# Inverse of Composite Bijection

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## Theorem

Let $f$ and $g$ be bijections such that $\Dom g = \Cdm f$.

Then:

- $\paren {g \circ f}^{-1} = f^{-1} \circ g^{-1}$

and $f^{-1} \circ g^{-1}$ is itself a bijection.

## Proof 1

$\left({g \circ f}\right)^{-1} = f^{-1} \circ g^{-1}$ is a specific example of Inverse of Composite Relation.

As $f$ and $g$ are bijections then by Bijection iff Inverse is Bijection, so are both $f^{-1}$ and $g^{-1}$.

By Composite of Bijections is Bijection, it follows that $f^{-1} \circ g^{-1}$ is a bijection.

$\blacksquare$

## Proof 2

Let $f: X \to Y$ and $g: Y \to Z$ be bijections.

Then:

\(\displaystyle \paren {g \circ f} \circ \paren {f^{-1} \circ g^{-1} }\) | \(=\) | \(\displaystyle g \circ \paren {\paren {f \circ f^{-1} } \circ g^{-1} }\) | Composition of Mappings is Associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle g \circ \paren {I_Y \circ g^{-1} }\) | Composite of Bijection with Inverse is Identity Mapping | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle g \circ g^{-1}\) | Identity Mapping is Left Identity | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle I_Z\) | Composite of Bijection with Inverse is Identity Mapping |

\(\displaystyle \paren {f^{-1} \circ g^{-1} } \circ \paren {g \circ f}\) | \(=\) | \(\displaystyle \paren {f^{-1} \circ \paren {g^{-1} \circ g} } \circ f\) | Composition of Mappings is Associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {f^{-1} \circ I_Y} \circ f\) | Composite of Bijection with Inverse is Identity Mapping | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle f^{-1} \circ f\) | Identity Mapping is Right Identity | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle I_X\) | Composite of Bijection with Inverse is Identity Mapping |

Hence the result.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{J}$ - 2011: Robert G. Bartle and Donald R. Sherbert:
*Introduction to Real Analysis*(4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions