Injective function

"Injective" redirects here. For other uses, seeInjective moduleandInjective object.

Inmathematics,aninjective function(also known asinjection,orone-to-one function) is afunctionfthat mapsdistinctelements of its domain to distinct elements; that is,x₁≠x₂impliesf(x₁) ≠f(x₂).(Equivalently,f(x₁) =f(x₂)impliesx₁=x₂in the equivalentcontrapositivestatement.) In other words, every element of the function'scodomainis theimageofat mostone element of itsdomain.The termone-to-one functionmust not be confused withone-to-one correspondencethat refers tobijective functions,which are functions such that each element in the codomain is an image of exactly one element in the domain.

Ahomomorphismbetweenalgebraic structuresis a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular forvector spaces,aninjective homomorphismis also called amonomorphism.However, in the more general context ofcategory 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; seeHomomorphism § Monomorphismfor more details.

A function$f$that is not injective is sometimes called many-to-one.

Definition

An injective function, which is not alsosurjective.

Further information on notation:Function (mathematics) § Notation

Let$f$be a function whose domain is a set$X.$The function$f$is said to beinjectiveprovided 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 thecontrapositivestatement.

Symbolically,$$\forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,$$ which is logically equivalent to thecontrapositive,$$\forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).$$

Examples

For visual examples, readers are directed to thegallery section.

• For any set$X$and any subset$S\subseteq X,$theinclusion map$S\to X$(which sends any element$s\in S$to itself) is injective. In particular, theidentity function$X\to X$is always injective (and in fact bijective).
• If the domain of a function is theempty set,then the function is theempty function,which is injective.
• If the domain of a function has one element (that is, it is asingleton set), then the function is always injective.
• The function$f:\mathbb {R} \to \mathbb {R}$defined by$f(x)=2x+1$is injective.
• The function$g:\mathbb {R} \to \mathbb {R}$defined by$g(x)=x^{2}$isnotinjective, because (for example)$g(1)=1=g(-1).$However, if$g$is redefined so that its domain is the non-negative real numbers [0,+∞), then$g$is injective.
• Theexponential function$\exp:\mathbb {R} \to \mathbb {R}$defined by$\exp(x)=e^{x}$is injective (but not surjective, as no real value maps to a negative number).
• Thenatural logarithmfunction$\ln:(0,\infty )\to \mathbb {R}$defined by$x\mapsto \ln x$is injective.
• The function$g:\mathbb {R} \to \mathbb {R}$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 thereal line$\mathbb {R},$then an injective function$f:\mathbb {R} \to \mathbb {R}$is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as thehorizontal line test.

Injections can be undone

Functions withleft inversesare 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. In this case,$g$is called aretractionof$f.$Conversely,$f$is called asectionof$g.$

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$f^{-1}[y]$(if it is non-empty) or to$a$(otherwise).

The left inverse$g$is not necessarily aninverseof$f,$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 necessarilyinvertible,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$\operatorname {In} _{J,Y}\circ g,$where$\operatorname {In} _{J,Y}$is theinclusion functionfrom$J$into$Y.$

More generally, injectivepartial functionsare calledpartial bijections.

Other properties

See also:List of set identities and relations § Functions and sets

The composition of two injective functions is injective.

• 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.$In other words, injective functions are precisely themonomorphismsin thecategorySetof sets.
• If$f:X\to Y$is injective and$A$is asubsetof$X,$then$f^{-1}(f(A))=A.$Thus,$A$can be recovered from itsimage$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 uniqueup to isomorphism,and$f$may be thought of as theinclusion functionof 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 ofcardinal 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 theCantor–Bernstein–Schroeder theorem.)
• If both$X$and$Y$arefinitewith 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 anembedding.
• 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: $$f(x)=2x+3$$

Proof: Let$f:X\to Y.$Suppose$f(x)=f(y).$So$2x+3=2y+3$implies$2x=2y,$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.If every horizontal line intersects the curve of$f(x)$in at most one point, then$f$is injective or one-to-one.

Gallery

• 
  Aninjectivenon-surjective function (injection, not a bijection)
• 
  Aninjectivesurjective function (bijection)
• 
  A non-injective surjective function (surjection, not a bijection)
• 
  A non-injective non-surjective function (also not a bijection)

• 
  Not an injective function. Here$X_{1}$and$X_{2}$are subsets of$X,Y_{1}$and$Y_{2}$are subsets of$Y$:for two regions where the function is not injective because more than one domainelementcan map to a single range element. That is, it is possible formore than one$x$in$X$to map to thesame$y$in$Y.$
• 
  Making functions injective. The previous function$f:X\to Y$can be reduced to one or more injective functions (say)$f:X_{1}\to Y_{1}$and$f:X_{2}\to Y_{2},$shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule$f$has not changed – only the domain and range.$X_{1}$and$X_{2}$are subsets of$X,Y_{1}$and$Y_{2}$are subsets of$Y$:for two regions where the initial function can be made injective so that one domain element can map to a single range element. That is, only one$x$in$X$maps to one$y$in$Y.$
• 
  Injective functions. Diagramatic interpretation in theCartesian plane,defined by themapping$f:X\to Y,$where$y=f(x),$$X=$domain of function,$Y=$range of function,and$\operatorname {im} (f)$denotes image of$f.$Every one$x$in$X$maps to exactly one unique$y$in$Y.$The circled parts of the axes represent domain and range sets— in accordance with the standard diagrams above

See also

• Bijection, injection and surjection– Properties of mathematical functions
• Injective metric space– Type of metric space
• Monotonic function– Order-preserving mathematical function
• Univalent function– Mathematical concept

References

• Bartle, Robert G. (1976),The Elements of Real Analysis(2nd ed.), New York:John Wiley & Sons,ISBN978-0-471-05464-1,p. 17ff.
• Halmos, Paul R.(1974),Naive Set Theory,New York: Springer,ISBN978-0-387-90092-6,p. 38ff.

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