Partially ordered set - Revision history

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	~~{{Distinguish2\|partial function of a [[multilinear map]] or the mathematical concept of a [[piecewise function]]}}~~		I'm Weldon and I live in a seaside city in northern Belgium, Heusy. I'm 23 and I'm will soon finish my study at Human Ecology.<br><br>Feel free to visit my blog post; FIFA 15 coin hack ([http://Proformaweb.it/fantacalcio/?p=219%FFdeer+Hunter+2014+cheats%FFnofollow Recommended Online site])
	~~{\| align="right"~~
	\|-
	~~\|[[Image:Partial function.svg\|thumb\|200px\|An example of partial function that is injective.]]~~
	\|-
	~~\|[[Image:Total function.svg\|thumb\|200px\|An example of total function that is not-injective.]]~~
	\|}

	~~In [[mathematics]], a~~ '~~''partial function''' from ''X'' to ''Y'' (written as ''f: X ↛ Y'') is~~ a ~~[[function (mathematics)\|function]] ''f: X' → Y''~~, where '' X' '' is a [[subset]] of ''X''. It generalizes the concept of a function ''f: X → Y'' by not forcing ''f'' to map ''every'' element of ''X'' to an element of ''Y'' (only some subset ''X''<nowiki>'</nowiki> of ''X''). ~~If '' X' '' = ''X'', then ''f'' is called a '''total function''~~' and ~~is equivalent to a function. Partial functions are often used when the exact [[domain of a function\|domain]], '' X~~' ~~'', is not known (e.g. many functions in [[computability theory]]).~~

	~~Specifically, we~~ will ~~say that for any ''x'' ∈ ''X'', either:~~
	* ''f''(''x'') = ''y'' ∈ ''Y'' (it is defined as a single element in ''Y'') or
	* ''f''(''x'') is undefined.

	~~For example we can consider the [[square root]] function restricted to the [[integer]]s~~
	:<~~math~~>~~g\colon \mathbb{Z} \to \mathbb{Z}~~<~~/math~~>
	~~:<math>g(n) = \sqrt{n}.</math>~~

	~~Thus ''g''(''n'') is only defined for ''n'' that are [[square number\|perfect squares]] (i.e. 0, 1, 4, 9, 16, ...). So, ''g''(25)&nbsp~~;~~= 5, but ''g''(26) is undefined.~~

	~~==Domain of a partial function==~~
	There are two distinct meanings in current mathematical usage for the notion of the [[domain of a function\|domain]] of a partial function. Most mathematicians, including [[recursion theory\|recursion theorists]], use the term "domain of ''f''" for the set of all values ''x'' such that ''f(x)'' is defined ('' X' '' above). But some, particularly [[category theory\|category theorists]], consider the domain of a partial function ''f'':''X''→''Y'' to be ''X'', and refer to '' X' '' as the '''domain of definition'''. Similarly, the term [[range]] can refer to either the [[codomain]] or the ''image'' of a function.

	~~Occasionally, a partial function with domain ''X'' and codomain ''Y'' is written as ''f'': ''X'' ⇸ ''Y'', using an arrow with vertical stroke.~~

	A partial function is said to be [[injective]] or [[surjective]] when the total function given by the restriction of the partial function to its domain of definition is. A partial function may be both injective and surjective, but the term [[bijection]] generally only applies to total functions.

	~~An injective partial function may be [[inverse relation\|inverted]] to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse.~~

	~~==Total function==~~
	~~Total function is a synonym for [[Function~~ (~~mathematics)\|function]]. The use of the prefix "total" is to suggest that it is a special case of a~~ [[partial function]]. For example, when considering the operation of [[morphism]] composition in [[Category theory\|Concrete Categories]], the composition operation <math>\circ : \operatorname{Hom}(C) \times \operatorname{Hom}(C) \to \operatorname{Hom}(C)</math> is a total function if and only if <math>\operatorname{Ob}(C)</math> has one element. The reason for this is that two morphisms <math>f:~~X\to Y</math> and <math>g:U\to V</math> can only be composed as <math>g \circ f<~~/~~math> if <math>Y=U<~~/~~math>, that is, the codomain of <math>f</math> must equal the domain of <math>g</math>~~.

	~~==Discussion and examples==~~
	The first diagram above represents a partial function that is '''not''' a total function since the element 1 in the left-hand set is not associated with anything in the right-hand set. Whereas, the second diagram represents a total function since every element on the left-hand set is associated with exactly one element in the right hand set.

	~~===Natural logarithm===~~
	Consider the [[natural logarithm]] function mapping the [[real number]]s to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a total function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a total function.

	~~===Subtraction of natural numbers===~~
	~~Subtraction of [[natural numbers]] (non-negative [[integers]]) can be viewed as a partial function:~~

	~~:<math>f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}<~~/~~math>~~
	~~:<math>f(x,y) = x - y.<~~/~~math>~~

	~~It is only defined when <math>x \ge y</math>.~~

	~~===Bottom type===~~

	In some [[automated theorem proving]] systems a partial function is considered as returning the [[bottom type]] when it is undefined. The [[Curry-Howard correspondence]] uses this to map proofs and computer programs to each other.

	In [[computer science]] a partial function corresponds to a subroutine that raises an exception or loops forever. The [[IEEE floating point]] standard defines a [[Not-a-number]] value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested.

	In a [[programming language]] where function parameters are [[statically typed]], a function may be defined as a partial function because the language's [[type system]] cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the true domain.

	=~~= See also ==~~
	* [[Bijection]]
	* [[Injective function]]
	* [[Surjective function]]
	* [[Multivalued function]]
	* [[Symmetric inverse semigroup]]
	* [[Densely defined operator]]

	~~== References ==~~
	*[[Martin Davis]] (1958)~~, ''Computability and Unsolvability'', McGraw-Hill Book Company, Inc, New York. Republished by Dover in 1982. ISBN 0-486-61471-9.~~
	*[[Stephen Kleene]] (1952), ''Introduction to Meta-Mathematics'', North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9.
	*[[Harold S. Stone]] (1972), ''Introduction to Computer Organization and Data Structures'', McGraw-Hill Book Company, New York.

	~~[[Category:Mathematical relations]]~~
	~~[[Category:Functions and mappings]]~~

en>Citation bot: [458]Add: author, first, postscript. | Josve05a

2013-12-17T18:12:01Z

[458]Add: author, first, postscript. | Josve05a

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en>David Eppstein: /* Inverse and order dual */ link Duality (order theory)

2012-09-01T04:06:57Z

Inverse and order dual: link Duality (order theory)

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