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Topic: Bijection

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In the News (Mon 22 Jul 19)

  Bijection - Wikipedia, the free encyclopedia
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.
Indeed, in axiomatic set theory, this is taken as the very definition of "same number of elements", and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
is not a bijection because π/3 and 2π/3 are both in the domain and both map to (√3)/2.
en.wikipedia.org /wiki/Bijection   (669 words)

 PlanetMath: mapping of period $n$ is a bijection
PlanetMath: mapping of period $n$ is a bijection
Cross-references: identity, injection, surjection, identity mapping, bijection, mapping, theorem
is a bijection, born on 2003-08-01, modified 2004-03-12.
planetmath.org /encyclopedia/MappingOfDegreeNIsASurjection.html   (53 words)

 PlanetMath: bijection
that is one-to-one and onto is called a bijection or bijective function from
It easy to see the inverse of a bijection is a bijection, and that a composition of bijections is again bijective.
This is version 10 of bijection, born on 2001-10-20, modified 2006-01-12.
planetmath.org /encyclopedia/Bijection.html   (102 words)

 Surjection - Wikipedia, the free encyclopedia
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.
By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain.
More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows.
en.wikipedia.org /wiki/Surjective   (532 words)

 Monday, April 17, 2000
f' is a bijection from A to mx{0} (it is the composition of f with the obvious bijection from m to mx{0}).
g' is a bijection from B to n x {1} (it is the composition of g with the obvious bijection from n to n x {1}).
Recall that f is a bijection from A to n, so f(P(b)) is an element of m, while g_(P(b)) is a bijection from P(b) to n, so g_(P(B)(b) is an element of n, and the ordered pair is an element of m x n.
math.boisestate.edu /~holmes/M387syllabus/node57.html   (539 words)

 Schroder-Bernstein Theorem
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.
If there is a 1-1 correspondence from A to B and a 1-1 corespondence from B to A, then there is a bijection between A and B. The proof is given elsewhere on these pages.
That is, the set of all positive real numbers is bijective with the set of all real numbers just from -1 to +1, including -1 and +1.
www.mathpath.org /concepts/Num/bernstein.htm   (652 words)

 Monday, April 10
This definition is merely ``proposed'' because there is something we have to prove before we can really regard it as a definition of the cardinality A.
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 said to be a bijection iff it is both an injection and a surjection.
math.boisestate.edu /~holmes/M387syllabus/node54.html   (650 words)

 Wikinfo | 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.
Generalising this to infinite sets leads to the concept of cardinal number, a way to distinguish the various infinite sizes of infinite sets.
Images, some of which are used under the doctrine of Fair use or used with permission, may not be available.
www.wikinfo.org /wiki.php?title=Bijective   (342 words)

 Chapter 2. The contextual component model   (Site not responding. Last check: 2007-10-22)
Components are stateful objects, usually EJBs, and an instance of a component is associated with a context, and given a name in that context.
Bijection provides a mechanism for aliasing internal component names (instance variables) to contextual names, allowing component trees to be dynamically assembled, and reassembled by Seam.
In essence, bijection lets you alias a context variable to a component instance variable, by specifying that the value of the instance variable is injected, outjected, or both.
docs.jboss.com /seam/reference/en/html/concepts.html   (3138 words)

 Padding and Bijection
It does appear that until recently there were zero bijective schemes for mapping a n-byte file to a file which is a multiple of the block size of an orthodox block cypher.
Another point is although Matts code is 1-1 (unadulterated bijective whatever) It does not mean that one byte of plaintext maps to one byte of cipher test.
But they suck in that they are not bijective to the orignal files iencrypted and a false key is very likely to lead to a file that could not be encrypted with the set of rules.
www.ciphersbyritter.com /NEWS6/PADDING.HTM   (14688 words)

 The local Langlands conjecture for GL(n) over a p-adic field, n < p, by Michael Harris   (Site not responding. Last check: 2007-10-22)
The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne group of F. This bijection is required to satisfy certain natural compatibilities, of which the most important is compatibility with local functional equations (preservation of L and epsilon factors of pairs).
It is enough to construct a bijection with these properties between supercuspidal representations of GL(n,F) and n-dimensional irreducible representations of the Weil group of F. In a previous paper, the author constructed a canonical bijection on the etale cohomology of the rigid-analytic coverings of the p-adic upper half space constructed by Drinfeld.
The present article uses a technique of non-Galois automorphic induction to show that the bijection previously constructed is compatible with epsilon factors of pairs of representations of GL(n,F) and GL(m,F) when n and m are prime to p (the tame case).
www.math.uiuc.edu /Algebraic-Number-Theory/0051   (214 words)

 [No title]   (Site not responding. Last check: 2007-10-22)
Find a bijection between the set of odd integers and the set of all integers {..., -5, -3, -1, 1, 3, 5,...} a) Describe this mapping: what is the domain?
Find a bijection between the set of all 6-letter strings from the letters CATDOG and the set of all 5-letter strings from the same letters, where the letters in strings are not allowed to repeat.
there exists a bijection between the integers and the naturals 2.
www.cs.umb.edu /~dqg/cs320/handouts/hw5.txt   (275 words)

 UMBC CMSC771 Knowledge Representation and Reasoning
For coloristic purposes we follow the tradition that calls the two sets MEN and WOMEN, so that bijection means marrying everyone off to someone of the opposite sex.
We often find it notationally convenient to regard a given bijection between MEN and WOMEN as the self-map on MEN U WOMEN (the union) which associates each person to his or her spouse.
That means we must avoid creating any unmarried man-woman pair (M,w) where M prefers w to his wife and w prefers M to her husband.
cgm.cs.mcgill.ca /~avis/courses/360/notes/stablemarriage.html   (1473 words)

