Bijective function

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.
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 a0 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 bisection.
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).
Related pages
Surjective function
Injective function
Inverse function
References
Other websites
interactive quiz
interactive
Functions and mappings
Algebra