra, Algebraic structure - Allowing axioms other than identities, Algebraic structure - Allowing additional structure, Algebraic structure - Categories

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