# Injective function

 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 : AB 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.[1][2][3]

The term injection and the related terms surjection and bijection 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.

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).[5]

 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

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  ${\displaystyle \forall a_{1},\,a_{2},\in A,\,\,\,\,f(a_{1})=f(a_{2})\,\,\Rightarrow \,\,a_{1}=a_{2}}$[2]

The element ${\displaystyle a}$ 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

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.[6]

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):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(xo)=f(x1) means xo=x1.

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 xo and x1 be real numbers. Suppose the line maps these two x-values to the same y-value. This means a·xo+b=a·x1+b. Subtract b from both sides. We get a·xo=a·x1. Now divide both sides by a (remember a≠0). We get xo=x1. 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)=x3 is an injection. However, the polynomial function of third degree: f(x)=x3 –3x is not an injection.

Discussion 1: Any horizontal line intersects the graph of

f(x)=x3 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) = x2 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)=x² 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) = 10x 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:  ${\displaystyle 100=10^{x-3}\,\,\Rightarrow \,\,2=x-3\,\,\Rightarrow \,\,x=5}$
 Injection: no horizontal line intersects more than one point of the graph Injection. f(x):ℝ→ℝ (and surjection) Injection. f(x):ℝ→ℝ (and surjection) Not an injection. f(x):ℝ→ℝ (is surjection) Not an injection. f(x):ℝ→ℝ (not surjection) Injection. f(x):ℝ→ℝ (not surjection) 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 10x.)

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.

## References

1. "The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2020-09-08.
2. Weisstein, Eric W. "Injection". mathworld.wolfram.com. Retrieved 2020-09-08.
3. C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Onto Mapping" (PDF). Addison-Wesley. p. 568. Retrieved 2014-01-01.
4. Miller, Jeff (2010). "Earliest Uses of Some of the Words of Mathematics". Tripod. Retrieved 2014-02-01.
5. Weisstein, Eric W. "One-to-One". mathworld.wolfram.com. Retrieved 2020-09-08.
6. Tanton, James (2005). Encyclopedia of Mathematics, Cardinality. Facts on File, New York. p. 60. ISBN 0-8160-5124-0. (in English)