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sets | A Wisdom Archive on sets | | sets A selection of articles related to sets | |
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sets
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| ARTICLES RELATED TO sets | | | |  sets: Encyclopedia II - Function mathematics - Mathematical definition of a functionA 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 - 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 - Mathematical definition of a function |
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| | | | | sets: 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 definitions is as follows:
Definition 1. A relation L over the sets X1, …, Xk is a subset of the cartesian product of those sets, written L ⊆ X1 × … × Xk. Under this definition, then, ...
See also:Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography Read more here: » Relation mathematics: Encyclopedia II - Relation mathematics - Formal definitions |
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| | | | | | sets: 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 both (see axiom of union). It is denoted by A ∪ B.
The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B.
Finally, the relative complement of B relative to A, also known as the set theoretic differenc ...
See also:Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Unions intersections and relative complements |
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| | | | | sets: 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 between (subsets of) real numbers, ordered in the natural way.
Inspired by the shape of the graph of a monotone function on the reals, such functions are also called monotonically increasing (or "non-decreasing" or, less precisely, just "increasing"). Likewise, a function is called monotonically decreasing (or "non-increasing" or "decreasing") if, whenever x ≤ y, then f< ...
See also:Monotonic function, Monotonic function - General definition, Monotonic function - Monotonicity in calculus and analysis, Monotonic function - Some basic applications and results, Monotonic function - Monotonicity in order theory, Monotonic function - Monotonic logic Read more here: » Monotonic function: Encyclopedia II - Monotonic function - Monotonicity in calculus and analysis |
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| | | | | sets: 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 arithmetics and who know what a binary relation is, but who are not familiar with order theoretic considerations so far.
Order theory - Partially ordered sets.
As already hinted at above, orders are special binary relations. Hence consider some set P and a relation ≤ on P. Then ≤ is a partial order if it is reflexive, antisymmetric, and transitive, ...
See also:Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature Read more here: » Order theory: Encyclopedia II - Order theory - Introduction to the basic definitions |
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| | | | | | sets: 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 (mathematics) gf of two bijections f XY and g YZ is a bijection. The inverse of gf is (gf)-1 = (f-1)(g-1).
On the other hand, if the composition g o f of two functions is bijective, we can only say ...
See also:Bijection, Bijection - Composition and inverses, Bijection - Bijections and cardinality, Bijection - Examples and counterexamples, Bijection - Properties, Bijection - Bijections and category theory, Bijection - Properties, Bijection - Category theory Read more here: » Bijection: Encyclopedia II - Bijection - Composition and inverses |
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| | | | | | | sets: 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 function value at x, or the image of x under f. The image of x can be written as f(x) or as y. Written mathematics sometimes omits the parentheses around the argument, thus: sin x, but calculators and computers require parentheses around the argument. In some branches of mathematics, such as automata theory, th ...
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 - The vocabulary of functions |
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| | | | | | sets: 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 symbolic forms for solutions are often difficult to obtain or cumbersome to manipulate with, and an approximate numerical solution may be in fact sufficient for use.
Equation solving - Inverse functions.
In the simple case of a function of one variable, say, h(x), we can solve an equation of the form
h(x)=c, c constant
by considering w ...
See also:Equation solving, Equation solving - Solution sets, Equation solving - Methods of solution, Equation solving - Inverse functions, Equation solving - Numerical methods, Equation solving - Solving other equations Read more here: » Equation solving: Encyclopedia II - Equation solving - Methods of solution |
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| | | | | sets: 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 message m:
where p(m) = Pr(M = m) is the probability of the message m taken from an implicit message space M.
Note: Strictly speaking, p(m) is the probability density func ...
See also:Information theory, Information theory - Mathematical preliminaries, Information theory - Basic Concepts for Discrete Channels, Information theory - Self-information, Information theory - Entropy, Information theory - Joint Entropy, Information theory - Conditional Entropy Equivocation, Information theory - Mutual Information Transinformation, Information theory - Continuous Channels, Information theory - Channel Capacity, Information theory - Source Theory, Information theory - Rate, Information theory - Fundamental Theorem, Information theory - Statement Noisy-Channel Coding Theorem, Information theory - Channel capacity of particular model channels, Information theory - Related Concepts, Information theory - Measure Theory, Information theory - Kolmogorov Complexity, Information theory - Applications, Information theory - Coding Theory, Information theory - Cryptography Cryptanalysis, Information theory - Relation with thermodynamic entropy, Information theory - Quantum Information Theory, Information theory - Detection and Estimation Theory, Information theory - Gambling, Information theory - Intelligence, Information theory - Music, Information theory - History Read more here: » Information theory: Encyclopedia II - Information theory - Basic Concepts for Discrete Channels |
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| | | | | | sets: 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 there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}.
With infinite sets such as the set of intege ...
See also:Continuum hypothesis, 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 Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - The size of a set |
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| | | | | | | sets: 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 whenever x and y are in S and x < z < y then z is in S.
As mentioned above, a particularly important case is when T = R, the set of real numbers.
Intervals of R are of the following eleven different types (where a and b are real numbers, with a < b):
< ...
See also:Interval mathematics, Interval mathematics - Algebra, Interval mathematics - Higher mathematics, Interval mathematics - Intervals in partial orders, Interval mathematics - Interval arithmetic, Interval mathematics - Relational operations, Interval mathematics - Alternative notation Read more here: » Interval mathematics: Encyclopedia II - Interval mathematics - Higher mathematics |
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| | | | | sets: Encyclopedia II - English National Opera - History In 1898, Lilian Baylis presented a series of opera recitals at the Old Vic theatre. Some ten years later, she established a theatre company there, initially performing 'cut-down' versions of Shakespeare's plays. Having added a small group of dancers to the company (a group which later separated from Vic-Wells and renamed: The Royal Ballet); Sadler's Wells Theatre opens and the Vic-Wells Opera Company is formed.
After closures during the Second World War, Sadler's Wells re-opened with Benjamin Britten's Peter Grimes, introducing ...
See also:English National Opera, English National Opera - History, English National Opera - Repertoire, English National Opera - Home, English National Opera - Education, English National Opera - Ring Cycle, English National Opera - Gilbert and Sullivan Opera Read more here: » English National Opera: Encyclopedia II - English National Opera - History |
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| | | | | sets: 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 of 1990. Engineer Patrick Naughton had become increasingly frustrated with the state of Sun's C++ and C APIs (application programming interfaces) and tools. While considering moving to NeXT, Naughton was offered a chance to work on new techno ...
See also:Java programming language, Java programming language - History, Java programming language - Early history, Java programming language - Java meets the Internet, Java programming language - Recent history, Java programming language - Version history, Java programming language - Philosophy, Java programming language - Object orientation, Java programming language - Platform independence, Java programming language - Automatic garbage collection, Java programming language - Syntax, Java programming language - Hello World example, Java programming language - Data structures, Java programming language - Interfaces and classes, Java programming language - Input/Output, Java programming language - APIs, Java programming language - Java Runtime Environment, Java programming language - Components of the JRE, Java programming language - Extensions and related architectures, Java programming language - Criticism, Java programming language - Language, Java programming language - Library, Java programming language - Performance, Java programming language - Notes Read more here: » Java programming language: Encyclopedia II - Java programming language - History |
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| | | | | sets: Encyclopedia II - Isomorphism - Definition Douglas Hofstadter provides an informal definition:
The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49)
Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, ...
See also:Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Definition |
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