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In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain.^{[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 non-injective non-surjective function (also not a bijection)
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]}
Definition
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 $f(a)=f(b)$, then $a=b$; that is, $f(a)=f(b)$ implies $a=b$. Equivalently, if $a\neq b$, then $f(a)\neq f(b)$.
For any set X and any subset S of X, the inclusion map (which sends any element s of S 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 g is redefined so that its domain is the non-negative real numbers [0,+?), then g is injective.
The function defined by is not injective, since, for example, .
More generally, when X and Y are both the real lineR, 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]}
Injective functions. Diagramatic interpretation in the Cartesian plane, defined by the mapping , where , X = domain of function, Y = range of function, and im(f) denotes image of f. Every one x in X maps to exactly one unique y in Y. The circled parts of the axes represent domain and range sets-- in accordance with the standard diagrams above.
Not an injective function. Here X_{1} and X_{2} are subsets of X, Y_{1} and Y_{2} are subsets of 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 onex in X to map to the samey in Y.
Making functions injective. The previous function can be reduced to one or more injective functions (say) and , shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule f has not changed - only the domain and range. X_{1} and X_{2} are subsets of X, Y_{1} and Y_{2} are subsets of 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 x in X maps to one y in Y.
Injections can be undone
Functions with left inverses are always injections. That is, given , if there is a function such that for every ,
g(f(x)) = x (f can be undone by g), then f is injective. In this case, g is called a retraction of f. Conversely, f is called a section of g.
Conversely, every injection f with non-empty domain has a left inverse g, which can be defined by fixing an element a in the domain of f so that g(x) equals the unique pre-image of x under f if it exists and g(x) = a otherwise.^{[6]}
The left inverse g is not necessarily an inverse of f, because the composition in the other order, , 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.
Injections may be made invertible
In fact, to turn an injective function into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range . That is, let such that for all x in X; then g is bijective. Indeed, f can be factored as , where is the inclusion function from J into Y.
The composition of two injective functions is injective.
If is injective, then f is injective (but g need not be).
is injective if and only if, given any functions g, whenever , then . In other words, injective functions are precisely the monomorphisms in the categorySet of sets.
If is injective and A is a subset of X, then . Thus, A can be recovered from its imagef(A).
If is injective and A and B are both subsets of X, then .
Every function can be decomposed as 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(W) of h as a subset of the codomain Y of h.
If 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 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.
Proving that functions are injective
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 , then .^{[7]}
Here is an example:
f = 2x + 3
Proof: Let . Suppose . So => => . 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(x) in at most one point, then f is injective or one-to-one.
^Unlike the corresponding statement that every surjective function has a right inverse, this does not require the axiom of choice, as the existence of 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 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}.