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Topic: Surjective function

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Injective function

Bijection

Surjection

	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)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-03)
		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).
		Since f is surjective, any element y of the co_domain must go back to at least one element x of the domain, and since f is a function, that element x cannot map to any other element y.
		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)

	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: 2007-11-03)
		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)

	The relationship between stratified -calculus and usual type theories
		The inhabitants of type n+1 are the functions from type n to type n, for each n; it is easy to define the surjective pair on type n+1 in a uniform manner in terms of the pair on type n.
		A function from type 1 to type 0 is represented by a function of type 2 taking a type 1 function to the (type 1) constant function of the type 0 value of the coded function.
		A function from type 0 to type 1 is coded by a function of type 2: values at constant functions of its intended type 0 arguments of the coded function are the intended type 1 values, while values at nonconstant functions are ignored (they may be taken to be a default value).
	math.boisestate.edu /~holmes/babydocs/node7.html (933 words)

	Surjective function Biography,info (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-11-03)
		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.
	www.danceage.com.cob-web.org:8888 /biography/sdmc_Onto (514 words)

	Functions (Site not responding. Last check: 2007-11-03)
		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)

	PlanetMath: surjective
		That is, by restricting the codomain, any function can be made into a surjection.
		The composition of surjective functions (when defined) is again a surjective function.
		This is version 3 of surjective, born on 2002-03-14, modified 2005-04-30.
	planetmath.org /encyclopedia/Surjection.html (79 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		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)

	[No title]
		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)

	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)

	Open Questions: Glossary of Mathematics
		The set A is called the "domain" of the function, while the "range" of the function consists of all b ∈ B that occur as the second element of a pair.
		A function is usually denoted symbolically with a name such as "f" and written in the form f(a) = b if (a,b) is a pair of corresponding elements.
		A function is "injective" or "1-to-1" if there is no b ∈ B that occurs more than once as the second element of a pair.
	www.openquestions.com /oq-magl.htm (335 words)

	lec11Sept
		Although computer scientists tend to use “input” and “output” for functions, mathematicians prefer to use “pre-image” for an input and “image” for the corresponding output.
		When describing a function, one must specify the set of all possible inputs (the domain) and the set of all possible outputs (the co-domain).
		This is a notation for functions whose outputs can be computed from the outputs of other functions by addition, mulitplication and other arithmetic functions.
	www.pitt.edu /~vanlehn/cs0441/lec23Sept.html (464 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 (677 words)

	Injective function - Wikipedia, the free encyclopedia
		In mathematics, an injective function is a function which associates distinct arguments to distinct values.
		More precisely, a function f is said to be injective if, for every y in the codomain, there is at most one x in the domain such that f(x) = y.
		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.
	en.wikipedia.org /wiki/Injective_function (572 words)

	Composition; Injective and Surjective Functions (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-11-03)
		A function f: A -> B is said to be injective (also known as one-to-one) if no two elements of A map to the same element in B. Note that some elements of B may remain unmapped in an injective function.
		A function f: A -> B is said to be surjective (also known as onto) if every element of B is mapped to by some element of A. In other words, no element of B is left out of the mapping.
		This function is an injection because every element in A maps to a different element in B. It is not a surjection because some elements in B aren't mapped to by the function.
	www.cs.appstate.edu.cob-web.org:8888 /~dap/classes/1100/sect3_3.html (687 words)

	[No title]
		A function is bijective iff it is both surjective and injective.
		The Boolean function ~ (``not'') is the function from {0,1} to {0,1} defined as follows: ~0 = 1; ~1 = 0.
		The Boolean function xor is the function from {0,1}x{0,1} to {0,1} defined as follows: 0 xor 0 = 0; 0 xor 1 = 1; 1 xor 0 = 1; 1 xor 1 = 0.
	cr.yp.to /2005-261.html (3048 words)

