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Encyclopedia > Injective function
An injective function. (Injection)
An injective function. (Injection)
Another injective function. (Bijection)
Another injective function. (Bijection)
A non-injective function. (Surjection)
A non-injective function. (Surjection)

In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y in the codomain, such that f(x) = y. One-to-one or one to one is an adjective which can qualify: in mathematics, a function, see injective function or bijective function in telecommunications, a communication, see one-to-one (communication) in computer science, an education concept, see one to one computing One to One can also refer to... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ... Image File history File links Injection. ... Image File history File links Injection. ... Image File history File links Bijection. ... Image File history File links Bijection. ... A bijective function. ... Image File history File links Surjection. ... Image File history File links Surjection. ... A surjective function. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the domain of a function is the set of all input values to the function. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...


Put another way, f is injective if f(a) = f(b) implies a = b (or ab implies f(a) ≠ f(b)), for any a, b in the domain.


An injective function is called an injection, and is also said to be an information-preserving or one-to-one function (however, the latter name is best avoided, since some authors understand it to mean a one-to-one correspondence, i.e. a bijective function). In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...


A function f that is not injective is sometimes called many-to-one. However, this name too is best avoided, since it is sometimes used to mean "single-valued", i.e. each argument is mapped to at most one value.

Contents

Examples and counter-examples

  • For any set X, the identity function 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,+∞), then g is injective.
  • The exponential function exp : mathbf{R} to mathbf{R} : x mapsto mathrm{e}^x is injective (but not surjective as no value maps to a negative number).
  • The natural logarithm function ln : (0,+infty) to mathbf{R} : x mapsto ln{x} is injective.
  • The function g : R → R defined by g(x) = xnx is not injective, since, for example, g(0) = g(1).

More generally, when X and Y are both the real line R, then an injective function f : R → R is one whose graph is never intersected by any horizontal line more than once. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, the real line is simply the set of real numbers. ...


Injections can be undone

Functions with left inverses (often called sections) are always injections. That is to say, for f : X → Y, if there exists a function g : Y → X such that, for every x in X In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In the mathematical field of category theory, a section is a morphism which has a left inverse, i. ...

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

then f is injective. Conversely, it is usually assumed that every injection with non-empty domain has a left inverse.


Note that g may not be a complete inverse of f because the composition in the other order, f o 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 (bijective). Injections are "reversible" but not always invertible. In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...


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 function. This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...


Injections may be made invertible

In fact, to turn an injective function f : X → Y into a bijective (hence invertible) 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,Yog, where inclJ,Yis the inclusion function from J into Y. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, inclusion is a partial order on sets. ...


Other properties

  • If f and g are both injective, then f o g is injective.
The composition of two injective functions is injective.
The composition of two injective functions is injective.
  • If g o 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 o g = f o h, then g = h.
  • If f : X → Y is injective and A is a subset 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 ∩ B) = f(A) ∩ f(B).
  • Every function h : W → Y can be decomposed as h = f o 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 numbers.
  • If both X and Y are finite with the same number of elements, then f : X → Y is injective if and only if f is surjective.
  • Every embedding is injective.

Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... “Superset” redirects here. ... Image of the Wikimedia Commons logo. ... In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ... In mathematics, inclusion is a partial order on sets. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...

Category theory view

In the language of category theory, injective functions are precisely the monomorphisms in the category of sets. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... For other uses, see Dimorphism (disambiguation) or Polymorphism (disambiguation). ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...


See also

Look up injective in Wiktionary, the free dictionary.

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