BIJECTION

Domain	Definition
Computing	Bijection 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). I.e. there is exactly one element of the domain which maps to each element of the codomain. For a general bijection f from the set A to the set B: f'(f(a)) = a where a is in A and f(f'(b)) = b where b is in B. A and B could be disjoint sets. See also injection, surjection, isomorphism, permutation. (2001-05-10). Source: The Free On-line Dictionary of Computing.
Source: compiled by the editor from various references; see credits.

Specialty Definition: 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.

In simple terms, a bijective function creates a one-to-one correspondence between its possible input values and possible output values. (In some references, the phrase "one-to-one" is used alone to mean bijective. Wikipedia does not follow this older usage.)

More formally, a function f: X → Y is bijective if for every y in the codomain Y there is exactly one x in the domain X with f(x) = y.

Surjective, not injective
Injective, not surjective

Bijective
Not surjective, not injective

When X and Y are both the real line R, then a bijective function f: R → R can be visualized as one whose graph is intersected exactly once by any horizontal line.

If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Generalising this to infinite sets leads to the concept of cardinal number, a way to distinguish the various infinite sizes of infinite sets.

Examples and counterexamples

Consider the function f: R → R defined by f(x) = 2x + 1. This function is bijective, since given an arbitrary real number y, we can solve y = 2x + 1 to get exactly one real solution x = (y − 1)/2.

On the other hand, the function g: R → R defined by g(x) = x² is not bijective, for two essentially different reasons. First, we have (for example) g(1) = 1 = g(−1), so that g is not injective; also, there is (for example) no real number x such that x² = −1, so that g is not surjective either. Either one of these facts is enough to show that g is not bijective.

However, if we define the function h: R⁺ → R⁺ by the same formula as g, but with the domain and codomain both restricted to only the nonnegative real numbers, then the function h is bijective. This is because, given an arbitrary nonnegative real number y, we can solve y = x² to get exactly one nonnegative real solution x = √y.

Properties

A function f: X → Y is bijective if and only if there exists a function g: Y → X such that g ^o f is the identity function on X and f ^o g is the identity function on Y. (In fancy language, bijections are precisely the isomorphisms in the category of sets.) In this case, g is uniquely determined by f and we call g the inverse function of f and write f⁻¹ = g. Furthermore, g is also a bijection, and the inverse of g is f again.
If f ^o g is bijective, then f is surjective and g is injective.
If f and g are both bijective, then f ^o g is also bijective.
If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (^o), form a group, the symmetric group of X, which is denoted variously by S(X), S_X, or X! (the last read "X factorial").

Usage Frequency: BIJECTION

Modern Translation: BIJECTION

Language		Translations for "BIJECTION"; alternative meanings/domain in parentheses.
German		Bijektion. (various references)

Japanese Kanji		双射 , 全単射 . (various references)

Japanese Katakana		ぜ�"た�"しゃ, そうしゃ (instrumentalist, man in prime, mowing down, operation, player, runner, strafing, sweeping with fire). (various references)

Pig Latin		ijectionbay

Russian		взаимно однозначное соответствие (one-to-one correspondence). (various references)
Source: compiled by the editor from various translation references.

Derivations: BIJECTION

Anagrams: BIJECTION

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "b-c-e-i-i-j-n-o-t"
	-2 letters: biontic.
	-3 letters: bionic, biotic, biotin, incite, inject, niobic, noetic, notice, object.
	-4 letters: beton, binit, biont, boite, cento, conte, ionic, jeton, joint, objet, ontic, tonic.
	-5 letters: bent, bice, bine, bint, bite, bone, cent, cine, cion, cite, coin, cone, coni, cote, ebon, etic, icon, inti, into, jeon, jibe, join, nice, nite, note, obit, once, otic, tine.
	Words containing the letters "b-c-e-i-i-j-n-o-t"
	+1 letter: bijections.
	+3 letters: objectifying.
	+5 letters: nonjusticiable, nonobjectivism, nonobjectivist, nonobjectivity.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Alternative Orthography: BIJECTION

Hexadecimal (or equivalents, 770AD-1900s) (references)

42 49 4A 45 43 54 49 4F 4E

Leonardo da Vinci (1452-1519; backwards) (references)

American Sign Language (origins from 1620-1817 in Italy and, especially, France) (references)

⁼

Semaphore (1791, in France) (references)

Braille (1829, in France) (references)

Morse Code (1836) (references)

-... .. .--- . -.-. - .. --- -.

Dancing Men (Sir Arthur Conan Doyle, 1903) (references)

Binary Code (1918-1938, probably earlier) (references)

01000010 01001001 01001010 01000101 01000011 01010100 01001001 01001111 01001110

HTML Code (1990) (references)

&#66 &#73 &#74 &#69 &#67 &#84 &#73 &#79 &#78

ISO 10646 (1991-1993) (references)

0042 0049 004A 0045 0043 0054 0049 004F 004E

British Sign Language (Fingerspelling, BSL; 1992, British Deaf Association Dictionary of British Sign Language) (references)

Encryption (beginner's substitution cypher): (references)

Parts of Speech	Percent	Usage per 100 Million Words	Rank in English
Noun (singular)	100%	3	202,518

BIJECTION

Specialty Definition: BIJECTION

Computing

Specialty Definition: Bijection

Examples and counterexamples

Properties

Crosswords: BIJECTION

Usage Frequency: BIJECTION

Modern Translation: BIJECTION

German

Japanese Kanji

Japanese Katakana

Pig Latin

Russian

Derivations: BIJECTION

Derivations