# Definition:Set of All Mappings

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## 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.

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

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.47$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products: Exercise $1$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: Further exercises: $2$ - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions