Injective function
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Inmathematics,aninjective function(also known asinjection,orone-to-one function[1]) 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.[2]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.[3]This is thus a theorem that they are equivalent for algebraic structures; seeHomomorphism § Monomorphismfor more details.
A functionthat is not injective is sometimes called many-to-one.[2]
Definition
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/02/Injection.svg/220px-Injection.svg.png)
Letbe a function whose domain is a setThe functionis said to beinjectiveprovided that for allandinifthen;that is,impliesEquivalently, ifthenin thecontrapositivestatement.
Symbolically, which is logically equivalent to thecontrapositive,[4]
Examples
[edit]For visual examples, readers are directed to thegallery section.
- For any setand any subsettheinclusion map(which sends any elementto itself) is injective. In particular, theidentity functionis 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 functiondefined byis injective.
- The functiondefined byisnotinjective, because (for example)However, ifis redefined so that its domain is the non-negative real numbers [0,+∞), thenis injective.
- Theexponential functiondefined byis injective (but not surjective, as no real value maps to a negative number).
- Thenatural logarithmfunctiondefined byis injective.
- The functiondefined byis not injective, since, for example,
More generally, whenandare both thereal linethen an injective functionis one whose graph is never intersected by any horizontal line more than once. This principle is referred to as thehorizontal line test.[2]
Injections can be undone
[edit]Functions withleft inversesare always injections. That is, givenif there is a functionsuch that for every,,thenis injective. In this case,is called aretractionofConversely,is called asectionof
Conversely, every injectionwith a non-empty domain has a left inverse.It can be defined by choosing an elementin the domain ofand settingto the unique element of the pre-image(if it is non-empty) or to(otherwise).[5]
The left inverseis not necessarily aninverseofbecause the composition in the other order,may differ from the identity onIn 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
[edit]In fact, to turn an injective functioninto a bijective (hence invertible) function, it suffices to replace its codomainby its actual imageThat is, letsuch thatfor all;thenis bijective. Indeed,can be factored aswhereis theinclusion functionfrominto
More generally, injectivepartial functionsare calledpartial bijections.
Other properties
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Injective_composition2.svg/300px-Injective_composition2.svg.png)
- Ifandare both injective thenis injective.
- Ifis injective, thenis injective (butneed not be).
- is injective if and only if, given any functionswheneverthenIn other words, injective functions are precisely themonomorphismsin thecategorySetof sets.
- Ifis injective andis asubsetofthenThus,can be recovered from itsimage
- Ifis injective andandare both subsets ofthen
- Every functioncan be decomposed asfor a suitable injectionand surjectionThis decomposition is uniqueup to isomorphism,andmay be thought of as theinclusion functionof the rangeofas a subset of the codomainof
- Ifis an injective function, thenhas at least as many elements asin the sense ofcardinal numbers.In particular, if, in addition, there is an injection fromtothenandhave the same cardinal number. (This is known as theCantor–Bernstein–Schroeder theorem.)
- If bothandarefinitewith the same number of elements, thenis injective if and only ifis surjective (in which caseis 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 functionis injective can be decided by only considering the graph (and not the codomain) of
Proving that functions are injective
[edit]A proof that a functionis 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 ifthen[6]
Here is an example:
Proof: LetSupposeSoimplieswhich impliesTherefore, it follows from the definition thatis injective.
There are multiple other methods of proving that a function is injective. For example, in calculus ifis 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, ifis a linear transformation it is sufficient to show that the kernel ofcontains only the zero vector. Ifis 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 functionof a real variableis thehorizontal line test.If every horizontal line intersects the curve ofin at most one point, thenis injective or one-to-one.
Gallery
[edit]-
Not an injective function. Hereandare subsets ofandare subsets of: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 oneinto map to thesamein
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Making functions injective. The previous functioncan be reduced to one or more injective functions (say)andshown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rulehas not changed – only the domain and range.andare subsets ofandare subsets of: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 oneinmaps to onein
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Injective functions. Diagramatic interpretation in theCartesian plane,defined by themappingwheredomain of function,range of function,anddenotes image ofEvery oneinmaps to exactly one uniqueinThe circled parts of the axes represent domain and range sets— in accordance with the standard diagrams above
See also
[edit]- 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
Notes
[edit]- ^Sometimesone-one function,in Indian mathematical education."Chapter 1:Relations and functions"(PDF).Archived(PDF)from the original on Dec 26, 2023 – via NCERT.
- ^abc"Injective, Surjective and Bijective".Math is Fun.Retrieved2019-12-07.
- ^"Section 7.3 (00V5): Injective and surjective maps of presheaves".The Stacks project.Retrieved2019-12-07.
- ^Farlow, S. J."Section 4.2 Injections, Surjections, and Bijections"(PDF).Mathematics & Statistics - University of Maine.Archived fromthe original(PDF)on Dec 7, 2019.Retrieved2019-12-06.
- ^Unlike the corresponding statement that every surjective function has a right inverse, this does not require theaxiom of choice,as the existence ofis implied by the non-emptiness of the domain. However, this statement may fail in less conventional mathematics such asconstructive mathematics.In constructive mathematics, the inclusionof the two-element set in the reals cannot have a left inverse, as it would violateindecomposability,by giving aretractionof the real line to the set {0,1}.
- ^Williams, Peter (Aug 21, 1996)."Proving Functions One-to-One".Department of Mathematics at CSU San Bernardino Reference Notes Page.Archived fromthe originalon 4 June 2017.
References
[edit]- 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
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