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Bijective function

one to one and onto mapping of a set X to a set Y
Gen bijection.svg
Bijection. There is exactly one arrow to every element in the codomain B (from an element of the domain A).

In mathematics, a bijective function or bijection is a function f : AB that is both an injection and a surjection. This means: for every element b in the codomain B there is exactly one element a in the domain A such that f(a)=b. Another name for bijection is 1-1 correspondence.[1][2]

The term bijection and the related terms surjection and injection were introduced by Nicholas Bourbaki.[3] In the 1930s, he and a group of other mathematicians published a series of books on modern advanced mathematics.

Gen not surjection not injection.svg
Not a bijection. (It is not a surjection. It is not an injection.)

Contents

Basic propertiesEdit

Formally:

   is a bijective function if     there is a unique     such that   

The element   is called the image of the element  .

  • The formal definition means: Every element of the codomain B is the image of exactly one element in the domain A.

The element   is called a pre-image of the element  .

  • The formal definition means: Every element of the codomain B has exactly one pre-image in the domain A.

Note: Surjection means minimum one pre-image. Injection means maximum one pre-image. So bijection means exactly one pre-image.

CardinalityEdit

Cardinality is the number of elements in a set. The cardinality of A={X,Y,Z,W} is 4. We write #A=4.[4]:60

  • Definition: Two sets A and B have the same cardinality if there is a bijection between the sets. So #A=#B means there is a bijection from A to B.

Bijections and inverse functionsEdit

Formally: Let f : AB be a bijection. The inverse function g : BA is defined by if f(a)=b, then g(b)=a. (See also Inverse function.)

  • The inverse function of the inverse function is the original function.[5]
  • A function has an inverse function if and only if it is a bijection.[6][7][8]

Note: The notation for the inverse function of f is confusing. Namely,

    denotes the inverse function of the function f, but    denotes the reciprocal value of the number x.

ExamplesEdit

Elementary functionsEdit

Let f(x):ℝ→ℝ be a real-valued function y=f(x) of a real-valued argument x. (This means both the input and output are numbers.)

  • Graphic meaning: The function f is a bijection if every horizontal line intersects the graph of f in exactly one point.
  • Algebraic meaning: The function f is a bijection if for every real number yo we can find at least one real number xo such that yo=f(xo) and if f(xo)=f(x1) means xo=x1 .

Proving that a function is a bijection means proving that it is both a surjection and an injection. So formal proofs are rarely easy. Below we discuss and do not prove. (See surjection and injection.)

Example: The linear function of a slanted line is a bijection. That is, y=ax+b where a≠0 is a bijection.

Discussion: Every horizontal line intersects a slanted line in exactly one point (see surjection and injection for proofs). Image 1.

Example: The polynomial function of third degree: f(x)=x3 is a bijection. Image 2 and image 5 thin yellow curve. Its inverse is the cube root function f(x)= ∛x and it is also a bijection f(x):ℝ→ℝ. Image 5: thick green curve.

Example: The quadratic function f(x) = x2 is not a bijection (from ℝ→ℝ). Image 3. It is not a surjection. It is not an injection. However, we can restrict both its domain and codomain to the set of non-negative numbers (0,+∞) to get an (invertible) bijection (see examples below).

Note: This last example shows this. To determine whether a function is a bijection we need to know three things:

  • the domain
  • the function machine
  • the codomain

Example: Suppose our function machine is f(x)=x².

  • This machine and domain=ℝ and codomain=ℝ is not a surjection and not an injection. However,
  • this same machine and domain=[0,+∞) and codomain=[0,+∞) is both a surjection and an injection and thus a bijection.

Bijections and their inversesEdit

Let f(x):AB where A and B are subsets of ℝ.

  • Suppose f is not a bijection. For any x where the derivative of f exists and is not zero, there is a neighborhood of x where we can restrict the domain and codomain of f to be a bisection.[4]:281
  • The graphs of inverse functions are symmetric with respect to the line y=x. (See also Inverse function.)

Example: The quadratic function defined on the restricted domain and codomain [0,+∞)

   defined by   

is a bijection. Image 6: thin yellow curve.

Example: The square root function defined on the restricted domain and codomain [0,+∞)

   defined by   

is the bijection defined as the inverse function of the quadratic function: x2. Image 6: thick green curve.

Example: The exponential function defined on the domain ℝ and the restricted codomain (0,+∞)

   defined by   

is a bijection. Image 4: thin yellow curve (a=10).

Example: The logarithmic function base a defined on the restricted domain (0,+∞) and the codomain ℝ

   defined by   

is the bijection defined as the inverse function of the exponential function: ax. Image 4: thick green curve (a=10).

Bijection: every vertical line (in the domain) and every horizontal line (in the codomain) intersects exactly one point of the graph.
 
1. Bijection. All slanted lines are bijections f(x):ℝ→ℝ.
 
2. Bijection. f(x):ℝ→ℝ. f(x)=x³.
 
3. Not a bijection. f(x):ℝ→ℝ. f(x)=x² is not a surjection. It is not an injection.
 
4. Bijections. f(x):ℝ→ (0,+∞). f(x)=10x (thin yellow) and its inverse f(x):(0,+∞)→ℝ. f(x)=log10x (thick green).
 
5. Bijections. f(x):ℝ→ℝ. f(x)=x³ (thin yellow) and its inverse f(x)=∛x (thick green).
 
6. Bijections. f(x):[0,+∞)→[0,+∞). f(x)=x² (thin yellow) and its inverse f(x)=√x (thick green).

Related pagesEdit

ReferencesEdit

  1. Weisstein, Eric. "Bijective function". From MathWorld--a Wolfram Web Resource. Retrieved February 2014. Check date values in: |accessdate= (help)
  2. C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Bijection" (PDF). addison-Wesley. p. 88. Retrieved February 2014. Check date values in: |accessdate= (help)CS1 maint: Uses authors parameter (link)
  3. Miller, Jeff (2010). "Earliest Uses of Some of the Words of Mathematics". Tripod. Retrieved February 2014. Check date values in: |accessdate= (help); |contribution= ignored (help)
  4. 4.0 4.1 Tanton, James (2005). Encyclopedia of Mathematics, Cardinality. Facts on File, New York. ISBN 0-8160-5124-0. (English)
  5. "Inverse of Bijection is Bijection". Retrieved February 2014. Check date values in: |accessdate= (help)
  6. "Injection iff Left Inverse". Retrieved February 2014. Check date values in: |accessdate= (help)
  7. "Surjection iff Right Inverse". Retrieved February 2014. Check date values in: |accessdate= (help)
  8. "Bijection iff Left and Right Inverse". Retrieved February 2014. Check date values in: |accessdate= (help)

Other websitesEdit