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Topic: Injective function

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Injective

Surjection

Bijection

Galois connection

Injective module

Equivalent

Identity function

Dominated convergence theorem

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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: 2007-10-20)
		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)}.
		Injective, surjective and bijective functions Several types of functions are very useful, deserve special names: * injective (one-to-one) functions send different arguments to different values; in other words, if x and y are members of the domain of f, then f(x) = f(y) if and only if x = y.
		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)

	Injective function - Wikipedia, the free encyclopedia
		In mathematics, an injective function is a function which associates distinct arguments to distinct values.
		This decomposition is unique up to isomorphism, and f may be thought of as the inclusion function of the range h(W) of h as a subset of the codomain Y of h.
		In the language of category theory, injective functions are precisely the monomorphisms in the category of sets.
	en.wikipedia.org /wiki/Injective_function (569 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-20)
		The set functions F and F(-1) (F inverse) are defined in terms of f as follows: F(A) = {f(x) such that x is an element of A} where A is an element of P(X).
		We must prove the equivalence of 3 statements: (a) f is injective (b) F is injective (c) F^(-1) is surjective We will show that (a) is equivalent to both (b) and (c).
		In this case, for f to be a function, X must also be empty, and f is the empty function.
	mathforum.org /library/drmath/view/62589.html (2544 words)

	RelationsAndFunctions - PineWiki
		Floor and ceiling functions: when x is a real number, floor(x) (usually written ⌊x⌋) is the largest integer less than or equal to x and ceiling(x) (usually written ⌈x⌉) is the smallest integer greater than or equal to x.
		Functions that are not bijections are not thought of as invertible, because they don't have inverse functions (they do, however have inverse relations).
		Various properties of f∘g (e.g., being injective) are related to the properties of f and g individually; details NotWrittenYet.
	pine.cs.yale.edu /pinewiki/RelationsAndFunctions (752 words)

	PlanetMath: function
		There is no universal agreement as to the definition of the range of a function.
		Some authors define the range of a function to be equal to the codomain, and others define the range of a function to be equal to the image.
		This is version 11 of function, born on 2001-10-19, modified 2005-06-01.
	planetmath.org /encyclopedia/Function.html (147 words)

	Functions
		The type of function we will run across here most frequently is a ``permutation'', defined precisely later, which is roughly speaking a rule which mixes up and swaps around the elements of a finite set.
		A function is also called a map, mapping, or transformation.
		function'' maps an infinite set to a finite set.
	web.usna.navy.mil /~wdj/book/node152.html (558 words)

	[No title]
		Equivalently, a function f from S to T is surjective if each t in T is the image of some s in S under f.
		Similarly, f2 is the function which sends a face f of C to its image r(f) under r (which is again a face).
		Relations A relation on a set is a generalization of the concept of a function from S to itself.
	web.usna.navy.mil /~wdj/sm485_1c.txt (949 words)

	Satisfaction
		So knowing what function or relation is expressed amounts to knowing what the value of the function is for every argument, and knowing which objects the relation relates.
		A function is said to be surjective if it is ``onto,'' that is, if every item in the range (or ``codomain'') of the function is a value of the function for some argument.
		A function is said to be bijective if it is both injective and surjective, that is, both one-to-one and onto.
	www.trinity.edu /cbrown/topics_in_logic/struct/node2.html (1126 words)

	Wikinfo \| Surjection
		In mathematics, a surjection is a type of function with the property that all possible output values of the function are generated as values of the function as the input to the function ranges over all possible input values.
		A function is bijective if and only if it is both surjective and injective.
		In other words, surjective functions are precisely the epimorphisms in the category of sets.
	www.wikinfo.org /wiki.php?title=Surjective (306 words)

	injection (Site not responding. Last check: 2007-10-20)
		Only injective functions have left inverses f' where f'(f(x)) = x, since if f were not an injection, there would be elements of B for which the value of f' was not unique.
		If an injective function is also a surjection then is it a bijection.
		The opposite of an injection function is a projection function which extracts a component of a constructed object, e.g.
	burks.bton.ac.uk /burks/foldoc/50/57.htm (160 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.
		Bijective functions play a fundamental role in many areas of mathematics, for instance in the definition of isomorphism (and related concepts such as homeomorphism and diffeomorphism), permutation group, projective map, and many others.
		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/Bijection (708 words)

	Time Began With a Timeless Point
		Symmetries are injective functions that map each element in a domain onto some element in the codomain, such that the relevant essential properties possessed by the element in the domain are also possessed by the element in the codomain to which the element in the domain is related.
		A bijective function relates each element in the domain to a unique element in the codomain, such that no two elements in the domain can be related to the same element in the codomain and there is no element in the codomain that is not related to an element in the domain.
		The wave function of the universe predicts that there is a big bang singularity, but since the singular point does not have enough structure to determine or constrain its effect to have a certain order, the wave function predicts that a maximally chaotic state will follow from the big bang singularity.
	www.qsmithwmu.com /time_began_with_a_timeless_point.htm (13335 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Function ", Cell[BoxData[ \(TraditionalForm\`h\)]], " is wildly non-injective, and function ", Cell[BoxData[ \(TraditionalForm\`h\)]], " is injective, and therefore bijective.
		\n\n\ ", StyleBox["Examples:", FontWeight->"Bold"], " \nThe inverse of the logarithm function is the exponential function.
		\nThe inverse of the successor function on the integers is the \ predecessor function (subtract 1).
	www.cs.cmu.edu /afs/andrew.cmu.edu/course/15/354/www/NBooks/Functions.nb (9924 words)

