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Injective function

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From Wikipedia, the free encyclopedia

Function that preserves distinctness

"Injective" redirects here. For other uses, seeInjective module andInjective object.

Function
x ↦f (x)
History of the function concept
Types bydomain andcodomain
`X` →𝔹 𝔹 →`X` 𝔹ⁿ →`X` `X` →ℤ ℤ →`X` `X` →ℝ ℝ →`X` ℝⁿ →`X` `X` →ℂ ℂ →`X` ℂⁿ →`X`
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions
v t e

Inmathematics, aninjective function (also known asinjection, orone-to-one function^[1]) is afunctionf that mapsdistinct elements of its domain to distinct elements of its codomain; that is,x₁ ≠x₂ impliesf(x₁) ≠f(x₂) (equivalently bycontraposition,f(x₁) =f(x₂) impliesx₁ =x₂). In other words, every element of the function'scodomain is theimage ofat most one element of itsdomain.^[2] The termone-to-one function must not be confused withone-to-one correspondence that refers tobijective functions, which are functions such that each element in the codomain is an image ofexactly one element in the domain.

Ahomomorphism betweenalgebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular forvector spaces, aninjective homomorphism is 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 § Monomorphism for more details.

A function $f {\displaystyle f}$ that is not injective is sometimes called many-to-one.^[2]

Definition

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The sets X = {1, 2, 3} and Y = {A, B, C, D}, and a function mapping 1 to D, 2 to B, and 3 to A. — An injective function, which is not alsosurjective

Further information on notation:Function (mathematics) § Notation

Let $f {\displaystyle f}$ be a function whose domain is a set⁠ $X {\displaystyle X}$ ⁠. The function $f {\displaystyle f}$ is said to beinjective provided that for all $a {\displaystyle a}$ and $b {\displaystyle b}$ in $X, {\displaystyle X,}$ 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 thecontrapositive statement.

Symbolically, $\forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,$ which is logically equivalent to thecontrapositive,^[4] $\forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).$ 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 aninclusion map.^[5]

Examples

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For visual examples, readers are directed to thegallery section.

For any set $X {\displaystyle X}$ 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 $g(x)=x^{2}$ isnot injective, because (for example) $g(1)=1=g(-1).$ However, if $g {\displaystyle g}$ is redefined so that its domain is the non-negative real numbers[0, +∞), then $g {\displaystyle g}$ is injective.
Theexponential function $\exp :\mathbb {R} \to \mathbb {R}$ defined by $\exp(x)=e^{x}$ is injective (but notsurjective, as no real value maps to a negative number).
Thenatural logarithm function $\ln :(0,\infty )\to \mathbb {R}$ 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 $X {\displaystyle X}$ and $Y {\displaystyle Y}$ 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.^[2]

Injections can be undone

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Functions withleft 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 $f {\displaystyle f}$ is injective. The proof is that $f(a)=f(b)\rightarrow g(f(a))=g(f(b))\rightarrow a=b.$

In this case, $g {\displaystyle g}$ is called aretraction of⁠ $f {\displaystyle f}$ ⁠. Conversely, $f {\displaystyle f}$ is called asection of⁠ $g {\displaystyle g}$ ⁠.For example: $f:\mathbb {R} \rightarrow \mathbb {R} ^{2},x\mapsto (1,m)^{\intercal }x$ is retracted by⁠ $g:y\mapsto {\frac {(1,m)}{1+m^{2}}}y$ ⁠.

Conversely, every injection $f {\displaystyle f}$ with a non-empty domain has a left inverse $g {\displaystyle g}$ . It can be defined by choosing an element $a {\displaystyle a}$ in the domain of $f {\displaystyle f}$ and setting $g(y)$ to the unique element of the pre-image $f^{-1}[y]$ (if it is non-empty) or to $a {\displaystyle a}$ (otherwise).^[6]

The left inverse $g {\displaystyle g}$ is not necessarily aninverse of $f, {\displaystyle f,}$ because the composition in the other order,⁠ $f\circ g$ ⁠, may differ from the identity on⁠ $Y {\displaystyle Y}$ ⁠. 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

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In fact, to turn an injective function $f:X\to Y$ into a bijective (hence invertible) function, it suffices to replace its codomain $Y {\displaystyle Y}$ 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 $g {\displaystyle g}$ is bijective. Indeed, $f {\displaystyle f}$ can be factored as⁠ $\operatorname {In} _{J,Y}\circ g$ ⁠, where $\operatorname {In} _{J,Y}$ is theinclusion function from $J {\displaystyle J}$ into⁠ $Y {\displaystyle Y}$ ⁠.

More generally, injectivepartial functions are calledpartial bijections.

Other properties

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The composition of two injective functions is injective.

