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Topic: Bijection, injection and surjection

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	Reference.com/Encyclopedia/Bijection
		Said another way, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective).
		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.
		is not a bijection because π/3 and 2π/3 are both in the domain and both map to (√3)/2.
	www.reference.com /browse/wiki/Bijection (720 words)

	Bijection, injection and surjection - Wikipedia, the free encyclopedia
		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.
		Bijective composition: the first function need not be surjective and the second function need not be injective.
	en.wikipedia.org /wiki/Bijection,_injection_and_surjection (1109 words)

	Schroder-Bernstein Theorem
		Two sets A and B have the same cardinality if there is a 1-1 and onto mapping or correspondence from A to B. Often, it is easier to find a 1-1 mapping rather than one that is 1-1 and onto.
		Similary, the word surjection is derived by combining jacere with the Latin word sur, meaning "upon." A mapping that is both an injection and a surjection is called a bijection.
		When there is a bijection between sets A and B, the sets are said to be equinumerous (having the same cardinality) or equipollent or equivalent.
	www.mathpath.org /concepts/Num/bernstein.htm (652 words)

	Surjection
		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.ebroadcast.com.au /lookup/encyclopedia/su/Surjection.html (189 words)

	Set Theory :: 3DSoftware.com
		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.
		Consider again the case of an injection where the cardinality of the second set is higher than the cardinality of the first set.
	www.3dsoftware.com /Math/Programming/SetTheory (2035 words)

	PlanetMath: ring homomorphism
		A ring isomorphism is a ring homomorphism which is a bijection.
		A ring monomorphism (respectively, ring epimorphism) is a ring homomorphism which is an injection (respectively, surjection).
		Cross-references: property, satisfy, multiplicative identity, surjection, injection, bijection, function, rings
	planetmath.org /encyclopedia/RingHomomorphism.html (169 words)

	Injection, Surjection, Bijection
		f(x) = 2x + 5 is a bijection.
		g(x) = arctan(x) is injective, not surjective on R.
		Since this function is periodic, it can not be one-to-one.
	pirate.shu.edu /projects/reals/logic/answers/jection2.html (190 words)

	Definitions of Computable
		The terms injection, surjection and bijection are all total functions defined as: Injection : one-to-one into : for every member of the domain, the function returns some member of the range, but not necessarily all members of the range will be returned.
		Surjection : many-to-one onto : for every member of the domain, the function returns some member of the range.
		Bijection : one-to-one onto : for every member of the domain, the function returns a unique member of the range.
	www.csee.umbc.edu /help/theory/computable.shtml (2082 words)

	Homework 13--Functions II
		For each of the following functions, tell whether it is an injection, a surjection, a bijection (both an injection and a surjection), or neither.
		surjection (not an injection because f(1)=1 and f(-1)=1)
		Our listing technique provides a surjection from N to N x N x N. Given an alphabet A, show that A* is countable.
	www.mathsci.appstate.edu /~dap/classes/1100/hmwk13sol.html (513 words)

	Permutations (Site not responding. Last check: 2007-11-03)
		It suffices to prove that in this case injection and surjection are equivalent.
		is an injection (prove it!), but not a surjection.
		are infinite then there exist injections which are not bijections and vice versa.
	ndp.jct.ac.il /~math/tutorials/Discrete/node80.html (160 words)

	Proof of Cantor-Bernstein Theorem
		B, given by k(x) = 2x is a bijection.
		and the bijection h sends x in S to 4x, and x outside S to x.
		Construct a square with (1/2,1/2) as center and side lenths of 1 unit so that A = [0,1] is on the bottom and B = [0,1] is the vertical side on the left.
	www.mathpath.org /proof/Sch-Bern/proofofS-B.htm (624 words)

	Set theory
		F be a function that assigns different elements in E to different elements in F. Then f is called a one-one function (injection).
		Let f be a function of E into F. Then f(E) If every element in F is an image of at least one element in E, then f is a function of E onto F. Then f(E)= F. Bijection is at the same time injection and surjection.
		That unique bijection g is called the inverse function of f.
	library.thinkquest.org /C0126820/functions.html (1052 words)

