Injective function

In mathematics, a **injective** function is a function *f*&nbsp;:&nbsp;*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).

## Basic properties
Formally:
:f:A \rightarrow B&nbsp; is an injective function if  &nbsp;\forall a_1, \,a_2 ,  \in A, \,\,\,\, a_1 \ne a_2 \,\, \Rightarrow \,\, f(a_1) \ne f(a_2)&nbsp; or equivalently
:f:A \rightarrow B&nbsp; is an injective function if &nbsp;\forall a_1, \,a_2 ,  \in A, \,\,\,\,f(a_1)=f(a_2) \,\, \Rightarrow \,\, a_1=a_2

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 
### Elementary functions
Let *f*(*x*):&#8477;→&#8477; 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*&ne;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*&middot;*x*o+*b*=*a*&middot;*x*1+*b*. Subtract *b* from both sides. We get *a*&middot;*x*o=*a*&middot;*x*1. Now divide both sides by *a* (remember *a*&ne;0).  We get *x*o=*x*1. So we have proved the formal definition and the function *y*=*ax*+*b* where *a*&ne;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 &ndash;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&gt;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*&sup2; to non-negative numbers (positive numbers and zero). Define 
:f_{/[0,+\infty)}(x):[0,+\infty) \rightarrow \mathbf{R}&nbsp; where &nbsp;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&gt;0 cut it in exactly one point. The horizontal lines *y*=c where c&le;0 do not cut the graph at any point.

Note: The fact that an exponential function is injective can be used in calculations. 
:a^{x_0}=a^{x_1} \,\, \Rightarrow  \,\, x_0=x_1, \, a>0 &nbsp;  
:Example: &nbsp;100=10^{x-3} \,\, \Rightarrow  \,\, 2=x-3 \,\, \Rightarrow  \,\,  x=5&nbsp;

### Other examples 
**Example:** The **logarithmic function base 10**  *f*(*x*):→&#8477; 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*:&#8469;→&#8469; 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
*Function (mathematics)
*Surjective function
*Bijective function

## References
## Other websites
*  interactive quiz
*  interactive