# bijective
## English
### Pronunciation

* Rhymes: -ɛktɪv

### Adjective

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. (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 $\beta$ 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.
* (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

* 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

* bijective numeration
* bijectively
* nonbijective
* bijectivity

#### Related terms

* bijection

#### See also

* injective
* surjective

### See also

* reversible

### Further reading

* Bijection, injection and surjection on Wikipedia.Wikipedia
* Bijective numeration on Wikipedia.Wikipedia
* Bijective proof on Wikipedia.Wikipedia
* Homomorphism on Wikipedia.Wikipedia

## French
### Pronunciation

* IPA: /bi.ʒɛk.tiv/
* Homophone: bijectives

### Adjective

bijective
* feminine singular of bijectif