In

, then is injective.
* The exponential function">,+∞) , then is injective.
* The exponential function">, then $g$ is injective.
* The exponential function $\backslash exp\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $\backslash exp(x)\; =\; e^x$ is injective (but not surjective, as no real value maps to a negative number).
* The natural logarithm function $\backslash ln\; :\; (0,\; \backslash infty)\; \backslash to\; \backslash R$ defined by $x\; \backslash mapsto\; \backslash ln\; x$ is injective.
* The function $g\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $g(x)\; =\; x^n\; -\; x$ is not injective, since, for example, $g(0)\; =\; g(1)\; =\; 0.$
More generally, when $X$ and $Y$ are both the

Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.

Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injective functions

{{Authority control Functions and mappings

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, an injective function (also known as injection, or one-to-one function) is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

that maps distinct elements to distinct elements; that is, implies . In other words, every element of the function's codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of one element of its domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

. The term must not be confused with that refers to bijective function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, which are functions such that each element in the codomain is an image of exactly one element in the domain.
A homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

between algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, an is also called a . However, in the more general context of category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details.
A function $f$ that is not injective is sometimes called many-to-one.
Definition

Let $f$ be a function whose domain is a set $X.$ The function $f$ is said to be injective provided that for all $a$ and $b$ in $X,$ if $f(a)\; =\; f(b),$ then $a\; =\; b$; that is, $f(a)\; =\; f(b)$ implies $a=b.$ Equivalently, if $a\; \backslash neq\; b,$ then $f(a)\; \backslash neq\; f(b).$ Symbolically, $$\backslash forall\; a,b\; \backslash in\; X,\; \backslash ;\backslash ;\; f(a)=f(b)\; \backslash Rightarrow\; a=b,$$ which is logically equivalent to thecontrapositive
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

,
$$\backslash forall\; a,\; b\; \backslash in\; X,\; \backslash ;\backslash ;\; a\; \backslash neq\; b\; \backslash Rightarrow\; f(a)\; \backslash neq\; f(b).$$
Examples

* For any set $X$ and any subset $S\; \backslash subseteq\; X,$ theinclusion map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$S\; \backslash to\; X$ (which sends any element $s\; \backslash in\; S$ to itself) is injective. In particular, the identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

$X\; \backslash to\; X$ is always injective (and in fact bijective).
* If the domain of a function is the empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

, then the function is the empty function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, which is injective.
* If the domain of a function has one element (that is, it is a singleton set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

), then the function is always injective.
* The function $f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $f(x)\; =\; 2\; x\; +\; 1$ is injective.
* The function $g\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $g(x)\; =\; x^2$ is injective, because (for example) $g(1)\; =\; 1\; =\; g(-1).$ However, if $g$ is redefined so that its domain is the non-negative real numbers real line
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$\backslash R,$ then an injective function $f\; :\; \backslash R\; \backslash to\; \backslash R$ is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the .
Injections can be undone

Functions with left inverses are always injections. That is, given $f\; :\; X\; \backslash to\; Y,$ if there is a function $g\; :\; Y\; \backslash to\; X$ such that for every $x\; \backslash in\; X,$ :$g(f(x))\; =\; x$ ($f$ can be undone by $g$), then $f$ is injective. In this case, $g$ is called aretraction
In academic publishing, a retraction is the action by which a published paper in an academic journal is removed from the journal. Online journals typically remove the retracted article from online access.
Procedure
A retraction may be initiate ...

of $f.$ Conversely, $f$ is called a section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...

of $g.$
Conversely, every injection $f$ with non-empty domain has a left inverse $g,$ which can be defined by fixing an element $a$ in the domain of $f$ so that $g(x)$ equals the unique pre-image of $x$ under $f$ if it exists and $g(x)\; =\; 1$ otherwise.
The left inverse $g$ is not necessarily an inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when add ...

of $f,$ because the composition in the other order, $f\; \backslash circ\; g,$ may differ from the identity on $Y.$ In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.
Injections may be made invertible

