Partial function: Difference between revisions
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In [[mathematics]] and [[theoretical computer science]], a '''partial function''' on a set is a function whose [[domain of a function|domain]] of definition need not be the whole set. | In [[mathematics]] and [[theoretical computer science]], a '''partial function''' on a set is a function whose [[domain of a function|domain]] of definition need not be the whole set. | ||
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A function on ''X'' is ''total'' if its domain of definition is the whole of ''f'': that is, if ''f'' is a function on ''X'' in the usual sense. | A function on ''X'' is ''total'' if its domain of definition is the whole of ''f'': that is, if ''f'' is a function on ''X'' in the usual sense. | ||
The set of partial functions from ''X'' to ''Y'' is [[partially ordered|partial order]] by '' | The set of partial functions from ''X'' to ''Y'' is [[partially ordered|partial order]] by ''extension'': we say that ''f'' extends ''g'' if the domain of definition of ''f'' contains that of ''g'' and ''f'' agrees with ''g'' on the domain of ''g''. | ||
==References== | ==References== | ||
* {{cite book | author=Zohar Manna | title=Mathematical Theory of Computation | publisher=McGraw-Hill | year=1974 | isbn=0-07-039910-7 | pages=44 }} | * {{cite book | author=Zohar Manna | title=Mathematical Theory of Computation | publisher=McGraw-Hill | year=1974 | isbn=0-07-039910-7 | pages=44 }} | ||
Latest revision as of 09:22, 19 June 2009
In mathematics and theoretical computer science, a partial function on a set is a function whose domain of definition need not be the whole set.
We may define a partial function f from X to Y by extending the codomain Y to Y* by an element ω for "undefined", assumed not to be in Y, so that f is a function in the usual sense from X to Y*, and we regard the domain of definition of f as the subset of X on which f does not take the value ω.
A function on X is total if its domain of definition is the whole of f: that is, if f is a function on X in the usual sense.
The set of partial functions from X to Y is partial order by extension: we say that f extends g if the domain of definition of f contains that of g and f agrees with g on the domain of g.
References
- Zohar Manna (1974). Mathematical Theory of Computation. McGraw-Hill, 44. ISBN 0-07-039910-7.