# Left and Right Inverses of Mapping are Inverse Mapping/Proof 3

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## Theorem

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

- $(1): \quad \exists g_1: T \to S: g_1 \circ f = I_S$
- $(2): \quad \exists g_2: T \to S: f \circ g_2 = I_T$

Then:

- $g_1 = g_2 = f^{-1}$

where $f^{-1}$ is the inverse of $f$.

## Proof

Because Composition of Mappings is Associative, brackets do not need to be used.

\((1):\quad\) | \(\displaystyle g_1 \circ f\) | \(=\) | \(\displaystyle I_S\) | ||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle g_1 \circ f \circ g_2\) | \(=\) | \(\displaystyle I_S \circ g_2\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle g_2\) | Definition of Identity Mapping |

\((2):\quad\) | \(\displaystyle f \circ g_2\) | \(=\) | \(\displaystyle I_T\) | ||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle g_1 \circ f \circ g_2\) | \(=\) | \(\displaystyle g_1 \circ I_T\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle g_1\) | Definition of Identity Mapping |

Thus $g_1 = g_2$.

Now suppose there exists $g_3: T \to S: g_3 \circ f = I_S$.

By the same argument as above, $g_3 = g_2$.

This means that $g_1 (= g_3)$ is the only left inverse of $f$.

Similarly, suppose there exists $g_4: T \to S: f \circ g_4 = I_T$.

By the same argument as above, $g_4 = g_1$.

This means that $g_2 (= g_4)$ is the only right inverse of $f$.

So $g_1 = g_2 = g_3 = g_4$ are all the same.

By Composite of Bijection with Inverse is Identity Mapping, it follows that this unique mapping is the inverse $f^{-1}$.

$\blacksquare$

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.7$: Inverses: Proposition $\text{A}.7.3$