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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''.

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 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 ).

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^n - x 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.


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

: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.

INJECTIONS MAY BE MADE INVERTIBLE


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 ''f'' o ''g'' 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