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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's
codomain In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
is the
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
between
algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
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 and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, 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, which is not also surjective. 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 \neq b, then f(a) \neq f(b) 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).An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, f:A\rightarrowtail B or f:A\hookrightarrow B), although some authors specifically reserve ↪ for an inclusion map.


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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
, then the function is the empty function, 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(x) = 2 x + 1 is injective. * The function g : \R \to \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 ,+∞), then g is injective. * The exponential function \exp : \R \to \R defined by \exp(x) = e^x is injective (but not Surjective function">surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
, as no real value maps to a negative number). * The natural logarithm function \ln : (0, \infty) \to \R defined by x \mapsto \ln x is injective. * The function g : \R \to \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
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
\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(f(x)) = x, then f is injective. The proof is that f(a) = f(b) \rightarrow g(f(a))=g(f(b)) \rightarrow a = b. In this case, g is called a retraction of f. Conversely, f is called a section of g. Conversely, every injection f with a non-empty domain has a left inverse g. It can be defined by choosing an element a in the domain of f and setting g(y) to the unique element of the pre-image f^ /math> (if it is non-empty) or to a (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 In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
, 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 image J = f(X). That is, let g : X \to J such that g(x) = f(x) 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 is the function (mathematics), function ι, \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota(x)=x. An inclusion map may also ...
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 the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
s are called partial bijections.


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 monomorphisms in the category Set of sets. * If f : X \to Y is injective and A is a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of X, then f^(f(A)) = A. Thus, A can be recovered from its
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
f(A). * If f : X \to Y is injective and A and B are both subsets of X, then f(A \cap B) = f(A) \cap f(B). * 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 is the function (mathematics), function ι, \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota(x)=x. An inclusion map may also ...
of the range h(W) 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 In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
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 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 (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...
. * 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 \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 the horizontal line test. If every horizontal line intersects the curve of f(x) in at most one point, then f is injective or one-to-one.


Gallery


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 Types of functions