⟦X→Y⟧

Since functions are things, we can consider sets of functions, and we can even create functions whose domain and codomain are themselves sets of functions

Definition: ⟦X→Y⟧

If [math]X [/math] and [math]Y [/math] are sets, then [math]⟦X→Y⟧ [/math] denotes the set of all functions with domain [math]X [/math] and codomain [math]Y [/math].

Note: I usually use the notation [X → Y], but it interferes with the wiki formatting, so I will use [math]⟦X→Y⟧ [/math] in the wiki. You may use either notation.

For example, if [math]X \href{/cs2800/wiki/index.php/Definition}{:=} \{1,2\} [/math] and [math]Y \href{/cs2800/wiki/index.php/Definition}{:=} \{a,b\} [/math], then [math]⟦X→Y⟧ [/math] would contain four functions:

• One taking both [math]1 [/math] and [math]2 [/math] to [math]a [/math]
• One taking both [math]1 [/math] and [math]2 [/math] to [math]b [/math]
• One taking both [math]1 [/math] to [math]a [/math] and [math]2 [/math] to [math]b [/math]
• One taking both [math]1 [/math] to [math]b [/math] and [math]2 [/math] to [math]a [/math]

It might be drawn this way (for example, to draw it as the domain of another function):