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# Topic: Surjection

###### In the News (Tue 16 Jul 19)

 Bijection, injection and surjection - Encyclopedia, History, Geography and Biography   (Site not responding. Last check: 2007-11-06) In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function is surjective (onto) if every element of the codomain is mapped to by some element (argument) of the domain; some images may be mapped to by more than one argument. Since every function is surjective when its codomain is restricted to its range, every injection induces a bijection onto its range. www.arikah.com /encyclopedia/Onto   (1335 words)

 Surjection - Wikipedia, the free encyclopedia In mathematics, a function f is said to be surjective if and only 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   (522 words)

 Bijection, injection and surjection - Encyclopedia, History, Geography and Biography   (Site not responding. Last check: 2007-11-06) An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows. Bijective composition: the first function need not be surjective and the second function need not be injective. www.arikah.net /encyclopedia/Onto   (1335 words)

 [No title] To prove f : A (B is a surjection, let b (B ---- (1), we need to prove there exists (i.e., to find) a (A such that f(a) = b ---- (2). Since g o f : A (C is a surjection by assumption, so (3) implies there exists a (A such that g o f (a) = g(b). Since f: A (B is a surjection by assumption, there exists x (A such that f(x) = y ------ (2). longwood.cs.ucf.edu /~sdelden/courses/a5s.doc   (1037 words)

 Read about Bijection, injection and surjection at WorldVillage Encyclopedia. Research Bijection, injection and ...   (Site not responding. Last check: 2007-11-06) An injection maps distinct arguments to distinct images, a surjection maps to all possible images and a bijection is simply a function that is both an injection and a surjection. A surjection, surjective function or onto function is a function which maps to all possible images. The composition of two surjections is again a surjection, but if the composition of two functions is a surjection, then it can only be concluded that the second applied is surjective. encyclopedia.worldvillage.com /s/b/Bijection   (612 words)

 Monday, April 10 We restate the definitions of injection, surjection, and bijection using function notation: Definition: A function (A,f,B) is an injection iff (Axy E A.(f(x) = f(y) -> x = y)) An injection is also said to be ``one-to-one''. Definition: A function (A,f,B) is a surjection iff (Ay E B.(Ex E A.(y = f(x)))) A surjection is also said to be ``onto''. Definition: A function (A,f,B) is said to be a bijection iff it is both an injection and a surjection. math.boisestate.edu /~holmes/M387syllabus/node54.html   (650 words)

 PlanetMath: a surjection between finite sets of the same cardinality is bijective PlanetMath: a surjection between finite sets of the same cardinality is bijective "a surjection between finite sets of the same cardinality is bijective" is owned by ratboy. This is version 2 of a surjection between finite sets of the same cardinality is bijective, born on 2005-07-12, modified 2005-07-13. planetmath.org /encyclopedia/ASurjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective.html   (88 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)

 List of mathematical functions - Wikipedia, the free encyclopedia Bijection: Is both an injective and a surjection. Superadditive function: The value of a sum is greater than or equal to the sum of the values of the summands. Surjection, surjective function: Every element of the codomain has a preimage. en.wikipedia.org /wiki/List_of_mathematical_functions   (823 words)

 Bijection, injection and surjection   (Site not responding. Last check: 2007-11-06) In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions) and images (output expressions) are related. Since trivially every function is surjective when its codomain is restricted to its range, every injection is actually a bijection to its range and thus there exists an inverse from its range to its domain. A surjection, surjective function or onto function is a function which maps at least one argument to every possible image. en.showmy.net /Onto   (804 words)

 Linear_Linear_Mappings_Key.nb Thus, the transform cannot be a surjection (or, by consequence, a bijection). This T is not one-to-one and is neither an injection or a surjection. Because it is both a surjection and an injection, we call T a bijection. www.solutionarchive.org /host/fjbrooks/002/old/MTH_UNGD_LinearAlgebra_Linear_Mappings   (539 words)

 Surjection: Definition and Links by Encyclopedian.com - All about Surjection   (Site not responding. Last check: 2007-11-06) This function is surjective, since given an arbitrary real number 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.encyclopedian.com /su/Surjection.html   (218 words)

 Composition; Injective and Surjective Functions   (Site not responding. Last check: 2007-11-06) An important observation about surjective functions is that a surjection from A to B means that the cardinality of A must be no smaller than the cardinality of B Since an injection means that A and a surjection means that A >= B, a bijection means that A Some functions are neither injective nor surjective. 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.mathsci.appstate.edu /~dap/classes/1100/sect3_3.html   (679 words)

