Bijective function

File:Gen bijection.svg

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 means: 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.

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.

File:Gen not surjection not injection.svg

Not a bijection. (It is not a surjection. It is not an injection.)

Basic properties

Formally:

[math]\displaystyle{ f:A \rightarrow B }[/math]  is a bijective function if  [math]\displaystyle{ \forall b \in B }[/math]  there is a unique  [math]\displaystyle{ a \in A }[/math]  such that  [math]\displaystyle{ f(a)=b \,. }[/math]

The element [math]\displaystyle{ b }[/math] is called the image of the element [math]\displaystyle{ a }[/math].

• The formal definition means: Every element of the codomain B is the image of exactly one element in the domain A.

The element [math]\displaystyle{ a }[/math] is called a pre-image of the element [math]\displaystyle{ b }[/math].

• The formal definition means: 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. We write #A=4.Template:Rp

• 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 are invertible by reversing the arrows. The new function is called the inverse function.

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,

 [math]\displaystyle{ f^{-1}(x) }[/math]  denotes the inverse function of the function f, but  [math]\displaystyle{ x^{-1}=\frac{1}{x} }[/math] 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 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(x₁) means x_o=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: Template:Nowrap beginf(x)=x³Template:Nowrap end is a bijection. Image 2 and image 5 thin yellow curve. Its inverse is the cube root function Template:Nowrap beginf(x)= ∛xTemplate:Nowrap end and it is also a bijection f(x):ℝ→ℝ. Image 5: thick green curve.

Example: The quadratic function Template:Nowrap beginf(x) = x²Template:Nowrap end 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.Template:Rp

• 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]\displaystyle{ f(x):[0,+\infty) \,\, \rightarrow \,\, [0,+\infty) }[/math]  defined by  [math]\displaystyle{ 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]\displaystyle{ f(x):[0,+\infty) \,\, \rightarrow \,\, [0,+\infty) }[/math]  defined by  [math]\displaystyle{ f(x) = \sqrt{x} }[/math]

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,+∞)

[math]\displaystyle{ f(x):\mathbf{R} \,\, \rightarrow \,\, (0,+\infty) }[/math]  defined by  [math]\displaystyle{ 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]\displaystyle{ f(x):(0,+\infty) \,\, \rightarrow \,\, \mathbf{R} }[/math]  defined by  [math]\displaystyle{ 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).

Bijection: every vertical line (in the domain) and every horizontal line (in the codomain) intersects exactly one point of the graph.
File:Line explicit ex.svg 1. Bijection. All slanted lines are bijections f(x):ℝ→ℝ.	File:Xto3.svg 2. Bijection. f(x):ℝ→ℝ. f(x)=x³.	File:Xto2.svg 3. Not a bijection. f(x):ℝ→ℝ. f(x)=x² is not a surjection. It is not an injection.
File:Logx inv.svg 4. Bijections. f(x):ℝ→ (0,+∞). f(x)=10x (thin yellow) and its inverse f(x):(0,+∞)→ℝ. f(x)=log10x (thick green).	File:Xto1over3.svg 5. Bijections. f(x):ℝ→ℝ. f(x)=x³ (thin yellow) and its inverse f(x)=∛x (thick green).	File:Xsqrt pos.svg 6. Bijections. f(x):[0,+∞)→[0,+∞). f(x)=x² (thin yellow) and its inverse f(x)=√x (thick green).

Related pages

• Function (mathematics)
• Surjective function
• Injective function
• Inverse function

Other websites

• Template:Cite web interactive quiz
• Template:Cite web interactive