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# 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 distinguish 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:

$f:A \rightarrow B$  is an injective function if  $\forall a_1, \,a_2 , \in A, \,\,\,\, a_1 \ne a_2 \,\, \Rightarrow \,\, f(a_1) \ne f(a_2)$  or equivalently
$f:A \rightarrow B$  is an injective function if  $\forall a_1, \,a_2 , \in A, \,\,\,\,f(a_1)=f(a_2) \,\, \Rightarrow \,\, a_1=a_2$[2]

The element $a$ is called a pre-image of the element $b$ if $f(a)=b$ . 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

$f_{/[0,+\infty)}(x):[0,+\infty) \rightarrow \mathbf{R}$  where  $f_{/[0,+\infty)}(x) = x^2$

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.

$a^{x_0}=a^{x_1} \,\, \Rightarrow \,\, x_0=x_1, \, a\gt 0$
Example:  $100=10^{x-3} \,\, \Rightarrow \,\, 2=x-3 \,\, \Rightarrow \,\, x=5$
 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" (in en-US). 2019-08-01.
2. Weisstein, Eric W.. "Injection" (in en).
3. C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Onto Mapping" (in English). Addison-Wesley. p. 568. Retrieved 2014-01-01.
4. Miller, Jeff (2010). "Earliest Uses of Some of the Words of Mathematics" (in English). Tripod. Retrieved 2014-02-01.
5. Weisstein, Eric W.. "One-to-One" (in en).
6. Tanton, James (2005). Encyclopedia of Mathematics, Cardinality. Facts on File, New York. p. 60. .  (in English)