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*(*x*_{1}) = *f*(*x*_{2}) implies *x*_{1} = *x*_{2}.^{[1]} 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 that is not injective is sometimes called many-to-one.^{[2]}

## Definition

Let be a function whose domain is a set The function is said to be **injective** provided that for all and in if then ; that is, implies Equivalently, if then

Symbolically,

^{[4]}

## Examples

- For any set and any subset the inclusion map (which sends any element to itself) is injective. In particular, the identity function 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 defined by is injective.
- The function defined by is
*not*injective, because (for example) However, if is redefined so that its domain is the non-negative real numbers [0,+∞), then is injective. - The exponential function defined by is injective (but not surjective, as no real value maps to a negative number).
- The natural logarithm function defined by is injective.
- The function defined by is not injective, since, for example,

More generally, when and are both the real line then an injective function 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

Functions with left inverses are always injections. That is, given if there is a function such that for every

- ( can be undone by ), then is injective. In this case, is called a retraction of Conversely, is called a section of

Conversely, every injection with non-empty domain has a left inverse which can be defined by fixing an element in the domain of so that equals the unique pre-image of under if it exists and otherwise.^{[5]}

The left inverse is not necessarily an inverse of because the composition in the other order, may differ from the identity on 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 into a bijective (hence invertible) function, it suffices to replace its codomain by its actual range That is, let such that for all ; then is bijective. Indeed, can be factored as where is the inclusion function from into

More generally, injective partial functions are called partial bijections.

## Other properties

- If and are both injective then is injective.

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

## Proving that functions are injective

A proof that a function 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 then ^{[6]}

Here is an example:

Proof: Let Suppose So implies which implies Therefore, it follows from the definition that is injective.

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

**^**"The Definitive Glossary of Higher Mathematical Jargon — One-to-One".*Math Vault*. 2019-08-01. Retrieved 2019-12-07.- ^
^{a}^{b}^{c}"Injective, Surjective and Bijective".*www.mathsisfun.com*. Retrieved 2019-12-07. **^**"Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project".*stacks.math.columbia.edu*. Retrieved 2019-12-07.**^**Farlow, S. J. "Injections, Surjections, and Bijections" (PDF).*math.umaine.edu*. Retrieved 2019-12-06.**^**Unlike the corresponding statement that every surjective function has a right inverse, this does not require the axiom of choice, as the existence of 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 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. "Proving Functions One-to-One". Archived from the original on 4 June 2017.

## 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*.

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