Inverse Mapping is Unique
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Theorem
Let $f: S \to T$ be a mapping.
If $f$ has an inverse mapping, then that inverse mapping is unique.
That is, if:
- $f$ and $g$ are inverse mappings of each other
and
- $f$ and $h$ are inverse mappings of each other
then $g = h$.
Proof 1
By the definition of inverse mapping:
\(\ds g \circ f\) | \(=\) | \(\ds I_S\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds h \circ f\) |
and:
\(\ds f \circ g\) | \(=\) | \(\ds I_T\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds f \circ h\) |
So:
\(\ds h\) | \(=\) | \(\ds h \circ I_T\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds h \circ \paren {f \circ g}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {h \circ f} \circ g\) | Composition of Mappings is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds I_S \circ g\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds g\) |
So $g = h$ and the inverse is unique.
$\blacksquare$
Proof 2
We need to show that:
- $\forall t \in T: \map g t = \map h t$
So:
\(\ds \map f {\map g t}\) | \(=\) | \(\ds t\) | Definition of Inverse Mapping | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map h t\) | \(=\) | \(\ds \map h {\map f {\map g t} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map h t\) | \(=\) | \(\ds \map g t\) | as $\forall s \in S: \map h {\map f s} = s$ |
$\blacksquare$
Hence the result.
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.3: \ 12 \ \text{(a)}$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{G}$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Inverse images and inverse functions
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Proposition $3.2$: Remark