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Injective function (one-to-one)

An injective (one-to-one) function maps distinct inputs to distinct outputs. This article explains definitions, equivalent formulations, basic properties, examples, non-examples, and historical notes.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 28, 2026

An injective function (also called an injection or one-to-one function) is a mapping between sets that never sends two different elements of the domain to the same element of the codomain. Informally, an injection preserves distinctness: if a1 and a2 are distinct domain elements then their images are distinct. Injective maps are fundamental in set theory, algebra and analysis because they express an embedding of one set into another without collisions.

Formal definitions and equivalent statements

Let f: A → B be a function from set A (the domain) to set B (the codomain). Common equivalent ways to say that f is injective include:

For all a1,a2 in A, if f(a1) = f(a2) then a1 = a2.
For every b in B there is at most one a in A such that f(a) = b.
There exists a left inverse g: f(A) → A (defined on the image of f) with g(f(a)) = a for every a in A.
f is left-cancellative under composition: if f·g1 = f·g2 then g1 = g2 for functions with appropriate domains.

Basic properties

Composition: the composition of two injective functions is injective.
Restriction: f is injective if and only if its restriction to any subset of the domain is injective.
Inverses: an injective function need not have a full two-sided inverse on all of B, but it has a left inverse defined on its image f(A). If it also surjects onto B it becomes a bijection and has a true inverse f^{-1}: B → A.
Cardinality consequences: for finite sets, an injection A → B implies |A| ≤ |B|. For infinite sets the existence of injections in both directions relates to cardinality comparisons and can lead to the Cantor–Bernstein theorem.

Examples and non-examples

Typical examples of injective functions include linear maps x → mx + c on the real line when m ≠ 0, the exponential map x → e^x on R, and the inclusion map of a subset into a larger set. Finite-to-one maps that are not one-to-one, constant maps, and many polynomial maps of even degree (over R) are not injective unless their domain is restricted.

Concrete non-examples: the function f(x) = x^2 on R is not injective because f(1) = f(-1). The sine function sin: R → R is not injective because it repeats values periodically. A constant map f(a)=b is injective only if the domain has at most one element.

Uses and importance

Injective functions are used to compare sizes of sets and to embed structures without identifying distinct elements. In algebra, injective homomorphisms identify substructures and are called embeddings. In computer science, injective encodings allow lossless representation of data. In analysis and geometry, injective continuous maps that are also continuous inverses on their images define homeomorphisms onto their images, providing ways to view one space inside another.

Distinctions, terminology and history

The phrase "one-to-one" is commonly used as a synonym for injective, but it can confuse readers who use "one-to-one correspondence" to mean bijection (both one-to-one and onto). The triad of terms injection, surjection and bijection was popularized in 20th century mathematics; the collective pseudonym Nicolas Bourbaki introduced this standardized vocabulary in modern texts produced in the 1930s. Related concepts are surjection (every element of the codomain has at least one preimage) and bijection (both injective and surjective). The codomain and image of a function are distinct notions sometimes discussed under the link labeled codomain.

Further reading and formal treatments are available in standard set theory and algebra texts; online references can be followed through the links above. The concise list of core facts about injections makes them one of the simplest yet most widely used structural tools in mathematics.

Illustration of an injection. Each element of Y has at most one original image: A, B, D one each, C none.

Examples and counterexamples

Extra-mathematical example: The function that assigns the number of the current identity card to each citizen of the Federal Republic of Germany with an identity card is injective, where the set of all possible identity card numbers is assumed as the target set (because identity card numbers are only assigned once).
$\mathbb {N}$ denote the set of natural numbers and $\mathbb {Z}$ the set of integers.

$f_1\colon \N \to \N,\, x \mapsto 2x$ is injective.

$f_2\colon \Z \to \Z,\, x \mapsto 2x$ is injective.

$f_3\colon \N \to \N,\, x \mapsto x^2$ is injective.

$f_4\colon \Z \to \Z,\, x \mapsto x^2$ is not injective, since, for example, f(1)=f(-1) holds.

