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Topic: Bijective functions

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	Function
		The mathematical notion of function is not limited to computations using single numbers, or even numbers at all - a function may be any of a wide variety of mappings, maps or transformations.
		As a mathematical term, "function" was coined by Leibniz, in 1694, to describe a quantity related to a curve; such as a curve's slope or a specific point of said curve.
		Functions related to curves are nowaday called differentiable functions and are still the most frequently type of functions encounted by non-mathematicians.
	www.ebroadcast.com.au /lookup/encyclopedia/pr/Preimage.html (1779 words)

	Wikinfo \| Function
		The most familiar kind of function is that where the argument and the function's value are both numbers, and the functional relationship is expressed by a formula, and the value of the function is obtained from the arguments by direct substitution.
		Those functions, first thought as purely imaginary and called collectively "monsters" as late as the turn of the 20th century, were later found to be important in the modelling of physical phenomena such as Brownian motion.
		The number of computable functions from integers to integers is countable, because number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers.
	www.wikinfo.org /wiki.php?title=function (2162 words)

	Function - Questionz.net , answers to all your questions (Site not responding. Last check: 2007-10-11)
		History As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve; such as a curve's slope or a specific point of said curve.
		Graph of a functions The graph of a function f is the collection of all points(x, f(x)), for all x in set X. In the example of the discrete function, the graph of f is {(1,a),(2,d),(3,c)}.
		Thus one obtains a function gÊoÊf: XÊ→ÊZ defined by (gÊoÊf)(x)Ê:= g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x).
	www.questionz.net /Family/Function.html (2514 words)

	Bijection
		Put another way, a bijection is a function which is both injective and surjective, and therefore bijections are also called one-to-one and onto.
		Generalising this to infinite sets leads to the concept of cardinal number, a way to distinguish the various infinite sizes of infinite sets.
		This function is bijective, since given an arbitrary real number
	www.ebroadcast.com.au /lookup/encyclopedia/bi/Bijective.html (251 words)

	Bijection - Wikipedia, the free encyclopedia
		In mathematics, a function f from a set X to a set Y is said to be bijective if for every y in Y there is exactly one x in X such that f(x) = y.
		A bijective function is also called a bijection or permutation.
		The function f from the real line R to R defined by f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y.
	en.wikipedia.org /wiki/Bijective (708 words)

	Quasi-Algoruthms / Quasi-Functions
		Before entering the F function the block is expanded from 32 to 48 bits and it is split in 4 sub-blocks each 6 bits.
		The quasi function, the arguments and the information about the primitive operation to be executed in each step are the arguments passed to the overlay function.
		The sigma function use the same argument as C. The direct consequence is that every value of x and k make the decision which function from the family C [the encryption Quasi Algorithm] they are going to be processed trough.
	members.tripod.com /kbajalc/algo/pme/quasi.html (3973 words)

	PlanetMath: permutation
		A permutation can also be seen as a bijective function of a set into itself.
		In fact, every bijection of a set into itself gives a permutation, and any permutation gives rise to a bijective function.
		Using the function approach, it can be proved that any permutation can be expressed as a composition of disjoint cycles and also as composition of (not necessarily disjoint) transpositions.
	planetmath.org /encyclopedia/Permutation.html (188 words)

	Function composition - Wikipedia, the free encyclopedia
		In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite.
		As a result the set of bijective functions f: X → X form a group with respect to the composition operator.
		As an example, suppose that an airplane's elevation at time t is given by the function h(t) and that the oxygen concentration at elevation x is given by the function c(x).
	en.wikipedia.org /wiki/Function_composition (544 words)

	The Weird World of Bi-Directional Programming \| Lambda the Ultimate
		There are a great many cases where it is valuable to be able to statically express and verify algorithms that are bijective or fit elsewhere in Pierce's "Lens Bestiary".
		Bijective AKA invertible functions can be used in logic programming to get smaller search spaces.
		What makes this interesting to me is how many things in the computing world "pretend" to be bijective, when this is not explicitly declared in the programming language -- data conversion, compression/decompression, data persistence, UI forms, etc. I wonder if we'll see programming languages that explicitly declare this relationship.
	lambda-the-ultimate.org /node/1526 (1397 words)

	Subroutines
		Since QCL allows for the inverse call of operators and can perform scratch-space management for quantum functions, the allowed side effects on the classical program state as well as on the quantum machine state have to be strictly specified.
		This means, that not only the program state must not be changed by these routines, but also that their execution may in no way depend on the global program state which includes global variables, options and the state of the internal random nuber generator.
		Functions are the most restrictive routine type and don't allow any interactions with the global state.
	tph.tuwien.ac.at /~oemer/doc/qcldoc/node11.html (1970 words)

	METU MATHEMATICS DEPARTMENT
		MATH 153 Calculus for Mathematics Students I (4-2)5 Functions, limit and derivative of a function of a single variable, A thorough discussion of the basic theorems of differential calculus: Intermediate value, extreme value, and the Mean Value Theorems, applications: Graph sketching and problems of extrema.
		Functions of several variables: Limit, continuity, partial derivatives, the chain rule, directional derivatives, tangent plane approximation and differentials, extreme values, Lagrange multipliers.
		Lebesgue integrable functions and the Lebesgue integral in Rn.
	www.math.metu.edu.tr /courses/undergrad.shtml (3246 words)

	Some definitions regarding Functions. (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-11)
		This type of functions is used in for example vector algebra and vector analysis.
		Each of them can be seen as a function of a two-tuple variable.
		In a Bijective function each element of A is mapped onto one and only one element in B and each element of B is mapped
	hemsidor.torget.se.cob-web.org:8888 /users/m/mauritz/math/gen/deff.htm (372 words)

