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.
\(\text {(1)}: \quad\) | \(\ds g_1 \circ f\) | \(=\) | \(\ds I_S\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds g_1 \circ f \circ g_2\) | \(=\) | \(\ds I_S \circ g_2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds g_2\) | Definition of Identity Mapping |
\(\text {(2)}: \quad\) | \(\ds f \circ g_2\) | \(=\) | \(\ds I_T\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds g_1 \circ f \circ g_2\) | \(=\) | \(\ds g_1 \circ I_T\) | |||||||||||
\(\ds \) | \(=\) | \(\ds g_1\) | 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 $\text{A}.7.3$