+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Mapping space +--{: .hide} [[!include mapping space - contents]] =-- =-- =-- # Function sets * table of contents {: toc} ## Definitions Given [[sets]] $A$ and $B$, the __function set__ $A^B$ is the set of all [[functions]] from $B$ to $A$. In the [[foundations]] of mathematics, the existence of such a set may be taken to follow from the existence of [[power sets]], from the axiom of [[subset collection]], or as an axiom (the __axiom of function sets__) in its own right. ## Generalisations Thinking of [[Set]] as a [[locally small category]], this is a special case of a [[hom-set]]. Thinking of $\Set$ as a [[cartesian closed category]], this is a special case of an [[exponential object]], which is a special case of an [[internal hom]]. ## Related concepts * [[function type]] * [[internal hom]] * [[bijection set]] category: foundational axiom [[!redirects function set]] [[!redirects function sets]] [[!redirects function-set]] [[!redirects function-sets]] [[!redirects axiom of function sets]] [[!redirects set of functions]]