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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
of one element of its domain. The term must not be confused with that refers to
bijective function In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
s, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures 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 and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s, an is also called a . However, in the more general context of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, 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

An injective function 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 $f\left(a\right) \neq f\left(b\right)$ in the contrapositive statement. Symbolically,$\forall a,b \in X, \;\; f(a)=f(b) \Rightarrow a=b,$ which is logically equivalent to the contrapositive,$\forall a, b \in X, \;\; a \neq b \Rightarrow f(a) \neq f(b).$

Examples

''For visual examples, readers are directed to the gallery section.'' * For any set $X$ and any subset $S \subseteq X,$ the inclusion map $S \to X$ (which sends any element $s \in S$ to itself) is injective. In particular, the identity function $X \to X$ is always injective (and in fact bijective). * If the domain of a function is the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, then the function is the
empty function In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the func ...
, which is injective. * If the domain of a function has one element (that is, it is a singleton set), 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 elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
$\R,$ 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 $x \in X$, $g\left(f\left(x\right)\right) = x$, then $f$ is injective. In this case, $g$ is called a
retraction Retraction or retract(ed) may refer to: Academia * Retraction in academic publishing, withdrawals of previously published academic journal articles Mathematics * Retraction (category theory) * Retract (group theory) * Retraction (topology) Huma ...
of $f.$ Conversely, $f$ is called a section 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 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 A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
from $J$ into $Y.$ More generally, injective
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
s are called
partial bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphi ...
s in the category Set of sets. * If $f : X \to Y$ is injective and $A$ is a subset of $X,$ then $f^\left(f\left(A\right)\right) = A.$ Thus, $A$ can be recovered from its
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
$f\left(A\right).$ * 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, and $f$ may be thought of as the
inclusion function In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
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 numbers. 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 infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
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, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
. * 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 In mathematics, the horizontal line test is a test used to determine whether a function is injective (i.e., one-to-one). In calculus A ''horizontal line'' is a straight, flat line that goes from left to right. Given a function f \colon \mathbb \t ...
. 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''.

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 Types of functions