	Objectives for Functions (Site not responding. Last check: 2007-11-03)
		Given a function, the student will be able to determine the domain, codomain, range, image and pre-image for the function and its inputs.
		Given a function, the student will be able to determine if the function is an injection, a surjection or a bijection.
		Given a function, the student will be able to determine its inverse if it has one.
	www.csl.mtu.edu /cs2311/www/Objectives/NewFunctions.htm (74 words)

	surjection from FOLDOC (Site not responding. Last check: 2007-11-03)
		A function f : A -> B is surjective or onto or a surjection if f A = B. I.e.
		Only surjections have right inverses, f' : B -> A where f (f' x) = x since if f were not a surjection there would be elements of B for which f' was not defined.
		Nearby terms: supervenience « supervenient « supremum « surjection » surjective » syllogism » symbolic logic
	www.swif.uniba.it /lei/foldop/foldoc.cgi?surjection (87 words)

	[No title]
		True False y = x + 1, x(N, y(N (natural numbers) injection surjection bijection y = x + 1, x(Z, y(Z (the integers) injection surjection bijection 3.
		(10 points) Find the indicated functions when f(x) and g(x) (from R to R) are defined as follows: f(x) = 2x2 + 1 g(x) = 3x + 4 (f(g)(x) ____________________________________ (f(f)(x) ____________________________________ (g(g)(x) ____________________________________ (g(f)(x) ____________________________________ (g-1)(x) ____________________________________ f(x) is an injection (one to one).
		True false g(x) is an injection (one to one).
	astro.temple.edu /~stafford/cis66sp03/pexam/mid1f02.doc (476 words)

	Semantic relations (Site not responding. Last check: 2007-11-03)
		a surjection (onto) if for each element of B there is an element of A that maps to it,
		a bijection if it is both 1-to-1 and onto,
		complete if the semantic mapping is a bijection (injection and surjection), and
	cs.wwc.edu /~aabyan/CII/BOOK/book/node63.html (590 words)

	Range (mathematics) Summary
		The codomain is sometimes taken to be the range, but more often is some standard set, such as the real numbers or the complex numbers, which contains the range.
		(Older books sometimes call what is now called the codomain the range, and what is now called the range the image set.) A function whose range equals its codomain is called onto or surjective.
		Let the function f be a function on the real numbers:
	www.bookrags.com /Range_(mathematics) (1081 words)

	[No title]
		(x, y)Îðf ¡Æÿ3þÿ3þçÿçÿçÿçÿçÿçÿªº1 & ó ¨!Injection, Surjection & Bijectionª" þf: A ’!
		B injection: "ðx1, x2ÎðA: f(x1) = f(x2) => x1=x2 surjection: "ðyÎðB $ðxÎðA: y = f(x) bijection: both injection and surjection¡Ö çÿçÿçÿçÿçÿçÿ ªº - ó ¨ Countabilityª üSet X is countable if: X is finite $ðf:N ’!
		B injection: "ðx1, x2ÎðA: f(x1) = f(x2) => x1=x2 surjection: "ðyÎðB $ðxÎðA: y = f(x) bijection: both injection and surjection¡Ö çÿçÿçÿçÿçÿçÿ ª - ó ¨ Countabilityª üSet X is countable if: X is finite $ðf:N ’!
	www.cs.joensuu.fi /pages/mozgovoy/tfcs/slides/lecture01.ppt (257 words)

	bijection from FOLDOC (Site not responding. Last check: 2007-11-03)
		A function is bijective or a bijection or a one-to-one correspondence if it is both injective (no two values map to the same value) and surjective (for every element of the codomain there is some element of the domain which maps to it).
		For a general bijection f from the set A to the set B:
		Nearby terms: Big Red Switch « Big Room « big win « bijection » Bill Gates » Bill Joy » binaries
	ftp.sunet.se /foldoc/foldoc.cgi?bijection (118 words)

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