Transplant (Abstract Algebra)/Examples/Addition on Positive Reals under Squaring
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Example of Transplant
Let $\struct {\R_{>0}, +}$ be the set of strictly positive real numbers under addition.
Let $f: \R_{>0} \to \R_{>0}$ be the permutation defined as:
- $\forall x \in \R_{>0}: \map f x = x^2$
The transplant $\oplus$ of $+$ under $f$ is given by:
- $x \oplus y = x + y + 2 \sqrt {x y}$
Proof
From Restriction of Real Square Mapping to Positive Reals is Bijection, $f$ is a bijection.
The inverse of $f$ is given as:
- $\forall x \in \R_{>0}: \map {f^{-1} } x = \sqrt x$
Hence from the Transplanting Theorem:
\(\ds \forall x, y \in \R_{>0}: \, \) | \(\ds x \oplus y\) | \(=\) | \(\ds \map f {\map {f^{-1} } x + \map {f^{-1} } y}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f {\sqrt x + \sqrt y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt x + \sqrt y}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x + y + 2 \sqrt {x y}\) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.8 \ \text {(b)}$