 Talk:Codomain - Wikipedia, the free encyclopedia While f and g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains. The codomain can affect whether or not the function is a surjection; in our example, g is a surjection while f is not. No doubt this is very confusing for those who are meeting functions for the first time; also I know of no use for the surjection concept (although I still have to read that article). en.wikipedia.org /wiki/Talk:Codomain   (648 words)

 Set Theory :: 3DSoftware.com In surjection (onto mapping) every element in the second set is mapped from at least one element of the first set. In surjective mapping, the cardinality of the second set is always less than or equal to the cardinality of the first set. On the other hand, with surjection (which is called “onto” mapping instead of “into” mapping), all of the elements are “forced onto” (imposed on) all the elements of the other set. www.3dsoftware.com /Math/Programming/SetTheory   (2035 words)

 Surjection - Psychology Central   (Site not responding. Last check: 2007-11-06) Image:Non-injective and surjective.png Image:Bijmap.png Image:Injective and non-surjective.png In mathematics, a function f is said to be surjective if and only 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. The natural logarithm function ln: (0..+∞) → R is surjective. If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. www.grohol.com /psypsych/Onto   (612 words)

 [No title] It is reasonable to expect that f should be in the image of OE* as long as OE is surjective in the dimensions that are relevant to f. Even better, if OE is surjective in rational cohomology only in dimension G(f) then f is in the image of OE*, at least up to a certain indeterminacy which doesn't contain f. Since g* induces a surjection in k-dimensional rational cohomology,Qthere is a phantom map f0 ~k+1 f such that f0 2 g*(Ph(Q Sk+1; Y)). hopf.math.purdue.edu /Ha-Strom/gray5.txt   (4561 words)

 [No title]   (Site not responding. Last check: 2007-11-06) Prove that if g is a surjection, and if X (B, Y (B, and X (Y, then g—1(X) (g—1(Y). Since g : A (B is a surjection and b (X (B by assumption, there exists a (A such that g(a) = b ---- (3). The function g : R (R is not a surjection because g(x) = 1 / (x2 + 1) > 0 for any x (R. Thus, there is no x (R such that g(x) = 1 / (x2 + 1) = (1. longwood.cs.ucf.edu /courses/cot3100.spr2000/section1/homework/key4sp00.doc   (1909 words)

 Reference.com/Encyclopedia/Bijection, injection and surjection In mathematics, the injection, surjection and bijection are classes of function distinguished by the manner in which arguments and images are mapped. A bijection is a function that is both injective and surjective. More formally a function f : A → B is surjective if, for every y in its codomain B, there exists an x in its domain A such that f(x)=y. www.reference.com /browse/wiki/Bijection   (688 words)

 The CTK Exchange Forums   (Site not responding. Last check: 2007-11-06) The empty set is in the powerset of any set (including itself) because the empty set is a subset of every set. Yes - A (the empty set) is a member of P(X), but it is not in the image of f - hence f is not a surjection. You may be curious to have a look at the old CTK Exchange archive. www.cut-the-knot.org /htdocs/dcforum/DCForumID6/346.shtml   (363 words)

 Countability This assignment is a mapping from the set of natural numbers to the set C, and since no element of C is omitted, the mapping is a surjection. No, all we need is a surjection and it is alright for elements in the codomain to be hit more than once in a surjection. It is an injection because no binary representation corresponds to more than one subset of the naturals, and it is a surjection because every binary represenation corresponds to some set of the naturals. www.mathsci.appstate.edu /~dap/classes/1100/sect3_4.html   (1278 words)

 [No title] B is a continuous surjection between completely regular spaces E and B, we may apply the Stone-Cech compactification functor fi to obtain a surjection fip : fiE ! B is surjective, then so is the map fip. B where E is a G-space, B is E=G and p is the canonical surjection. hopf.math.purdue.edu /Feldman-Wilce/fibdegen.txt   (5513 words)

 surjection - OneLook Dictionary Search Tip: Click on the first link on a line below to go directly to a page where "surjection" is defined. surjection : The American Heritage® Dictionary of the English Language [home, info] Surjection : Eric Weisstein's World of Mathematics [home, info] www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=surjection   (138 words)

 Surjection from LiveJournal   (Site not responding. Last check: 2007-11-06) Further, the Zariski topology on the space of Poisson primitive ideals of \$R\$ agrees with the quotient topology induced by the natural surjection from the maximal ideal space of \$R\$ onto the Poisson primitive ideal space. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2,C)-representations of the fundamental group of M and prove that the... Made slides at the last minute, met with Barb at the last minute, was pressured by her to schedule a practice talk at the last minute (tomorrow morning at 9 freakin' AM), asked lab people to go at the last minute. www.ljseek.com /search/Surjection   (802 words)

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