Partial function: Difference between revisions

Revision as of 22:05, 14 August 2022

Revision as of 18:08, 14 December 2022

Line 44:

== 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> denoted by <math>[X \rightharpoonup Y],</math> is the union 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>

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).