Bijective function facts for kids
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. 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).
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.
Not a bijection. (It is not a surjection. It is not an injection.) 
Contents
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 preimage 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 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.
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 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.
 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: x^{2}. 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: a^{x}. Image 4: thick green curve (a=10).
Related pages
interactive quiz interactive
See also
In Spanish: Función biyectiva para niños