All Topics  
Injective function

 

 
   Email Print
   Bookmark   Link
 

 

Injective function
 
 
  Topics Injective function Topic Home

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an injective function is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 which associates distinct arguments with distinct values.

An injective function is called an injection, and is also said to be a one-to-one function (not to be confused with one-to-one correspondence, i.e.






Discussion
Ask a question about 'Injective function'
Start a new discussion about 'Injective function'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Injection
Bijection
Surjection
In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an injective function is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 which associates distinct arguments with distinct values.

An injective function is called an injection, and is also said to be a one-to-one function (not to be confused with one-to-one correspondence, i.e. a bijective function).

A function f that is not injective is sometimes called many-to-one. (However, this terminology is also sometimes used to mean "single-valued", i.e. each argument is mapped to at most one value.)

A monomorphism
Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an Injective function homomorphism. A monomorphism from X to Y is often denoted with the notation ....
 is a generalization of an injective function in category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
.

Definition

Let f be a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 whose domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
 is a set A. The function f is injective if for all a and b in A, if f(a) = f(b), then a = b; that is, f(a) = f(b) implies a = b.  Equivalently, if a ? b, then f(a) ? f(b).

Examples

  • For any set X, the identity function
    Identity function

    In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument....
     on X is injective.
  • The function f : R ? R defined by f(x) = 2x + 1 is injective.
  • The function g : R ? R defined by g(x) = x2 is not injective, because (for example) g(1) = 1 = g(-1). However, if g is redefined so that its domain is the non-negative real numbers [0,+8), then g is injective.
  • The exponential function
    Exponential function

    The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
     exp : R → R defined by exp(x) = ex is injective (but not surjective as no value maps to a negative number).
  • The natural logarithm
    Natural logarithm

    The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828....
     function ln : (0, ∞) → R defined by x ↦ ln x is injective.
  • The function g : R ? R defined by g(x) = xn − x is not injective, since, for example, g(0) = g(1).


More generally, when X and Y are both the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
 R, then an injective function f : R ? R is one whose graph is never intersected by any horizontal line more than once.

Injections can be undone


Functions with left inverses
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
 are always injections. That is, given f : X ? Y, if there is a function g : Y ? X such that, for every x ∈ X

g(f(x)) = x (f can be undone by g)


then f is injective. In this case, f is called a section of g and g is called a retraction of f.

Conversely, every injection f with non-empty domain has a left inverse g (in conventional mathematics). Note that g may not be a complete inverse
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
 of f because the composition in the other order, f ° g, may not be the identity on Y. In other words, a function that can be undone or "reversed", such as f, is not necessarily invertible
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
 (bijective). Injections are "reversible" but not always invertible.

Although it is impossible to reverse a non-injective (and therefore information-losing) function, you can at least obtain a "quasi-inverse" of it, that is a multiple-valued
Multivalued function

In mathematics, a multivalued function is a total relation; i.e. every input is associated with one or more outputs. Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input....
 function.

Injections may be made invertible


In fact, to turn an injective function f : X ? Y into a bijective (hence invertible
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
) function, it suffices to replace its codomain Y by its actual range J = f(X). That is, let g : X ? J such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as inclJ,Y ° g, where inclJ,Y is the inclusion function from J into Y.

Other properties

  • If f and g are both injective, then f ° g is injective.
  • If g ° f is injective, then f is injective (but g need not be).
  • f : X ? Y is injective if and only if, given any functions g, h : W ? X, whenever f ° g = f ° h, then g = h. In other words, injective functions are precisely the monomorphism
    Monomorphism

    In the context of abstract algebra or universal algebra, a monomorphism is an Injective function homomorphism. A monomorphism from X to Y is often denoted with the notation ....
    s in the category
    Category theory

    In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
     Set
    Category of sets

    In mathematics, the category of sets, denoted as Set, is the Category theory whose Category theory are all Set and whose morphisms are all function s....
     of sets.
  • If f : X ? Y is injective and A is a subset
    Subset

    In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
     of X, then f −1(f(A)) = A. Thus, A can be recovered from its image f(A).
  • If f : X ? Y is injective and A and B are both subsets of X, then f(A n B) = f(A) n f(B).
  • Every function h : W ? Y can be decomposed as h = f ° 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(W) of h as a subset of the codomain Y of h.
  • If f : X ? Y is an injective function, then Y has at least as many elements as X, in the sense of cardinal number
    Cardinal number

    In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of Set ....
    s. In particular, if, in addition, there is an injection from to , then and has the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem
    Cantor–Bernstein–Schroeder theorem

    In axiomatic set theory, the Cantor?Bernstein?Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schr?der, states that, if there exist injective functions f : A ? B and g : B ? A between the Set A and B, then there exists a bijection function h : A ? B....
    .)
  • If both X and Y are finite
    Finite set

    In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
     with the same number of elements, then f : X ? Y is injective if and only if f is surjective.
  • Every embedding
    Embedding

    In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
     is injective.


See also

  • surjective function
  • Injective module
    Injective module

    In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers....
  • Horizontal line test
    Horizontal line test

    In mathematics, the horizontal line test is a test used to determine if a function is injective, surjective or bijective.Suppose there is a function f : X ? Y with a graph., and you have a horizontal line of X x Y ....
  • Injective metric space
    Injective metric space

    In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of Chebyshev distance in higher-dimensional vector spaces....