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Bijective function
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 : A → B that is both an injection and a surjection.^{[1]} This is equivalent to the following statement: 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 11 correspondence (read "onetoone correspondence).^{[2]}^{[3]}
The term bijection and the related terms surjection and injection were introduced by Nicholas Bourbaki.^{[4]} In the 1930s, he and a group of other mathematicians published a series of books on modern advanced mathematics.
Not a bijection. (It is not a surjection. It is not an injection.) 
Contents
Basic properties
Formally:
 [math]f:A \rightarrow B[/math] is a bijective function if [math]\forall b \in B[/math], there is a unique [math]a \in A[/math] such that [math]f(a)=b \,.[/math]
where the element [math]b[/math] is called the image of the element [math]a[/math], and the element [math]a[/math] the preimage of the element [math]b[/math].
The formal definition can also be interpreted in two ways:
 Every element of the codomain B is the image of exactly one element in the domain A.
 Every element of the codomain B has exactly one preimage in the domain A.
Note: Surjection means minimum one preimage. Injection means maximum one preimage. So bijection means exactly one preimage.
Cardinality
Cardinality is the number of elements in a set. The cardinality of A={X,Y,Z,W} is 4. This can be written as #A=4.^{[5]}^{:60}
By 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 functions
Bijections and inverse functions are related to each other, in that a bijection is invertible, can be turned into its inverse function by reversing the arrows.
Formally: Let f : A → B be a bijection. The inverse function g : B → A 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.^{[6]}
 A function has an inverse function if and only if it is a bijection.^{[7]}^{[8]}^{[9]}
Note: The notation for the inverse function of f is confusing. Namely,
 [math]f^{1}(x)[/math] denotes the inverse function of the function f, but [math]x^{1}=\frac{1}{x}[/math] denotes the reciprocal value of the number x.
Examples
Elementary functions
Let f(x):ℝ→ℝ be a realvalued function y=f(x) of a realvalued 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 y_{o} we can find at least one real number x_{o} such that y_{o}=f(x_{o}) and if f(x_{o})=f(x_{1}) means x_{o}=x_{1} .
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)=x^{3} 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) = x^{2} 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 nonnegative 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 inverses
Let f(x):A→B 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.^{[5]}^{: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,+∞)
 [math]f(x):[0,+\infty) \,\, \rightarrow \,\, [0,+\infty)[/math] defined by [math]f(x) = x^2[/math]
is a bijection. Image 6: thin yellow curve.
Example: The square root function defined on the restricted domain and codomain [0,+∞)
 [math]f(x):[0,+\infty) \,\, \rightarrow \,\, [0,+\infty)[/math] defined by [math]f(x) = \sqrt{x}[/math]
is the bijection defined as the inverse function of the quadratic function: x^{2}. Image 6: thick green curve.
Example: The exponential function defined on the domain ℝ and the restricted codomain (0,+∞)
 [math]f(x):\mathbf{R} \,\, \rightarrow \,\, (0,+\infty)[/math] defined by [math]f(x) = a^x \, ,\,\, a\gt 1[/math]
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
 [math]f(x):(0,+\infty) \,\, \rightarrow \,\, \mathbf{R}[/math] defined by [math]f(x) = \log_a x \, ,\,\, a\gt 1[/math]
is the bijection defined as the inverse function of the exponential function: a^{x}. Image 4: thick green curve (a=10).
Related pages
References
 ↑ "The Definitive Glossary of Higher Mathematical Jargon" (in enUS). 20190801. https://mathvault.ca/mathglossary/.
 ↑ Weisstein, Eric W.. "Bijective" (in en). https://mathworld.wolfram.com/Bijective.html.
 ↑ "Oxford Concise Dictionary of Mathematics, Bijection" (in English). addisonWesley. 2009. p. 88. http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf. Retrieved 20140201.
 ↑ Miller, Jeff (2010). "Earliest Uses of Some of the Words of Mathematics" (in English). Tripod. http://jeff560.tripod.com/i.html. Retrieved 20140201.
 ↑ ^{5.0} ^{5.1} Tanton, James (2005). Encyclopedia of Mathematics, Cardinality. Facts on File, New York. . (in English)
 ↑ "Inverse of Bijection is Bijection" (in English). http://www.proofwiki.org/wiki/Inverse_of_Bijection_is_Bijection. Retrieved 20140201.
 ↑ "Injection iff Left Inverse" (in English). http://www.proofwiki.org/wiki/Injection_iff_Left_Inverse. Retrieved 20140201.
 ↑ "Surjection iff Right Inverse" (in English). http://www.proofwiki.org/wiki/Surjection_iff_Right_Inverse. Retrieved 20140201.
 ↑ "Bijection iff Left and Right Inverse" (in English). http://www.proofwiki.org/wiki/Bijection_iff_Left_and_Right_Inverse. Retrieved 20140201.
Other websites
 "Injective, Surjective, Bijective" (in English). 2013. http://www.mathsisfun.com/sets/injectivesurjectivebijective.html. Retrieved 20131201. interactive quiz
 "Injectivity, Surjectivity" (in English). Wolfram alpha. https://www.wolframalpha.com/examples/InjectivitySurjectivity.html. Retrieved 20131201. interactive
