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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 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 $f\left(a\right) = f\left(b\right),$ then $a = b$; that is, $f\left(a\right) = f\left(b\right)$ implies $a=b.$ Equivalently, if $a \neq b,$ then $f\left(a\right) \neq f\left(b\right).$ Symbolically, $\forall a,b \in X, \;\; f(a)=f(b) \Rightarrow a=b,$ which is logically equivalent to the
contrapositive 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 ...
, $\forall a, b \in X, \;\; a \neq b \Rightarrow f(a) \neq f(b).$

# Examples

* For any set $X$ and any subset $S \subseteq X,$ the
inclusion 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 ...
$S \to X$ (which sends any element $s \in S$ 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 ...

$X \to X$ is always injective (and in fact bijective). * If the domain of a function is the
empty set #REDIRECT Empty set#REDIRECT Empty set 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, cha ...

, 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 $f : \R \to \R$ defined by $f\left(x\right) = 2 x + 1$ is injective. * The function $g : \R \to \R$ defined by $g\left(x\right) = x^2$ is injective, because (for example) $g\left(1\right) = 1 = g\left(-1\right).$ However, if $g$ is redefined so that its domain is the non-negative real numbers ,+∞), then $g$ is injective. * The exponential function $\exp : \R \to \R$ defined by $\exp\left(x\right) = e^x$ is injective (but not surjective, as no real value maps to a negative number). * The natural logarithm function $\ln : \left(0, \infty\right) \to \R$ defined by $x \mapsto \ln x$ is injective. * The function $g : \R \to \R$ defined by $g\left(x\right) = x^n - x$ is not injective, since, for example, $g\left(0\right) = g\left(1\right) = 0.$ More generally, when $X$ and $Y$ are both the
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 an ...
$\R,$ then an injective function $f : \R \to \R$ 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 $f : X \to Y,$ if there is a function $g : Y \to X$ such that for every $x \in X,$ :$g\left(f\left(x\right)\right) = x$ ($f$ can be undone by $g$), then $f$ is injective. In this case, $g$ is called a
retraction In academic publishing Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal articles, books or thesis' form. The part of academic written o ...
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 $g\left(x\right)$ equals the unique pre-image of $x$ under $f$ if it exists and $g\left(x\right) = 1$ 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, $f \circ g,$ 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 $f : X \to Y$ into a bijective (hence invertible) function, it suffices to replace its codomain $Y$ by its actual range $J = f\left(X\right).$ That is, let $g : X \to J$ such that $g\left(x\right) = f\left(x\right)$ for all $x \in X$; then $g$ is bijective. Indeed, $f$ can be factored as $\operatorname_ \circ g,$ where $\operatorname_$ is 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, \ ...
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, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly one ...
s.

# Other properties

* If $f$ and $g$ are both injective then $f \circ g$ is injective. * If $g \circ f$ is injective, then $f$ is injective (but $g$ need not be). * $f : X \to Y$ is injective if and only if, given any functions $g,$ $h : W \to X$ whenever $f \circ g = f \circ h,$ then $g = h.$ In other words, injective functions are precisely the
monomorphism 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
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 $f : X \to Y$ is injective and $A$ is a
subset 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). ...

of $X,$ then $f^\left(f\left(A\right)\right) = A.$ 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 ...
$f\left(A\right).$ * If $f : X \to Y$ is injective and $A$ and $B$ are both subsets of $X,$ then $f\left(A \cap B\right) = f\left(A\right) \cap f\left(B\right).$ * Every function $h : W \to Y$ can be decomposed as $h = f \circ g$ 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 $h\left(W\right)$ of $h$ as a subset of the codomain $Y$ of $h.$ * If $f : X \to Y$ 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 $f : X \to Y$ 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 $f\left(x\right) = f\left(y\right),$ then $x = y.$ Here is an example: $f(x) = 2 x + 3$ Proof: Let $f : X \to Y.$ Suppose $f\left(x\right) = f\left(y\right).$ 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 the
horizontal 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 $f\left(x\right)$ in at most one point, then $f$ is injective or one-to-one.

* * * *

# References

* , p. 17 ''ff''. * , p. 38 ''ff''.