Injective Article Index for
Injective 		Website Links For
Function 		 

Information About

Injective
  CATEGORIES ABOUT INJECTIVE FUNCTION
   
functions and mappings
   
basic concepts in set theory
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



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 And Only If , for every ''y'' in the Codomain , there is at most one ''x'' in the Domain such that ''f''(''x'') = ''y''.

Put another way, ''f'' is injective if and only if ''f''(''a'') = ''f''(''b'') implies ''a'' = ''b'' (or ''a''
eq ''b'' implies ''f''(''a'') 
eq ''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 .)

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.


EXAMPLES AND COUNTER-EXAMPLES

  • For any set ''X'', the Identity Function on ''X'' is injective.

  • The function ''f'' : R → R defined by ''f''(''x'') = 2''x'' + 1 is injective.

  • The function ''g'' : R → R defined by ''g''(''x'') = ''x''2 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} o \mathbf{R}^+ : x \mapsto \mathrm{e}^x is injective.

  • The Natural Logarithm function \ln : (0,+\infty) o \mathbf{R} : x \mapsto \ln{x} is injective.

  • The function ''g'' : R → R defined by g(x) = x^3 - x is not injective, since, for example, ''g''(-1) = ''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.


INJECTIONS ARE INVERTIBLE

Another definition of injection is a function whose effect can be undone. More precisely, ''f'' : ''X'' → ''Y'' is injective if and only 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''.

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 fact, to turn an injective function ''f'' : ''X'' → ''Y'' into a from ''J'' into ''Y''.


OTHER PROPERTIES

  • If ''f'' and ''g'' are both injective, then ''g'' o ''f'' 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 Number s.

  • If both ''X'' and ''Y'' are .

  • Every Embedding is injective.



CATEGORY THEORY VIEW

In the language of Category Theory , injective functions are precisely the Monomorphism s in the Category Of Sets .


SEE ALSO