Partial function: Difference between revisions

Revision as of 13:22, 16 December 2022

Revision as of 18:01, 16 December 2022

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== Function spaces ==

TheFor convenience, denote the set of all partial functions <math>f : X \rightharpoonup Y</math> from a set <math>X</math> to a set <math>Y,</math> denoted by <math>[X \rightharpoonup Y],.</math>{{cn|reason=3 different notationsThis haveset been proposed recently by different Wikipedia editors, viz. "X→Y", "[X→Y]", and "{X→Y}". Only those that are supported by citations should be kept (resp. added tois the currentunion version).|date=December 2022}} isof the unionsets of all functions defined on subsets of <math>X</math> with same codomain <math>Y</math>:

: <math>[X \rightharpoonup Y] = \bigcup_{D \subseteq{ X}} [D \to Y],</math>

the latter also 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>