Image of Element under Inverse Mapping/Corollary 2
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Theorem
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping such that its inverse $f^{-1}: T \to S$ is also a mapping.
Then:
- $\forall y \in T: \map f {\map {f^{-1} } y} = y$
Proof
\(\ds \forall x \in S, y \in T: \, \) | \(\ds \map f x\) | \(=\) | \(\ds y\) | |||||||||||
\(\, \ds \iff \, \) | \(\ds \map {f^{-1} } y\) | \(=\) | \(\ds x\) | Image of Element under Inverse Mapping | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall y \in T: \, \) | \(\ds \map f {\map {f^{-1} } y}\) | \(=\) | \(\ds y\) | substituting $\map {f^{-1} } y$ for $x$ |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse: 1. (of a function)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse: 1. (of a function)