 Doug's Expositions   (Site not responding. Last check: 2007-10-22)
Then the B_n are disjoint, and f is a bijection between the union of the A_n and the union of the B_n.
Thus, f on the union of the A_n and the inverse of g on A' is a bijection from A to B. This was the claim.
A set S is infinite if and only if there is a bijection between S and a proper subset of S. For example, the nonnegative integers are infinite because the map x-->x+1 is a bijection with the positive integers.
www.math.columbia.edu /~zare/cardinality.html   (835 words)

 Joe Hurd: Research: The Schroeder-Bernstein Theorem   (Site not responding. Last check: 2007-10-22)
Claim 1: f is a bijection from X to f(X).
Proof of Claim 2: This follows from the fact that f(X) is a subset of B, and X is the disjoint union of A - B and f(X).
Therefore the required bijection is the function j : A -> B defined as j(a) = if a in X then f(a) else a.
www.cl.cam.ac.uk /~jeh1004/research/formalize/schroeder.html   (339 words)

 Math 3000 Sample Final Exam Questions
The function g is no longer a bijection since it is now not onto, but it is still one-to-one.
, and being the composition of two bijections, it is itself a bijection.
One may assume that it is true, i.e., take it as an additional axiom of set theory, and obtain what is now called "standard" set theory, or assume that it is not true and obtain a "non-standard" set theory.
www-math.cudenver.edu /~wcherowi/courses/m3000/abexfs.html   (1112 words)

 CiteULike: Tag bijection   (Site not responding. Last check: 2007-10-22)
Bijective counting of tree-rooted maps and shuffles of parenthesis systems
A Strahler bijection between Dyck paths and planar trees
A Foata bijection for the alternating group and for q analogues
www.citeulike.org /tag/bijection   (157 words)

 M3000 Homework #20
We have then that f is a bijection, so A is equivalent to A ×; {x}.
If B is finite, it would have an infinite subset, namely A. This contradicts Theorem 5.5, every subset of a finite set is finite.
This proof does not correctly construct the bijection needed to show that this is true.
www-math.cudenver.edu /~wcherowi/courses/m3000/abhw20.html   (620 words)

 [No title]
Try to set up a bijection between the interval [0, 1] and the set F of all real functions f(x) y (fy(x) (6) where y ([0, 1].
As the attempted bijection (6) is in no way specified, we can require that the function related to (1+y)/2 ([0, 1] f(1+y)/2(x) = g(x) (7) is always the same as the diagonal function g(y) constructed up to the point y.
But once the system has been fixed, the bijective mapping of the points N of the continuum on the n-tuples (x1,x2,..., xn) is fixed too N ((x1,x2,..., xn) ((n = ((((((((.
www.fh-augsburg.de /~mueckenh/Infinity/MA2-040405.doc   (2221 words)

 [No title]   (Site not responding. Last check: 2007-10-22)
DEFINITION: Two sets A and B are called *equinumerous* or are said to have the same *cardinality* if there is a bijection f : A-->B. [bijection or 1-1 correspondence means that f is 1-1 and onto.
It's a bit counterintuitive to say that these two sets are equinumerous since the second is a proper subset of the first.
Sometimes it's hard to establish a bijection between two sets but there is a nice theorem to the rescue.
www.columbia.edu /~mpj9/4236/card.txt   (537 words)

 [No title]
The function f defined by f(x) = x + 2 is such a bijection.
The function f defined by f(x) = 3x + 2 is such a bijection.
The function f defined by  EMBED Equation.3  is such a bijection.
www.math.metu.edu.tr /cahitarf/Solutions.doc   (1168 words)

 Cardinality of Sets
Definition: Two sets A and B have the same cardinality if there is a bijection f:A->B. Example: The sets A={red, violet, green, yellow} and B={1, 2, 3, 4} have the same cardinality, since there is a bijection from A to B (find one).
N) if there is a bijection from the set {0, 1,..., n-1} to A, for some natural number n.
So f is not onto, and it follows that f is not a bijection.
www.cs.utexas.edu /~eberlein/cs336/cardinality.html   (748 words)

 CmSc 365 Bijections   (Site not responding. Last check: 2007-10-22)
A1, is a bijection from A1 to A2 /sqrt means square root/
The bijection is obtained a composition of of f5 and the inverse of f4:
The bijection may be directly mapping objects, or it may be a composition of other bijections.
www.simpson.edu /~sinapova/cmsc365-02/L02-Bijections.htm   (318 words)

 DMTCS vol 1 no 1 (1997), pp. 53-67   (Site not responding. Last check: 2007-10-22)
This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape.
The next part of the paper presents the proof of the bijectivity of our construction.
Jean-Christophe Novelli and Igor Pak and Alexander V. Stoyanovskii (1997), A direct bijective proof of the hook-length formula, Discrete Mathematics and Theoretical Computer Science 1, pp.
dmtcs.loria.fr /volumes/abstracts/dm010104.abs.html   (175 words)

 Dual Schroder-Bernstein Theorem
A, then there exists a bijection between and B. The dual of the theorem is this: if for two sets A and B there are surjective maps A
The proof given above only shows that a bijection exists but does not provide a way of constructing it unlike in our proof of the Cantor-Bernstein Theorem.
The middle school student should note that the dual is trivially valid for finite sets.
www.mathpath.org /concepts/dual.Sch-Bern.htm   (252 words)

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