Definition:Set of All Mappings

Definition

Let $S$ and $T$ be sets.

The set of (all) mappings from $S$ to $T$ is:

$T^S := \set {f \subseteq S \times T: f: S \to T \text { is a mapping} }$

Also known as

Some sources refer to $T^S$ as the power set, but such sources are not always clear as to what of.

Beware of confusing with this term used in the sense of the set of all subsets.

It is sometimes unwieldy to write $T^S$, particularly when $T$ and/or $S$ have themselves superscripts or subscripts attached.

In these cases, it is convenient to write $\sqbrk {S \to T}$ for the set of mappings from $S$ to $T$.

Some sources give $\map {\mathscr F_T} S$ or $\map {\mathscr F} {S, T}$ for $T^S$.