Bijective function

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.^[1] 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").^[2][3]

The term bijection and the related terms surjection and injection were introduced by Nicholas Bourbaki.^[4] 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.)

Formally:

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

where the element ${\displaystyle b}$ is called the image of the element ${\displaystyle a}$, and the element ${\displaystyle a}$ the pre-image of the element ${\displaystyle b}$.

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 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.^[5]: 60

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 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.^[6]
• A function has an inverse function if and only if it is a bijection.^[7][8][9]

Note: The notation for the inverse function of f is confusing. Namely,

 ${\displaystyle f^{-1}(x)}$  denotes the inverse function of the function f, but  ${\displaystyle x^{-1}={\frac {1}{x}}}$ denotes the reciprocal value of the number x.

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 .

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).

• 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.

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.^[5]: 281
• 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,+∞)

${\displaystyle f(x):[0,+\infty )\,\,\rightarrow \,\,[0,+\infty )}$  defined by  ${\displaystyle f(x)=x^{2}}$

is a bijection. Image 6: thin yellow curve.

Example: The square root function defined on the restricted domain and codomain [0,+∞)

${\displaystyle f(x):[0,+\infty )\,\,\rightarrow \,\,[0,+\infty )}$  defined by  ${\displaystyle f(x)={\sqrt {x}}}$

is the bijection defined as the inverse function of the quadratic function: x². Image 6: thick green curve.

${\displaystyle f(x):\mathbf {R} \,\,\rightarrow \,\,(0,+\infty )}$  defined by  ${\displaystyle f(x)=a^{x}\,,\,\,a>1}$

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

${\displaystyle f(x):(0,+\infty )\,\,\rightarrow \,\,\mathbf {R} }$  defined by  ${\displaystyle f(x)=\log _{a}x\,,\,\,a>1}$

is the bijection defined as the inverse function of the exponential function: a^x. Image 4: thick green curve (a=10).

• Surjective function
• Injective function
• Inverse function

• "Injective, Surjective, Bijective". 2013. Retrieved 2013-12-01. interactive quiz
• "Injectivity, Surjectivity". Wolfram alpha. Archived from the original on 2013-11-25. Retrieved 2013-12-01. interactive