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Encyclopedia > Injectivity
An injective function.
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An injective function.
Another injective function.
Another injective function.
A non-injective function.
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A non-injective function.

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, for every y in the codomain, there is at most one x in the domain such that f(x) = y. Image File history File links Diagram to illustrate a function that is an injection but not a surjection. ... Image File history File links Diagram to illustrate a function that is an injection but not a surjection. ... A bijection. ... A bijection. ... Image File history File links A diagram illustrating a function that is not injective but is surjective. ... Image File history File links A diagram illustrating a function that is not injective but is surjective. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Partial plot of a function f. ... Partial plot of a function f. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... In mathematics, the domain of a function is the set of all input values to the function. ...


Put another way, f is injective if f(a) = f(b) implies a = b (or a b 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 information-preserving or, sometimes, one-to-one function. (However, this 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 is injective.
  • The natural logarithm function is injective.
  • The function g : R → R defined by g(x) = x3x 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. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ... In mathematics, the real line is simply the set of real numbers. ...


Injections are invertible

Another definition of injection is a function whose effect can be undone. More precisely, f : X → Y is injective if there exists a function g : Y → X such that g(f(x)) = x for every x in ´ X; that is, g o f  equals the identity function on X. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...


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 mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...


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 g o f is injective.
Injective composition.
Injective composition.
  • 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 An illustration of two functions/mappings: the left is an injective and non-surjective function, the right is neither injective nor surjective. ... Image File history File links An illustration of two functions/mappings: the left is an injective and non-surjective function, the right is neither injective nor surjective. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... 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. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... 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. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...


See also


  Results from FactBites:
 
Injective cogenerator - Wikipedia, the free encyclopedia (551 words)
In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality.
For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group).
The Tietze extension theorem can be used to show that an interval is an injective cogenerator in a category of topological spaces subject to separation axioms.
Bijection, injection and surjection - Wikipedia, the free encyclopedia (1039 words)
A function is injective (one-to-one) if each image is mapped to by at most one element of the domain.
A note on terminology: a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective.
A function f : A → B is injective if and only if A is empty or f is left-invertible, that is, there is a function g: B → A such that g o f = identity function on A.
  More results at FactBites »

 
 

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