different between bijection vs bijective

English

Etymology

From French bijection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

Pronunciation

• IPA^(key): /ba?.d??k.??n/

Noun

bijection (plural bijections)

1. (set theory) A one-to-one correspondence, a function which is both a surjection and an injection.
   • 2002, Yves Nievergelt, Foundations of Logic and Mathematics, page 214,

     The present text has defined a set to be finite if and only if there exists a bijection onto a natural number, and infinite if and only if there does not exist any such bijection.

   • 2007, C. J. Date, Logic and Databases: The Roots of Relational Theory, page 167,

     Note in particular that a function is a bijection if and only if it's both an injection and a surjection.

   • 2013, William F. Basener, Topology and Its Applications, unnumbered page,

     The basic idea is that two sets A and B have the same cardinality if there is a bijection from A to B. Since the domain and range of the bijection is not relevant here, we often refer to a bijection from A to B as a bijection between the sets, or a one-to-one correspondence between the elements of the sets.

Synonyms

• (function that is both a surjection and an injection): one-to-one correspondence

Related terms

• injection
• surjection

Translations

Anagrams

• objicient

French

Pronunciation

• IPA^(key): /bi.??k.sj??/

Noun

bijection f (plural bijections)

1. (set theory) bijection

English

Pronunciation

• Rhymes: -?kt?v

Adjective

bijective (not comparable)

1. (mathematics, of a map) 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 ${\displaystyle \beta }$ 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.

2. (mathematics) Having a component that is (specified to be) a bijective map; that specifies a bijective map.
   • 2002, Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM Press, page 774,

     Proving the conjecture is equivalent to constructing a PCP that reads 2 symbols and accepts iff these symbols satisfy a bijective constraint.

Usage notes

• If a bijective map exists from one set to another, the reverse is necessarily true, and the sets are said to be in bijective (also one-to-one) correspondence.
• 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

• bijectively
• bijectivity

Related terms

• bijection

Translations

Further reading

• Bijection, injection and surjection on Wikipedia.Wikipedia
• Bijective numeration on Wikipedia.Wikipedia
• Bijective proof on Wikipedia.Wikipedia
• Homomorphism#Isomorphism on Wikipedia.Wikipedia

See also

• reversible

French

Pronunciation

• IPA^(key): /bi.??k.tiv/
• Homophone: bijectives

Adjective

bijective

1. feminine singular of bijectif