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Encyclopedia > Identity function

An identity function f is a function which doesn't have any effect: it always returns the same value that was used as its argument.


Formally, if M is a set, we define the identity function idM on M to be that function with domain and codomain M which satisfies

idM(x) = x    for all elements x in M.

If f : M → N is any function, then we have f o idM = f = idN o f. In particular, idM is the identity element of the monoid of all functions from M to M.


When choosing M equal to the positive integers, one obtains the identity function Id(n), which is a multiplicative function considered in number theory.


  Results from FactBites:
 
Identity function - Wikipedia, the free encyclopedia (196 words)
An identity function f is a function which does not have any effect: it always returns the same value that was used as its argument.
is the identity element of the monoid of all functions from M to M.
The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
Bijection, injection and surjection - Wikipedia, the free encyclopedia (1098 words)
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
A function is surjective (onto) if every element of the codomain is mapped to by some element (argument) of the domain; some images may be mapped to by more than one argument.
A bijective function is a bijection (one-to-one correspondence).
  More results at FactBites »

 
 

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