bijective
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English
[edit]Pronunciation
[edit]- Rhymes: -ɛktɪv
Adjective
[edit]bijective (not comparable)
- (mathematics, of a function) Associating to each element of the codomain exactly one element of the domain; establishing a perfect (one-to-one) correspondence between the elements of the domain and the codomain; (formally) both injective and surjective.
- 1987, James S. Royer, A Connotational Theory of Program Structure, Springer, LNCS 273, page 15,
- Then, by a straightforward, computable, bijective numerical coding, this idealized FORTRAN determines an EN.[effective numbering] (Note: In this FORTRAN example, we could have omitted restrictions on I/O and instead used a computable, bijective, numerical coding for inputs and outputs to get another EN determined by FORTRAN.)
- 1993, Susan Montgomery, Hopf Algebras and Their Actions on Rings, American Mathematical Society, CBMS, Regional Conference Series in Mathematics, Number 83, page 124,
- Recent experience indicates that for infinite-dimensional Hopf algebras, the “right” definition of Galois is to require that be bijective.
- 2008, B. Aslan, M. T. Sakalli, E. Bulus, Classifying 8-Bit to 8-Bit S-Boxes Based on Power Mappings, Joachim von zur Gathen, José Luis Imana, Çetin Kaya Koç (editors), Arithmetic of Finite Fields: 2nd International Workshop, Springer, LNCS 5130, page 131,
- Generally, there is a parallel relation between the maximum differential value and maximum LAT value for bijective S-boxes.
- 2010, Kang Feng, Mengzhao Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer, page 39:
- An isomorphism is a bijective homomorphism.
- 2012 [Introduction to Graph Theory, McGraw-Hill], Gary Chartrand, Ping Zhang, A First Course in Graph Theory, 2013, Dover, Revised and corrected republication, page 64,
- The proof that isomorphism is an equivalence relation relies on three fundamental properties of bijective functions (functions that are one-to-one and onto): (1) every identity function is bijective, (2) the inverse of every bijective function is also bijective, (3) the composition of two bijective functions is bijective.
- 1987, James S. Royer, A Connotational Theory of Program Structure, Springer, LNCS 273, page 15,
- (mathematics) Having a component that is (specified to be) a bijective map; that specifies a bijective map.
Usage notes
[edit]- Bijective functions are invertible, and their inverses are themselves bijective functions. In particular, if a bijective map exists from one set to another, the reverse is necessarily true. Pairs of sets which admit a bijection from one to the other are said to be in bijection, in bijective correspondence, or (in the context of cardinality) equinumerous.
- A bijective map is often called a bijection.
- A bijective map from a set (usually, but not exclusively, a finite set) to itself may be called a permutation.
Derived terms
[edit]Related terms
[edit]Translations
[edit]both injective and surjective
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having a bijective map
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See also
[edit]See also
[edit]Further reading
[edit]- Bijection, injection and surjection on Wikipedia.Wikipedia
- Bijective numeration on Wikipedia.Wikipedia
- Bijective proof on Wikipedia.Wikipedia
- Homomorphism on Wikipedia.Wikipedia
French
[edit]Pronunciation
[edit]- IPA(key): /bi.ʒɛk.tiv/
- Homophone: bijectives
Adjective
[edit]bijective