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| **Injection**. Maximum one arrow to each element in the codomain B (from an element in domain A). |

In mathematics, a **injective** function is a function *f* : *A* → *B* with the following property. For every element *b* in the codomain *B*, there is **at most** one element *a* in the domain *A* such that *f*(*a*)=*b*, or equivalently, distinct elements in the domain map to distinct elements in the codomain.

The term *injection* and the related terms *surjection* and *bijection* were introduced by Nicholas Bourbaki. In the 1930s, he and a group of other mathematicians published a series of books on modern advanced mathematics.

An injective function is often called a **1-1** (read "one-to-one") function. However, this is to be distinguished from a **1-1 correspondence**, which is a bijective function (both injective and surjective).

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| **Not an injection**. Two elements {X} and {Y} in the domain A are mapped to the same element {1} in the codomain B. |

Basic properties
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Formally:

{\displaystyle f:A\rightarrow B}  is an injective function if  {\displaystyle \forall a\_{1},\,a\_{2},\in A,\,\,\,\,a\_{1}\neq a\_{2}\,\,\Rightarrow \,\,f(a\_{1})\neq f(a\_{2})}  or equivalently
{\displaystyle f:A\rightarrow B}  is an injective function if  {\displaystyle \forall a\_{1},\,a\_{2},\in A,\,\,\,\,f(a\_{1})=f(a\_{2})\,\,\Rightarrow \,\,a\_{1}=a\_{2}}
The element a is called a **pre-image** of the element {\displaystyle b} if {\displaystyle f(a)=b} . Injections have one or none pre-images for every element *b* in *B*.

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 is written as #*A*=4.

If the cardinality of the codomain is less than the cardinality of the domain, then the function cannot be an injection. For example, there is no injection from 6 elements to 5 elements, since it is impossible to map 6 elements to 5 elements without a duplicate.

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 real numbers.)

* **Graphic meaning**: The function *f* is an injection if every horizontal line intersects the graph of *f* in **at most** one point.
* **Algebraic meaning**: The function *f* is an injection if *f*(*x*o)=*f*(*x*1) means *x*o=*x*1.

**Example:** The linear function of a slanted line is 1-1. That is, *y*=*ax*+*b* where *a*≠0 is an injection. (It is also a surjection and thus a bijection.)

Proof: Let *x*o and *x*1 be real numbers. Suppose the line maps these two *x*-values to the same *y*-value. This means *a*·*x*o+*b*=*a*·*x*1+*b*. Subtract *b* from both sides. We get *a*·*x*o=*a*·*x*1. Now divide both sides by *a* (remember *a*≠0). We get *x*o=*x*1. So we have proved the formal definition and the function *y*=*ax*+*b* where *a*≠0 is an injection.
**Example:** The polynomial function of third degree: 
*f*(*x*)=*x*3 is an injection. However, the polynomial function of third degree: 
*f*(*x*)=*x*3 –3x is not an injection.

Discussion 1: Any horizontal line intersects the graph of 
*f*(*x*)=*x*3 exactly once. (Also, it is a surjection.)

Discussion 2. Any horizontal line between *y*=-2 and *y*=2 intersects the graph in three points so this function is **not** an injection. (However, it is a surjection.)
**Example:** The quadratic function 
*f*(*x*) = *x*2 is **not** an injection. 

Discussion: Any horizontal line *y*=c where c>0 intersects the graph in two points. So this function is **not** an injection. (Also, it is not a surjection.)
Note: One can make a non-injective function into an injective function by eliminating part of the domain. We call this **restricting** the domain. For example, restrict the domain of *f*(*x*)=*x*² to non-negative numbers (positive numbers and zero). Define 

{\displaystyle f\_{/[0,+\infty )}(x):[0,+\infty )\rightarrow \mathbf {R} }  where  {\displaystyle f\_{/[0,+\infty )}(x)=x^{2}}
This function is now an injection. (See also restriction of a function.)

**Example:** The exponential function 
*f*(*x*) = 10*x* is an injection. (However, it is not a surjection.)

Discussion: Any horizontal line intersects the graph in at most one point. The horizontal lines *y*=c where c>0 cut it in exactly one point. The horizontal lines *y*=c where c≤0 do not cut the graph at any point.
Note: The fact that an exponential function is injective can be used in calculations. 

{\displaystyle a^{x\_{0}}=a^{x\_{1}}\,\,\Rightarrow \,\,x\_{0}=x\_{1},\,a>0} 
Example:  {\displaystyle 100=10^{x-3}\,\,\Rightarrow \,\,2=x-3\,\,\Rightarrow \,\,x=5} 

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| Injection: no horizontal line intersects more than one point of the graph |
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 Injection. *f*(*x*):ℝ→ℝ (and surjection) | 

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 Injection. *f*(*x*):ℝ→ℝ (and surjection) | 

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 Not an injection. *f*(*x*):ℝ→ℝ (is surjection) |
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 Not an injection. *f*(*x*):ℝ→ℝ (not surjection) | 

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Injection. *f*(*x*):ℝ→ℝ (not surjection) | 

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Injection. *f*(*x*):(0,+∞)→ℝ (and surjection) |

### Other examples

**Example:** The **logarithmic function base 10** *f*(*x*):(0,+∞)→ℝ defined by *f*(*x*)=log(*x*) or *y*=log10(*x*) is an injection (and a surjection). (This is the inverse function of 10*x*.)

**Example:** The function *f*:ℕ→ℕ that maps every natural number *n* to *2n* is an injection. Every even number has exactly one pre-image. Every odd number has no pre-image.

Related pages
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* Function (mathematics)
* Surjective function
* Bijective function

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