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| **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 **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.

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

Basic properties
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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
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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
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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,

 {\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*.

Examples
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### 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*(*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 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*)=*x*². 

* 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,+∞)

{\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: *x**2*. Image 6: thick green curve.

**Example:** The exponential function defined on the domain ℝ and the restricted codomain (0,+∞) 

{\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).


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| Bijection: every vertical line (in the domain) and every horizontal line (in the codomain) intersects exactly one point of the graph. |
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1. Bijection. All slanted lines are bijections *f*(*x*):ℝ→ℝ. | 

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2. Bijection. *f*(*x*):ℝ→ℝ. *f*(*x*)=*x*³. | 

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3. Not a bijection. *f*(*x*):ℝ→ℝ. *f*(*x*)=*x*² is not a surjection. It is not an injection. |
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4. Bijections. *f*(*x*):ℝ→ (0,+∞). *f*(*x*)=10*x* (thin yellow) and its inverse *f*(*x*):(0,+∞)→ℝ. *f*(*x*)=log10*x* (thick green). | 

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5. Bijections. *f*(*x*):ℝ→ℝ. *f*(*x*)=*x*³ (thin yellow) and its inverse *f*(*x*)=∛*x* (thick green). | 

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6. Bijections. *f*(*x*):[0,+∞)→[0,+∞). *f*(*x*)=*x*² (thin yellow) and its inverse *f*(*x*)=√*x* (thick green). |

Related pages
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* Surjective function
* Injective function
* Inverse function

Other websites
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* "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