Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements to distinct elements; that is, 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.^[1] 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.^[2] 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.^[1]

Definition[]

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 ${\displaystyle f(a)\neq f(b).}$

Symbolically,

$${\displaystyle \forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,}$$

$${\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).}$$

Examples[]

Injective functions. Diagramatic interpretation in the Cartesian plane, defined by the mapping ${\displaystyle f:X\to Y,}$ where ${\displaystyle y=f(x),}$ ${\displaystyle X=}$ domain of function, ${\displaystyle Y=}$ range of function, and ${\displaystyle \operatorname {im} (f)}$ denotes image of ${\displaystyle f.}$ Every one ${\displaystyle x}$ in ${\displaystyle X}$ maps to exactly one unique ${\displaystyle y}$ in ${\displaystyle Y.}$ The circled parts of the axes represent domain and range sets— in accordance with the standard diagrams above.

• 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.^[1]

Not an injective function. Here ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are subsets of ${\displaystyle X,Y_{1}}$ and ${\displaystyle Y_{2}}$ are subsets of ${\displaystyle Y}$: for two regions where the function is not injective because more than one domain element can map to a single range element. That is, it is possible for more than one ${\displaystyle x}$ in ${\displaystyle X}$ to map to the same ${\displaystyle y}$ in ${\displaystyle Y.}$

Making functions injective. The previous function ${\displaystyle f:X\to Y}$ can be reduced to one or more injective functions (say) ${\displaystyle f:X_{1}\to Y_{1}}$ and ${\displaystyle f:X_{2}\to Y_{2},}$ shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule ${\displaystyle f}$ has not changed – only the domain and range. ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are subsets of ${\displaystyle X,Y_{1}}$ and ${\displaystyle Y_{2}}$ are subsets of ${\displaystyle Y}$: 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 ${\displaystyle x}$ in ${\displaystyle X}$ maps to one ${\displaystyle y}$ in ${\displaystyle Y.}$

Injections can be undone[]

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 ${\displaystyle x\in X,}$

${\displaystyle g(f(x))=x}$ (${\displaystyle f}$ can be undone by ${\displaystyle g}$), then ${\displaystyle f}$ is injective. In this case, ${\displaystyle g}$ is called a retraction of ${\displaystyle f.}$ Conversely, ${\displaystyle f}$ is called a section of ${\displaystyle g.}$

Conversely, every injection ${\displaystyle f}$ with non-empty domain has a left inverse ${\displaystyle g,}$ which can be defined by fixing an element ${\displaystyle a}$ in the domain of ${\displaystyle f}$ so that ${\displaystyle g(x)}$ equals the unique pre-image of ${\displaystyle x}$ under ${\displaystyle f}$ if it exists and ${\displaystyle g(x)=1}$ otherwise.^[4]

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[]

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 range ${\displaystyle J=f(X).}$ 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[]

• If ${\displaystyle f}$ and ${\displaystyle g}$ are both injective then ${\displaystyle f\circ g}$ is injective.

The composition of two injective functions 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[]

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 ${\displaystyle x=y.}$^[5]

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.

See also[]

• Bijection, injection and surjection – Properties of mathematical functions
• Injective metric space – Type of metric space
• Monotonic function
• Univalent function

Notes[]

References[]

• 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[]

Wikimedia Commons has media related to Injectivity.

Look up injective in Wiktionary, the free dictionary.

• 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