Partial function: Difference between revisions

Revision as of 18:08, 14 December 2022

Revision as of 19:10, 14 December 2022

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== Function spaces ==

The set of all partial functions <math>f : X \rightharpoonup Y</math> from a set <math>X</math> to a set <math>Y,</math> may be denoted by <math>\{X \rightharpoonup Y\},</math> and is the unionset of all functions definedwith on[[codomain]] subsets{{mvar|Y}} which have a subset of <math>{{mvar|X</math>}} withas samea codomain___domain. If the set of all functions from {{mvar|A}} to {{mvar|B}} is denoted <math>Y\{A\to B\}, </math>: then the set of all partial functions from {{mvar|X}} to {{mvar|Y}} is

: <math>\{X \rightharpoonup Y\} = \bigcup_{D \subseteq{X}} (\{D \to Y),\};</math>

theit lattercan also be written as <math display="inline">\bigcup_{D\subseteq{X}} Y^D.</math> In finite case, its cardinality is

: <math>|X \rightharpoonup Y| = (|Y| + 1)^{|X|},</math>

because any partial function can be extended to a function by any fixed value <math>c</math> not contained in <math>Y,</math> so that the codomain is <math>Y \cup \{ c \},</math> an operation which is injective (unique and invertible by restriction).