	MTH 220 - Monroe Community College Web Site
		Determine which of these characteristics is associated with a given function.
		8.3 Prove that a given function in one-to-one, or give a counterexample to show that it is not.
		8.4 Prove that a given function is onto, or give a counterexample to show that it is not.
	www.monroecc.edu /depts/math/mth220.htm (607 words)

	Math Forum - Ask Dr. Math
		I would be most obliged if perhaps you could provide me with the definitions of the terms injective, surjective, and bijective as they apply to functions in set theory.
		(This is often called a "one-to-one" function.) f is a surjective function if for all t in T, there is an s in S such that f(s) = t.
		(This is often called an "onto" function.) f is a bijective function if it is both injective and surjective.
	mathforum.org /library/drmath/view/52448.html (288 words)

	Quantum Operations
		QCL allows to invert operators and can perform scratch-space management for quantum functions, thus allowed side effects on the classical program state as well as on the quantum machine state have to be strictly specified.
		The 4 QCL routine types form a call hierarchy, which means that a routine may invoke only subroutines of the same or a lower level (see table 3.6).
		This means, that not only the program state must not be changed by these routines, but also that their execution may in no way depend on the global program state which includes global variables, options and the state of the internal random number generator.
	tph.tuwien.ac.at /~oemer/doc/quprog/node14.html (1622 words)

	METU MATHEMATICS DEPARTMENT
		Content: Functions, Limit and Derivative of a Function of a Single Variable, A Thorough Discussion of the Basic Theorems of Differential Calculus: Intermediate Value, Extreme Value, and the Mean Value Theorems, Applications: Graph Sketching and Problems of Extrema.
		Construction of the exponential and the logarithmic functions.
		Definition of the Limit and Continuity of Functions of 2,3 Variables, Examples on Approaching to a Point P from various Directions, Geometric interpretation of Continuous Surfaces.
	www.math.metu.edu.tr /courses/core.shtml (2899 words)

	Injective, surjective and bijective functions: Definition and Links by Encyclopedian.com (via CobWeb/3.1 ... (Site not responding. Last check: 2007-10-11)
		The volume of water in the Amazon river is greater than the next eight largest rivers in the world combined.
		This page functions similarly to a disambiguation page; that is, it just points to other pages.
		Post a link to definition / meaning of " Injective, surjective and bijective functions " on your site.
	www.encyclopedian.com.cob-web.org:8888 /in/Injective,-surjective-and-bijective-functions.html (137 words)

	Basic definitions and facts
		We will use the convention that such a function f is written to the right of its argument, i.e.
		The main advantage of this notation is, that composition of functions is written in the `right order', i.e.
		The group package of MAPLE also provides functions to multiply permutations that are given in disjoint cycle notation.
	www.kent.ac.uk /ims/personal/pf10/comp_all/node3.html (953 words)

	Methodology
		The area of space T is equal to the area of S and therefore each side has a length of
		Of course, there is an entire family of bijective functions which satisfy this property, but few transformations can guarantee a small distance between (x
		These equations are fragmented with many indicator functions because, though the points along the circumference of a circle are easily defined (
	home1.gte.net /~WILLIMUR/writing/morph5/node2.html (297 words)

	A remark over the converse of Hölder inequality
		be the set of measurable nonnegative real functions defined on
		Keywords and phrases: Measure space; Positive homogenous functional; Hölder inequality; Conjugate functions.
		Manuscript submissions are open for the new special issue on Ultra-Wideband Antennas
	www.hindawi.com /GetArticle.aspx?doi=10.1155/S1025583499000089 (106 words)

	Applied Maths 5
		Expected values, variance, moments, moment generating functions, Bernoullis trials, Binomial, Poisson, normal distributions for detailed study with proof, Other common distributions,
		Operations on sets, relations and their functions, partial order and equivalence relations, Peanos axioms and mathematical induction, Injective, Surjective, Bijective functions, Pigeonhole principle and its applications.
		Formal mathematical systems, elements of theory of some algebras such as rings, integral domains, group, fields Boolean algebra, semigroup.
	members.tripod.com /~saumi/courses/appliedmaths5.htm (163 words)

	intro2math
		injective functions, surjective functions, bijective functions, composition of functions, identity function, inverse functions, piecewise defined functions, induced functions on powersets, equality of sets, permutations, projections from products, direct products of functions
		bijective sets, finite, infinite, countable, uncountable, cardinal number, cardinality, Cantor-Schroeder-Bernstein Theorem, Cantor's Theorem, cardinalities of the number systems, counting products, counting permutations, counting combinations
		characteristic functions, powersets, Hasse diagrams of powersets, factor sets, kernels, direct products, direct powers, functions as elements of direct powers, permutations, exponents and function composition, orbits of permutations, cycles
	www.shsu.edu /~mth_jws11/intro2math.html (530 words)

	Unbounded pre-images of bijective continuos functions. (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-11)
		IÂ´d like to find an example of a continuous function
		f:X->Y (X,Y Banach spaces) that is a bijection and there exist a point y in Y such that the pre-image of all balls centered in y is an unbounded set.
		The time now is Sat Dec 23, 2006 12:36 am
	sci4um.com.cob-web.org:8888 /about1315.html (137 words)

	[No title]
		Fill in the blanks [A set with no members at all is called an __________].
		True/False [log (xy) = log x — log y] Evaluate an expression [with set operations, exponential, log or summations] Given a function with its domain and range, find if it is onto, one-to-one, bijective or none of them.
		Give a table showing the sequence of computation.
	www.cs.colostate.edu /~malaiya/166/review1.doc (137 words)

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