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Encyclopedia > Bijective

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one_to_one and onto. Intuitively, a bijective function creates a correspondence that associates each input value with exactly one output value and each output value with exactly one input value. (In some references, the phrase "one_to_one" is used alone to mean bijective. This encyclopedia does not follow this older usage.)


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

Image:bijMap.png

Bijective (injective and surjective)

Image:mathmap.png

Injective, not surjective

Image:ontoMap.png

Surjective, not injective

Image:mathmap2.png

Not surjective, not injective

When X and Y are both the real line R, then a bijective function fR → R can be visualized as one whose graph is intersected exactly once by any horizontal line. (This is a special case of the horizontal line test.)


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. 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.


Examples and counterexamples

Consider the function fR → 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 gR → 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: [0, ∞) → [0, ∞) 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 fX → Y is bijective if and only if there exists a function gY → 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 Set 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), SX, or X! (the last read "X factorial").

See also





  Results from FactBites:
 
Bijection - Wikipedia, the free encyclopedia (669 words)
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.
PlanetMath: mapping of period $n$ is a bijection (53 words)
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.
  More results at FactBites »

 
 

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