In fact, to turn an injective function $f\; :\; X\; \backslash to\; Y$ into a bijective (hence invertible) function, it suffices to replace its codomain $Y$ by its actual range $J\; =\; f(X).$ That is, let $g\; :\; X\; \backslash to\; J$ such that $g(x)\; =\; f(x)$ for all $x\; \backslash in\; X$; then $g$ is bijective. Indeed, $f$ can be factored as $\backslash operatorname\_\; \backslash circ\; g,$ where $\backslash operatorname\_$ is theinclusion function
In mathematics, if is a subset of , then the inclusion map (also inclusion function, insertion, or canonical injection) is the function (mathematics), function that sends each element of to , treated as an element of :
:\iota: A\rightarrow B, \ ...

from $J$ into $Y.$
More generally, injective partial function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s are called partial bijection
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s.
Other properties

* If $f$ and $g$ are both injective then $f\; \backslash circ\; g$ is injective. * If $g\; \backslash circ\; f$ is injective, then $f$ is injective (but $g$ need not be). * $f\; :\; X\; \backslash to\; Y$ is injective if and only if, given any functions $g,$ $h\; :\; W\; \backslash to\; X$ whenever $f\; \backslash circ\; g\; =\; f\; \backslash circ\; h,$ then $g\; =\; h.$ In other words, injective functions are precisely themonomorphism
In the context of abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...

s in the category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

Set of sets.
* If $f\; :\; X\; \backslash to\; Y$ is injective and $A$ is a subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of $X,$ then $f^(f(A))\; =\; A.$ Thus, $A$ can be recovered from its image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

$f(A).$
* If $f\; :\; X\; \backslash to\; Y$ is injective and $A$ and $B$ are both subsets of $X,$ then $f(A\; \backslash cap\; B)\; =\; f(A)\; \backslash cap\; f(B).$
* Every function $h\; :\; W\; \backslash to\; Y$ can be decomposed as $h\; =\; f\; \backslash circ\; g$ for a suitable injection $f$ and surjection $g.$ This decomposition is unique up to isomorphism Two mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is a ...

, and $f$ may be thought of as the inclusion function
In mathematics, if is a subset of , then the inclusion map (also inclusion function, insertion, or canonical injection) is the function (mathematics), function that sends each element of to , treated as an element of :
:\iota: A\rightarrow B, \ ...

of the range $h(W)$ of $h$ as a subset of the codomain $Y$ of $h.$
* If $f\; :\; X\; \backslash to\; Y$ is an injective function, then $Y$ has at least as many elements as $X,$ in the sense of cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

s. In particular, if, in addition, there is an injection from $Y$ to $X,$ then $X$ and $Y$ have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
* If both $X$ and $Y$ are finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

with the same number of elements, then $f\; :\; X\; \backslash to\; Y$ is injective if and only if $f$ is surjective (in which case $f$ is bijective).
* An injective function which is a homomorphism between two algebraic structures is an embedding
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

.
* Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function $f$ is injective can be decided by only considering the graph (and not the codomain) of $f.$
Proving that functions are injective

A proof that a function $f$ is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if $f(x)\; =\; f(y),$ then $x\; =\; y.$ Here is an example: $$f(x)\; =\; 2\; x\; +\; 3$$ Proof: Let $f\; :\; X\; \backslash to\; Y.$ Suppose $f(x)\; =\; f(y).$ So $2\; x\; +\; 3\; =\; 2\; y\; +\; 3$ implies $2\; x\; =\; 2\; y,$ which implies $x\; =\; y.$ Therefore, it follows from the definition that $f$ is injective. There are multiple other methods of proving that a function is injective. For example, in calculus if $f$ is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if $f$ is a linear transformation it is sufficient to show that the kernel of $f$ contains only the zero vector. If $f$ is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list. A graphical approach for a real-valued function $f$ of a real variable $x$ is thehorizontal line test
Horizontal may refer to:
* Horizontal plane, in astronomy, geography, geometry and other sciences and contexts
*Horizontal coordinate system
The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon ...

. If every horizontal line intersects the curve of $f(x)$ in at most one point, then $f$ is injective or one-to-one.
See also

* * * *Notes

References

* , p. 17 ''ff''. * , p. 38 ''ff''.External links

Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.

Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injective functions

{{Authority control Functions and mappings

Basic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...

Types of functions