Any function $f\colon X \to Y,\, x \mapsto f(x)$ from a two-element set $X=\{a,b\}$ into a one-element set $Y=\{c\}$ is not injective, because necessarily both elements of $c\in Y$ are mapped to the single element

f(a)=f(b)=c despite $a\neq b$

Properties

Note that the injectivity of a function depends $f\colon A\to B$ only on the function graph $\{(x, f(x)) \mid x \in A\}$ (unlike surjectivity, which also depends on the target set read from the function graph).
function $f\colon A\to B$ is injective exactly if for all subsets $X, Y \subseteq A$ holds: $f(X \cap Y)=f(X) \cap f(Y)$
function $f\colon A\to B$ is injective if and only if $f^{-1}(f(T))=T$ for all $T \subseteq A$ (where denotes $f^{-1}\colon {\mathcal {P}}(B)\to {\mathcal {P}}(A)$ the primal image function).
If the functions $f\colon A\to B$ and $g\colon B\to C$ are injective, then the composition (concatenation) is also $g \circ f\colon A \to C$ injective.
From the injectivity of $g\circ f$ it follows that is injective.
function $f\colon A\to B$ with non-empty definition set is injective if and only if a left inverse, that is a function $g\colon B \to A$ with $g \circ f = \operatorname{id}_A$ (where denotes $\operatorname {id} _{A}$ the identical mapping to ).
function $f\colon A\to B$ is injective if it is left truncable, i.e. if for any function $g, h\colon C \to A$ $f \circ g = f \circ h$ the equality follows from (This property motivates the term monomorphism used in category theory, but for general morphisms injective and left truncable are no longer equivalent).
Any function $f\colon A\to B$ is $f=h\circ g$ representable as a concatenation , where surjective and injective (namely an inclusion mapping).
A continuous real-valued function on a real interval is injective if it is strictly monotonically increasing or strictly monotonically decreasing in its entire domain of definition, i.e. if for any two numbers and from the domain of definition the applies: From f ( a (increasing), or from f ( a (falling).

A group or vector space homomorphism is injective exactly when its kernel is trivial, i.e. consists only of the neutral element or the zero vector.

Three injective strictly monotonically increasing real functions.

Three injective strictly monotonically decreasing real functions.

Mightiness of sets

The notion of injection plays an important role in set theory in the definition and comparison of powers, a notion that generalises the number of elements from finite sets to arbitrary sets. Two sets $X,\,Y$ are called "of equal power" if there is both an injection from to and one from to (In this case, bijections from one set to the other also exist). On the other hand, is called of smaller power than if there is an injection from to but none from to .

Drawer closure

A frequent inference scheme in proofs, especially in number theory, uses the statement that a mapping a finite set into a set with fewer elements cannot be injective, that there are elements $a,b\in X$ with $a\neq b$ and equal image f(a)=f(b) Because of the notion of many objects in fewer drawers, this is called "drawer closure".

Number of injective mappings

The number of injective mappings from a defining set into a given finite target set with property $|B| \geq |A|$ is given by:

$|B|\cdot (|B|-1) \cdot \ldots \cdot (|B|-|A|+1) = \frac{|B|!}{(|B|-|A|)!} = |A|! \cdot \binom{|B|}{|A|}$

In combinatorics, this corresponds to a variation without repetition.

Questions and Answers

Q: What is an injective function in mathematics?

A: An injective function is a function f: A → B with the property that distinct elements in the domain map to distinct elements in the codomain.

Q: What is the relation between elements in the domain and codomain of an injective function?

A: For every element b in the codomain B, there is at most one element a in the domain A such that f(a)=b.

Q: Who introduced the terms injection, surjection, and bijection?

A: Nicholas Bourbaki and a group of other mathematicians introduced the terms injection, surjection, and bijection.

Q: What does an injective function mean?

A: An injective function means that each element in the domain A maps to a unique element in the codomain B.

Q: How is an injective function different from a 1-1 correspondence?

A: An injective function is often called a 1-1 (one-to-one) function but is distinguished from a 1-1 correspondence, which is a bijective function (both injective and surjective).

Q: What is the property of an injective function?

A: The property of an injective function is that distinct elements in the domain map to distinct elements in the codomain.

Q: What is the significance of injective functions in mathematics?

A: Injective functions play an important role in many mathematical fields, including topology, analysis, and algebra, due to their property of having distinct elements in the domain map to distinct elements in the codomain.

Author

AlegsaOnline.com - Injective function - Leandro Alegsa

Creation date: November 13, 2022
Last updated: May 28, 2026

URL: https://en.alegsaonline.com/art/47369