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In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, x_{1} ≠ x_{2} implies f(x_{1}) ≠ f(x_{2}). (Equivalently, f(x_{1}) = f(x_{2}) implies x_{1} = x_{2} in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of at most one element of its domain.^{[1]} 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.^{[2]} 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.^{[1]}
Further information on notation: Function (mathematics) § Notation |
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,
For visual examples, readers are directed to the gallery section.
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.^{[1]}
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).^{[4]}
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.
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.
See also: List of set identities and relations § Functions and sets |
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 ^{[5]}
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.
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 has not changed – only the domain and range. and are subsets of and are subsets of : 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 in maps to one in
Not an injective function. Here and are subsets of and are subsets of : 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 one in to map to the same in
Injective functions. Diagramatic interpretation in the Cartesian plane, defined by the mapping where domain of function, range of function, and denotes image of Every one in maps to exactly one unique in The circled parts of the axes represent domain and range sets— in accordance with the standard diagrams above