If $f {\displaystyle f}$ and $g {\displaystyle g}$ are both injective then $f\circ g$ is injective.
If $g\circ f$ is injective, then $f {\displaystyle f}$ is injective (but $g {\displaystyle g}$ need not be).
$f:X\to Y$ is injective if and only if, given any functions⁠ $g {\displaystyle g}$ ⁠, $h:W\to X$ whenever⁠ $f\circ g=f\circ h$ ⁠, then⁠ $g=h$ ⁠. In other words, injective functions are precisely themonomorphisms in thecategorySet of sets.
If $f:X\to Y$ is injective and $A {\displaystyle A}$ is asubset of⁠ $X {\displaystyle X}$ ⁠, then⁠ $f^{-1}(f(A))=A$ ⁠. Thus, $A {\displaystyle A}$ can be recovered from itsimage⁠ $f(A)$ ⁠.
If $f:X\to Y$ is injective and $A {\displaystyle A}$ and $B {\displaystyle B}$ are both subsets of⁠ $X {\displaystyle X}$ ⁠, 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 $f {\displaystyle f}$ and surjection⁠ $g {\displaystyle g}$ ⁠. This decomposition is uniqueup to isomorphism, and $f {\displaystyle f}$ may be thought of as theinclusion function of the range $h(W)$ of $h {\displaystyle h}$ as a subset of the codomain $Y {\displaystyle Y}$ of⁠ $h {\displaystyle h}$ ⁠.
If $f:X\to Y$ is an injective function, then $Y {\displaystyle Y}$ has at least as many elements as $X, {\displaystyle X,}$ in the sense ofcardinal numbers. In particular, if, in addition, there is an injection from⁠ $Y {\displaystyle Y}$ ⁠ to⁠ $X {\displaystyle X}$ ⁠, then $X {\displaystyle X}$ and $Y {\displaystyle Y}$ have the same cardinal number. (This is known as theCantor–Bernstein–Schroeder theorem.)
If both $X {\displaystyle X}$ and $Y {\displaystyle Y}$ arefinite with the same number of elements, then $f:X\to Y$ is injective if and only if $f {\displaystyle f}$ is surjective (in which case $f {\displaystyle f}$ 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 $f {\displaystyle f}$ is injective can be decided by only considering the graph (and not the codomain) of⁠ $f {\displaystyle f}$ ⁠.

Proving that functions are injective

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A proof that a function $f {\displaystyle f}$ 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 $x=y$ ⁠.^[7]

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 $f {\displaystyle f}$ is injective.

There are multiple other methods of proving that a function is injective. For example, in calculus if $f {\displaystyle f}$ 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 $f {\displaystyle f}$ is a linear transformation it is sufficient to show that the kernel of $f {\displaystyle f}$ contains only the zero vector. If $f {\displaystyle f}$ 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 $f {\displaystyle f}$ of a real variable $x {\displaystyle x}$ is thehorizontal line test. If every horizontal line intersects the curve of $f(x)$ in at most one point, then $f {\displaystyle f}$ is injective or one-to-one.

Gallery

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Aninjective non-surjective function (injection, not a bijection)
Aninjective surjective 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 $Y {\displaystyle Y}$ ⁠: for two regions where the function is not injective because more than one domainelement can map to a single range element. That is, it is possible formore than one $x {\displaystyle x}$ in $X {\displaystyle X}$ to map to thesame $y {\displaystyle y}$ in⁠ $Y {\displaystyle Y}$ ⁠.
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 $f {\displaystyle f}$ 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 $Y {\displaystyle Y}$ ⁠: 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 $x {\displaystyle x}$ in $X {\displaystyle X}$ maps to one $y {\displaystyle y}$ in⁠ $Y {\displaystyle Y}$ ⁠.
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 $f {\displaystyle f}$ ⁠. Every one $x {\displaystyle x}$ in $X {\displaystyle X}$ maps to exactly one unique $y {\displaystyle y}$ in⁠ $Y {\displaystyle Y}$ ⁠. The circled parts of the axes represent domain and range sets — in accordance with the standard diagrams above

Notes

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^Sometimesone-one function in Indian mathematical education."Chapter 1: Relations and functions"(PDF).Archived(PDF) from the original on December 26, 2023 – via NCERT.
^^a ^b ^c"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.
^"What are usual notations for surjective, injective and bijective functions?".Mathematics Stack Exchange. Retrieved2024-11-24.
^Unlike the corresponding statement that every surjective function has a right inverse, this does not require theaxiom of choice, as the existence of $a {\displaystyle a}$ is implied by the non-emptiness of the domain. However, this statement may fail in less conventional mathematics such asconstructive mathematics. In constructive mathematics, the inclusion $\{0,1\}\to \mathbb {R}$ of the two-element set in the reals cannot have a left inverse, as it would violateindecomposability, by giving aretraction of 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 original on 4 June 2017.

References

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Bartle, Robert G. (1976),The Elements of Real Analysis (2nd ed.), New York:John Wiley & Sons,ISBN 978-0-471-05464-1, p. 17ff.
Halmos, Paul R. (1974),Naive Set Theory, New York: Springer,ISBN 978-0-387-90092-6, p. 38ff.

External links

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Wikimedia Commons has media related toInjectivity.

Look upinjective in Wiktionary, the free dictionary.

Mathematical logic

General

Theorems
(list),
paradoxes

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types ofsets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps, cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive