Injective function

In mathematics, an injective function (also known as injection, or one-to-one function^[1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x₁ ≠ x₂ implies f(x₁) ≠ f(x₂) (equivalently by contraposition, f(x₁) = f(x₂) implies x₁ = x₂). In other words, every element of the function's codomain is the image of at most one element of its domain.^[2] The term one-to-one function must not be confused with one-to-one correspondence 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 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 spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category 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; see Homomorphism § Monomorphism for more details.

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

Definition#edit

An injective function, which is not also surjective

Let ${\displaystyle f}$ be a function whose domain is a set ⁠ ${\displaystyle X}$ ⁠. The function ${\displaystyle f}$ is said to be injective provided that for all ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle X,}$ if ⁠ ${\displaystyle f(a)=f(b)}$ ⁠, then ⁠ ${\displaystyle a=b}$ ⁠; that is, ${\displaystyle f(a)=f(b)}$ implies ⁠ ${\displaystyle a=b}$ ⁠. Equivalently, if ⁠ ${\displaystyle a\neq b}$ ⁠, then ${\displaystyle f(a)\neq f(b)}$ in the contrapositive statement.

Symbolically, ${\displaystyle \forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,}$ which is logically equivalent to the contrapositive,^[4] ${\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).}$ An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, ${\displaystyle f:A\rightarrowtail B}$ or ⁠ ${\displaystyle f:A\hookrightarrow B}$ ⁠), although some authors specifically reserve ↪ for an inclusion map.^[5]

Examples#edit

For visual examples, readers are directed to the gallery section.

For any set ${\displaystyle X}$ and any subset ⁠ ${\displaystyle S\subseteq X}$ ⁠, the inclusion map ${\displaystyle S\to X}$ (which sends any element ${\displaystyle s\in S}$ to itself) is injective. In particular, the identity function ${\displaystyle X\to X}$ is always injective (and in fact bijective).
If the domain of a function is the empty set, 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 ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ defined by ${\displaystyle f(x)=2x+1}$ is injective.
The function ${\displaystyle g:\mathbb {R} \to \mathbb {R} }$ defined by ${\displaystyle g(x)=x^{2$ is not injective, because (for example) ${\displaystyle g(1)=1=g(-1).}$ However, if ${\displaystyle g}$ is redefined so that its domain is the non-negative real numbers [0, +∞), then ${\displaystyle g}$ is injective.
The exponential function ${\displaystyle \exp :\mathbb {R} \to \mathbb {R} }$ defined by ${\displaystyle \exp(x)=e^{x$ is injective (but not surjective, as no real value maps to a negative number).
The natural logarithm function ${\displaystyle \ln :(0,\infty )\to \mathbb {R} }$ defined by ${\displaystyle x\mapsto \ln x}$ is injective.
The function ${\displaystyle g:\mathbb {R} \to \mathbb {R} }$ defined by ${\displaystyle g(x)=x^{n}-x}$ is not injective, since, for example, ⁠ ${\displaystyle g(0)=g(1)=0}$ ⁠.

More generally, when ${\displaystyle X}$ and ${\displaystyle Y}$ are both the real line ⁠ ${\displaystyle \mathbb {R} }$ ⁠, then an injective function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the horizontal line test.^[2]

Injections can be undone#edit

Functions with left inverses are always injections. That is, given ⁠ ${\displaystyle f:X\to Y}$ ⁠, if there is a function ${\displaystyle g:Y\to X}$ such that for every ⁠ ${\displaystyle x\in X}$ ⁠, ⁠ ${\displaystyle g(f(x))=x}$ ⁠, then ${\displaystyle f}$ is injective. The proof is that ${\displaystyle f(a)=f(b)\rightarrow g(f(a))=g(f(b))\rightarrow a=b.}$

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

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

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

In fact, to turn an injective function ${\displaystyle f:X\to Y}$ into a bijective (hence invertible) function, it suffices to replace its codomain ${\displaystyle Y}$ by its actual image ${\displaystyle J=f(X).}$ That is, let ${\displaystyle g:X\to J}$ such that ${\displaystyle g(x)=f(x)}$ for all ⁠ ${\displaystyle x\in X}$ ⁠; then ${\displaystyle g}$ is bijective. Indeed, ${\displaystyle f}$ can be factored as ⁠ ${\displaystyle \operatorname {In} _{J,Y}\circ g}$ ⁠, where ${\displaystyle \operatorname {In} _{J,Y$ is the inclusion function from ${\displaystyle J}$ into ⁠ ${\displaystyle Y}$ ⁠.