	[Inquiry] Re: Logic Of Relatives (Site not responding. Last check: 2007-11-03)
		Note 52 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o In the case of a 2-adic relation F c X x Y that has the qualifications of a function f : X -> Y, there are a number of further differentia that arise:
		For example, the range of the function f above is Y'= {0, 2, 5, 6, 7, 8, 9}.
		Thus, if we form a new function g : X -> Y' that looks just like f on the domain X but is assigned the codomain Y', then g is surjective, and is described as mapping "onto" Y'.
	stderr.org /pipermail/inquiry/2003-April/000290.html (361 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Injective functions (one-to-one): Every element of domain is mapped to a unique co-domain element.
		Surjective function (onto): The range and co-domain are the same.
		Use a hash function to convert string “lions” to a number N. Now put the information on lions in position N of an array.
	www.cs.nmsu.edu /~mii/focus_groups/math_301/doc/301_worksheet_4.doc (448 words)

	Homework 2.6
		Determine whether each relation is a function from X = {1,2,3,4} to Y = {a,b,c,d}.
		Give two examples of a function that is one-to-one but not onto.
		Give an example of a function that is onto but not one-to-one.
	www.omegamath.com /Discrete/hwk2.6.html (274 words)

	[No title]
		A homomorphism is a function f that maps from one mathematical structure (the source) to another (the target) and preserves the structure of the first.
		An injective function f:A -> B is one that maps elements of the set A to the set B such that for every x in A, f(x) is in B. There may be elements in B that are not reachable by f(x).
		A surjective function f:A -> B is one that maps elements of the set A to the set B such that for every f(x) in B, there is a x in A. There may be more elements in A that do not map to an element of B.
	lal.cs.byu.edu /logics/glossary.html (1149 words)

	Math Forum Discussions
		For example in the category of sets the coproduct of two sets is given by their disjoint union and the canonical injections are not surjective.
		If you restrcit to the category of sets with surjective functions, there is no coproduct.
		Categoricaly that is the mais reason to considering all functions and not just the surjective ones.
	mathforum.org /kb/thread.jspa?threadID=1328732&messageID=4262657 (386 words)

	Amazon.co.uk: Lie Groups and Subsemigroups with Surjective Exponential Function (Memoirs of the AMS): Books: Karl ... (Site not responding. Last check: 2007-11-03)
		Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjective.
		It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered.
		Under nature reductions setting aside the "group part" of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir.
	amazon.co.uk.cob-web.org:8888 /exec/obidos/ASIN/0821806416 (565 words)

	Topological space (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-11-03)
		A function between topological spaces is said to be continuous if the inverse image of every open set is open.
		The category of topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in mathematics.
		Many sets of operators in functional analysis are endowed with topologies that are defined by specifying When a particular sequence of functions converges to the zero function.
	topological-space.iqnaut.net.cob-web.org:8888 (2393 words)

	Cardinality Constraints
		These properties are strictly related: a multirelation is total if and only if its transpose is surjective, and it is monodrome if and only if its transpose is injective; finally, it is a relation if and only if its transpose is.
		forces the corresponding leg to be an surjective function; this definition is equivalent to the one above.
		Thus, if a multirelation has to be an injective relation, its transpose has to be a partial function, and viceversa.
	erw.dsi.unimi.it /ERW/x421.html (663 words)

	A connected complex simple centerfree Lie group whose exponential function is not surjective (ResearchIndex)
		A connected complex simple centerfree Lie group whose exponential function is not surjective (1995)
		We discuss the example of a complex simple Lie group G, with trivial center, whose exponential map is not surjective and with dim C G = 10.
		A semisimple complex Lie groups has surjective exponential function if and only if it is isomorphic to a finite product of groups PSl(n(i); C...
	citeseer.ist.psu.edu /28486.html (246 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Functions Functions are special types of binary relations.
		Terminology The set S is the domain of the function.
		The range or image of the function is a subset of T, containing all t with f(s) = t for some s(S. More Terminology A function is injective iff f(s) = t and f(s’) = t then s=s’.
	www.cs.umanitoba.ca /~ckemke/74.303-Formal-Languages/2005SS/Mathematical-Background-SS.doc (1745 words)

	Surjective function - Wikipedia, the free encyclopedia
		If f and g are both surjective, then f
		Every function h: X → Z can be decomposed as h = g
		: A/~ → B be the well-defined function given by f
	en.wikipedia.org /wiki/Surjection (524 words)

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