Editing Partial function (section)

== Basic concepts ==
{| align="right"
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|[[Image:Partial function.svg|thumb|200px|An example of a partial function that is [[injective]].]] 
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|[[Image:Total function.svg|thumb|200px|An example of a [[#Function|function]] that is not injective.]]
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A partial function arises from the consideration of maps between two sets {{mvar|X}} and {{mvar|Y}} that may not be defined on the entire set {{mvar|X}}.  A common example is the square root operation on the real numbers <math>\mathbb{R}</math>: because negative real numbers do not have real square roots, the operation can be viewed as a partial function from <math>\mathbb{R}</math> to <math>\mathbb{R}.</math>  The ''domain of definition'' of a partial function is the subset {{mvar|S}} of {{mvar|X}} on which the partial function is defined; in this case, the partial function may also be viewed as a function from {{mvar|S}} to {{mvar|Y}}.  In the example of the square root operation, the set {{mvar|S}} consists of the nonnegative real numbers <math>[0, +\infty).</math>

The notion of partial function is particularly convenient when the exact domain of definition is unknown or even unknowable. For a computer-science example of the latter, see ''[[Halting problem]]''.

In case the domain of definition {{mvar|S}} is equal to the whole set {{mvar|X}}, the partial function is said to be ''total''.  Thus, total partial functions from {{mvar|X}} to {{mvar|Y}} coincide with functions from {{mvar|X}} to {{mvar|Y}}.

Many properties of functions can be extended in an appropriate sense of partial functions.  A partial function is said to be [[Injective function|injective]], [[Surjective function|surjective]], or [[Bijection|bijective]] when the function given by the restriction of the partial function to its domain of definition is injective, surjective, bijective respectively.

Because a function is trivially surjective when restricted to its image, the term [[partial bijection]] denotes a partial function which is injective.<ref name="Hollings2014-251">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=251}}</ref>

An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse. Furthermore, a function which is injective may be inverted to a bijective partial function.

The notion of [[Transformation (function)|transformation]] can be generalized to partial functions as well. A '''partial transformation''' is a function <math>f : A \rightharpoonup B,</math> where both <math>A</math> and <math>B</math> are [[subset]]s of some set <math>X.</math><ref name="Hollings2014-251"/>