More generally, injective partial functions are called partial bijections.

Other properties#edit

The composition of two injective functions is injective.

If ${\displaystyle f}$ and ${\displaystyle g}$ are both injective then ${\displaystyle f\circ g}$ is injective.
If ${\displaystyle g\circ f}$ is injective, then ${\displaystyle f}$ is injective (but ${\displaystyle g}$ need not be).
${\displaystyle f:X\to Y}$ is injective if and only if, given any functions ⁠ ${\displaystyle g}$ ⁠, ${\displaystyle h:W\to X}$ whenever ⁠ ${\displaystyle f\circ g=f\circ h}$ ⁠, then ⁠ ${\displaystyle g=h}$ ⁠. In other words, injective functions are precisely the monomorphisms in the category Set of sets.
If ${\displaystyle f:X\to Y}$ is injective and ${\displaystyle A}$ is a subset of ⁠ ${\displaystyle X}$ ⁠, then ⁠ ${\displaystyle f^{-1}(f(A))=A}$ ⁠. Thus, ${\displaystyle A}$ can be recovered from its image ⁠ ${\displaystyle f(A)}$ ⁠.
If ${\displaystyle f:X\to Y}$ is injective and ${\displaystyle A}$ and ${\displaystyle B}$ are both subsets of ⁠ ${\displaystyle X}$ ⁠, then ⁠ ${\displaystyle f(A\cap B)=f(A)\cap f(B)}$ ⁠.
Every function ${\displaystyle h:W\to Y}$ can be decomposed as ${\displaystyle h=f\circ g}$ for a suitable injection ${\displaystyle f}$ and surjection ⁠ ${\displaystyle g}$ ⁠. This decomposition is unique up to isomorphism, and ${\displaystyle f}$ may be thought of as the inclusion function of the range ${\displaystyle h(W)}$ of ${\displaystyle h}$ as a subset of the codomain ${\displaystyle Y}$ of ⁠ ${\displaystyle h}$ ⁠.
If ${\displaystyle f:X\to Y}$ is an injective function, then ${\displaystyle Y}$ has at least as many elements as ${\displaystyle X,}$ in the sense of cardinal numbers. In particular, if, in addition, there is an injection from ⁠ ${\displaystyle Y}$ ⁠ to ⁠ ${\displaystyle X}$ ⁠, then ${\displaystyle X}$ and ${\displaystyle Y}$ have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
If both ${\displaystyle X}$ and ${\displaystyle Y}$ are finite with the same number of elements, then ${\displaystyle f:X\to Y}$ is injective if and only if ${\displaystyle f}$ is surjective (in which case ${\displaystyle f}$ is bijective).
An injective function which is a homomorphism between two algebraic structures is an embedding.
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 ${\displaystyle f}$ is injective can be decided by only considering the graph (and not the codomain) of ⁠ ${\displaystyle f}$ ⁠.

Proving that functions are injective#edit

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

Here is an example: ${\displaystyle f(x)=2x+3}$

Proof: Let ⁠ ${\displaystyle f:X\to Y}$ ⁠. Suppose ⁠ ${\displaystyle f(x)=f(y)}$ ⁠. So ${\displaystyle 2x+3=2y+3}$ implies ⁠ ${\displaystyle 2x=2y}$ ⁠, which implies ⁠ ${\displaystyle x=y}$ ⁠. Therefore, it follows from the definition that ${\displaystyle f}$ is injective.

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

Notes#edit

^ Sometimes one-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. Retrieved 2019-12-07.
^ "Section 7.3 (00V5): Injective and surjective maps of presheaves". The Stacks project. Retrieved 2019-12-07.
^ Farlow, S. J. "Section 4.2 Injections, Surjections, and Bijections" (PDF). Mathematics & Statistics - University of Maine. Archived from the original (PDF) on Dec 7, 2019. Retrieved 2019-12-06.
^ "What are usual notations for surjective, injective and bijective functions?". Mathematics Stack Exchange. Retrieved 2024-11-24.
^ Unlike the corresponding statement that every surjective function has a right inverse, this does not require the axiom of choice, as the existence of ${\displaystyle a}$ is implied by the non-emptiness of the domain. However, this statement may fail in less conventional mathematics such as constructive mathematics. In constructive mathematics, the inclusion ${\displaystyle \{0,1\}\to \mathbb {R} }$ of the two-element set in the reals cannot have a left inverse, as it would violate indecomposability, by giving a retraction 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 from the original on 4 June 2017.

References#edit

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

External links#edit

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