# Definition:Inverse of Mapping

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## Contents

## Definition

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping

The **inverse** of $f$ is the relation defined as:

- $f^{-1} := \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$

That is:

- $f^{-1} := \left\{{\left({t, s}\right): \left({s, t}\right) \in f}\right\}$

That is, $f^{-1} \subseteq T \times S$ is the relation which satisfies:

- $\forall s \in S: \forall t \in T: \left({t, s}\right) \in f^{-1} \iff \left({s, t}\right) \in f$

## Also known as

The **inverse** of a mapping is also known as its **converse**.

Some authors use the notation $f^\gets$ instead of $f^{-1}$.

## Also see

- Definition:Preimage of Mapping (also known as
**inverse image**)

- Inverse of Mapping is One-to-Many Relation where it is demonstrated that $f^{-1}$ is in general not itself a mapping.

## Sources

- Paul R. Halmos:
*Naive Set Theory*(1960)... (previous)... (next): $\S 10$: Inverses and Composites - Seth Warner:
*Modern Algebra*(1965)... (previous)... (next): $\S 5$ - George McCarty:
*Topology: An Introduction with Application to Topological Groups*(1967): $\text{I}$, Problem $\text {AA}$ - Robert H. Kasriel:
*Undergraduate Topology*(1971)... (previous)... (next): $\S 1.11$: Relations