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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 $f$ that is not injective is sometimes called many-to-one.^[2]

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

Symbolically,$\forall a,b\in X,f\left(a\right)=f\left(b\right)\Rightarrow a=b,$ which is logically equivalent to the contrapositive,^[4]$\forall a,b\in X,a\ne b\Rightarrow f\left(a\right)\ne f\left(b\right).$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 an inclusion map.^[5]

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

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

More generally, when $X$ and $Y$ are both the real line ⁠$ℝ$⁠, then an injective function $f:ℝ\to ℝ$ 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]

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

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

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

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

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

More generally, injective partial functions are called partial bijections.

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

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

Here is an example: $f\left(x\right)=2x+3$

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

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

• 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

• 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.

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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