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:
where the element formula_5 is called the image of the element formula_6, and the element formula_6 the pre-image of the element formula_5. 
The formal definition can also be interpreted in two ways: 
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.) 
Note: The notation for the inverse function of "f" is confusing. Namely,
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.)
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.
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 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:
Example: Suppose our function "machine" is "f"("x")="x"². 
Bijections and their inverses.
Let "f"("x"):"A"→"B" where "A" and "B" are subsets of ℝ.
Example: The quadratic function defined on the restricted domain and codomain [0,+∞)
is a bijection. Image 6: thin yellow curve.
Example: The square root function defined on the restricted domain and codomain [0,+∞) 
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,+∞) 
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
is the bijection defined as the inverse function of the exponential function: "a""x". Image 4: thick green curve ("a"=10).