	Math 417 Homework 1
		Thus f is injective and surjective, hence bijective.
		Thus fg is surjective and injective, hence bijective.
		Thus the set of injective functions from A to A is the same as the set of bijective functions.
	www.math.unl.edu /~bharbourne1/M417Spr03/M417Hmwk1Sols.html (1306 words)

	nashNET
		Instead of a simple 1:1 function of note to key, as in the last section, the flute’s key mechanism boasts a far more complex mapping, where, more often than not, several keys must be pressed to produce even one solitary pitch.
		The array will return the first note, so, in the event that a multiphonic signal, such as piano music, is passed to the function, it has an outside chance of extracting the normally higher melody line (or ostinato) from the right hand’s notes.
		functions), the variety of flute rod positions and rotations coupled with those of the individual flute keys (which also add scalings) would, this time, make a similar approach far more difficult and costly.
	www.nashnet.co.uk /english/mivi/instruments.htm (6242 words)

	Surjective function - Wikipedia, the free encyclopedia
		In mathematics, a function f is said to be surjective if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y.
		This decomposition is unique up to isomorphism, and f may be thought of as a function with the same values as h but with its codomain restricted to the range h(W) of h, which is only a subset of the codomain Z of h.
		In the language of category theory, surjective functions are precisely the epimorphisms in the category of sets.
	en.wikipedia.org /wiki/Surjection (524 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		>This is the function: > >(sigma^2,lambda,z^2) -> >(>z^2lambda((1-exp(-2at))/(2a))+sigma^2(1-exp(-2at))/(2a), > >3(sigma^2((1-exp(-2at))/(2a)))^2+6sigma^2(((1-exp(-2at))/(2a)))(z >^2lambda((1-exp(-2at))/(2a)))+lambda(z^4((1-exp(-4at))/(4a)))+3(z >^2lambda((1-exp(-2at))/(2a)))^2 > >15(sigma^2((1-exp(-2at))/(2a)))^3+153(sigma^2((1-exp(-2at))/(2a)) >)^2z^2lambda(((1-exp(-2at))/(2a)))+153sigma^2((1-exp(-2at))/(2a) >)(3z^4lambda(((1-exp(-4at))/(4a)))+3(lambdaz^2((1-exp(-2at))/(2 >a)))^2)+15z^6lambda(((1-exp(-6at))/(6a)))+15z^2lambda((1-exp(-2at >))/(2a))(z^4lambda((1-exp(-4at))/(4a))+3(z^2lambda(1-exp(-2at))/ >(2a))^2)+15lambda^3z^6((1-exp(-2at))/(2a))^3 >) Ugh!
		I'll use Maple too: o2:=simplify(output,{sigma^2=x,lambda=y,z^2=z,exp(at)=c}); gives a vector of rational functions* with constant denominators, so that for any a,t your map is now a _polynomial_ function.
		So no, this function is not even close to injective.
	www.math.niu.edu /~rusin/known-math/99/injective (329 words)

	Math Is Fun Forum / Need help with a injective (one-to-one) proof
		A) If f is injective, then for every element in Im f (the range), there must be one and only one path to get back to f.
		Remember, two elements in Im f can't go to the same element in A, because then f wouldn't even be a function, as it wouldn't be well defined.
		Since f is a function, for every element a∈A, there exists a b∈B, such that f(a) = b.
	www.mathsisfun.com /forum/viewtopic.php?id=2873 (946 words)

	Bijection
		A bijection (or bijective function) is a mathematical function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto.
		In simple terms, a bijective function creates a one-to-one correspondence between its possible input values and possible output values.
		Generalising this to infinite sets leads to the concept of cardinal number, a way to distinguish the various infinite sizes of infinite sets.
	www.fact-index.com /b/bi/bijection_1.html (286 words)

	Good Math, Bad Math : Categories and SubThings
		We can take the set of all injective functions to A (an injective function from X to Y is a function that maps each member of X to a unique member of Y).
		If there are injective functions from the domain of functions in I2 to the domain of functions in I1, then the members of I2 are also subsets of I1.
		What was confusing me is that you say that the domain of a function (the domain set of the mappings you refer to) "is a function" instead of a set (or an equivalence class).
	scienceblogs.com /goodmath/2006/07/categories_and_subthings.php (1310 words)

	Injective function
		A mathematical function is called injective (or one-to-one or an injection) if the function maps no more than one possible input value to each possible output value.
		This function is injective, since given arbitrary real numbers
		In other words, injective functions are precisely the monomorphisms in the category of sets.
	www.fact-index.com /i/in/injective_function.html (198 words)

	[No title]
		In fact, for every "function" E from cardinals to cardinals that satisfies these two requirements (weak monotonicity, and the cf(E(x)) > x) there is a model of ZFC that has E(x)=2^x.
		It doesn't imply GCH, though; for example if for every alpha one has 2^{Aleph_alpha}=Aleph_{alpha+2}, (which is consistent with ZFC by Easton forcing), one still has that the continuum function is injective.
		Later, Easton in "Powers of regular cardinals" extended this result to all regular cardinals simultaneously: the only restriction on phi -> 2^phi for phi regular is that the function be nondecreasing and not have cofinality(2^phi) = phi.
	www.math.niu.edu /~rusin/known-math/99/luzin_easton (1129 words)

	Projective and Injective Modules (Site not responding. Last check: 2007-10-20)
		A module j is injective if, for any pair of modules a and b, and any monomorphism f from a to b, and any homomorphism g from a to j, there is at least one homomorphism h from b to j such that fh = g.
		is injective, the function g(a) into p defines, and is defined by, component functions from a into each c
		Put these together to build a function h, and p is injective.
	www.mathreference.com /mod-pit,intro.html (460 words)

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