Editing Partial function (section)

== Function spaces ==

For 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> by <math>[X \rightharpoonup Y].</math>  This set is the union of the sets of 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).