Bijective function

Image:Gen bijection.svgBijection. 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. 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 1-1 correspondence (read "one-to-one correspondence").

The term bijection and the related terms surjection and injection were introduced by Nicholas Bourbaki. In the 1930s, he and a group of other mathematicians published a series of books on modern advanced mathematics.
Image:Gen not surjection not injection.svgNot a bijection. (It is not a surjection. It is not an injection.) 

Basic properties
Formally:
 is a bijective function if , there is a unique  such that  
where the element  is called the image of the element , and the element  the pre-image of the element . 

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 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.

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. 

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.
A function has an inverse function if and only if it is a bijection.

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.

Examples 
Elementary functions
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 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 bijection.
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. 135px1. Bijection. All slanted lines are bijections f(x):ℝ→ℝ.  220px2. Bijection. f(x):ℝ→ℝ. f(x)=x³. 220px3. Not a bijection. f(x):ℝ→ℝ. f(x)=x² is not a surjection. It is not an injection.220px4. Bijections. f(x):ℝ→ (0,+∞). f(x)=10x (thin yellow) and its inverse f(x):(0,+∞)→ℝ. f(x)=log10x (thick green).   220px5. Bijections. f(x):ℝ→ℝ. f(x)=x³ (thin yellow) and its inverse f(x)=∛x (thick green). 220px6. Bijections. f(x):[0,+∞)→[0,+∞). f(x)=x² (thin yellow) and its inverse f(x)=√x (thick green).  

Related pages
Surjective function
Injective function
Inverse function

References

Other websites
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Category:Functions and mappings
Category:Algebra