Transplant (Abstract Algebra)/Examples/Multiplication on Reals under 1-x

Example of Transplant

Let $\struct {\R, \times}$ be the set of real numbers under multiplication.

Let $f: \R \to \R$ be the permutation defined as:

$\forall x \in \R: \map f x = 1 - x$

The transplant $\otimes$ of $\times$ under $f$ is given by:

$x \otimes y = x + y - x y$

Proof

The fact that $f$ is a bijection is taken as read.

The inverse of $f$ is given as:

$\forall x \in \R: \map {f^{-1} } x = 1 - x$

Hence from the Transplanting Theorem:

\(\ds \forall x, y \in \R: \, \)

\(\ds x \otimes y\)

\(=\)

\(\ds \map f {\map {f^{-1} } x \times \map {f^{-1} } y}\)

\(\ds \)

\(=\)

\(\ds \map f {\paren {1 - x} \times \paren {1 - y} }\)

\(\ds \)

\(=\)

\(\ds \map f {1 - x - y + x y}\)

\(\ds \)

\(=\)

\(\ds 1 - \paren {1 - x - y + x y}\)

\(\ds \)

\(=\)

\(\ds x + y - x y\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.8 \ \text {(a)}$