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; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with 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 is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details.
A function that is not injective is sometimes called many-to-one.
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 in the contrapositive statement.
Symbolically, which is logically equivalent to the contrapositive,
Examples
For visual examples, readers are directed to the gallery section. 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 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 .
Injections can be undone
Functions with left inverses are always injections. That is, given if there is a function such that for every , , then is injective. In this case, is called a retraction of Conversely, is called a section of
Conversely, every injection with a non-empty domain has a left inverse . It can be defined by choosing an element in the domain of and setting to the unique element of the pre-image (if it is non-empty) or to (otherwise).
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.