# Definition:Set of All Mappings

## Contents

## 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 $T^S$.

## Also see

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: 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 - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 1.5$: Ordered Sets: Further exercises: $2$