# Definition:Inverse Mapping/Definition 2

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## Definition

Let $f: S \to T$ and $g: T \to S$ be mappings.

Let:

- $g \circ f = I_S$
- $f \circ g = I_T$

where:

- $g \circ f$ and $f \circ g$ denotes the composition of $f$ with $g$ in either order
- $I_S$ and $I_T$ denote the identity mappings on $S$ and $T$ respectively.

That is, $f$ and $g$ are both left inverse mappings and right inverse mappings of each other.

Then:

- $g$ is
**the inverse (mapping) of $f$** - $f$ is
**the inverse (mapping) of $g$**.

## Also known as

If $f$ has an **inverse mapping**, then $f$ is an **invertible mapping**.

Hence, when the inverse (relation) of $f^{-1}$ is a mapping, we say that **$f$ has an inverse mapping**.

Some sources, in distinguishing this from a left inverse and a right inverse, refer to this as the **two-sided inverse**.

Some sources use the term **converse mapping** for **inverse mapping**.

## Also see

- Results about
**inverse mappings**can be found**here**.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.6$. Products of bijective mappings. Permutations: $\text{(iii)}$: Lemma - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Theorem $5.9$ - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $2$: Maps and relations on sets: Definition $2.12$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.7$: Inverses - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.8$