Definition:Set of All Mappings

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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.

Some sources write this as $^S T$.

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 $T^S$.

Also see