Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements to distinct elements; that is, f(x1) = f(x2) implies x1 = x2.[1] In other words, every element of the function's codomain is the image of at most one element of its domain.[2] 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.[3] 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.[2]

DefinitionEdit

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  

Symbolically,

 
which is logically equivalent to the contrapositive,[4]
 

ExamplesEdit

 
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.
  • 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 not 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 horizontal line test.[2]

 
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  
 
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  

Injections can be undoneEdit

Functions with left inverses are always injections. That is, given   if there is a function   such that for every  

  (  can be undone by  ), then   is injective. In this case,   is called a retraction of   Conversely,   is called a section of  

Conversely, every injection   with non-empty domain has a left inverse   which can be defined by fixing an element   in the domain of   so that   equals the unique pre-image of   under   if it exists and   otherwise.[5]

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 invertibleEdit

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.

Other propertiesEdit

  • If   and   are both injective then   is injective.
 
The composition of two injective functions is injective.
  • If   is injective, then   is injective (but   need not be).
  •   is injective if and only if, given any functions     whenever   then   In other words, injective functions are precisely the monomorphisms in the category Set of sets.
  • If   is injective and   is a subset of   then   Thus,   can be recovered from its image  
  • If   is injective and   and   are both subsets of   then  
  • Every function   can be decomposed as   for a suitable injection   and surjection   This decomposition is unique up to isomorphism, and   may be thought of as the inclusion function of the range   of   as a subset of the codomain   of  
  • If   is an injective function, then   has at least as many elements as   in the sense of cardinal numbers. In particular, if, in addition, there is an injection from   to   then   and   have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
  • If both   and   are finite with the same number of elements, then   is injective if and only if   is surjective (in which case   is bijective).
  • An injective function which is a homomorphism between two algebraic structures is an embedding.
  • Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function   is injective can be decided by only considering the graph (and not the codomain) of  

Proving that functions are injectiveEdit

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  [6]

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.

See alsoEdit

NotesEdit

  1. ^ "The Definitive Glossary of Higher Mathematical Jargon — One-to-One". Math Vault. 2019-08-01. Retrieved 2019-12-07.
  2. ^ a b c "Injective, Surjective and Bijective". www.mathsisfun.com. Retrieved 2019-12-07.
  3. ^ "Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project". stacks.math.columbia.edu. Retrieved 2019-12-07.
  4. ^ Farlow, S. J. "Injections, Surjections, and Bijections" (PDF). math.umaine.edu. Retrieved 2019-12-06.
  5. ^ Unlike the corresponding statement that every surjective function has a right inverse, this does not require the axiom of choice, as the existence of   is implied by the non-emptiness of the domain. However, this statement may fail in less conventional mathematics such as constructive mathematics. In constructive mathematics, the inclusion   of the two-element set in the reals cannot have a left inverse, as it would violate indecomposability, by giving a retraction of the real line to the set {0,1}.
  6. ^ Williams, Peter. "Proving Functions One-to-One". Archived from the original on 4 June 2017.

ReferencesEdit

External linksEdit