Determine whether the given map $\phi$ is an isomorphism of the following binary structures. Justify your answers.
$\langle\mathbb{R}, +\rangle$ with $\langle\mathbb{R}, +\rangle$, where $\phi(x) = x^5$, $\forall\: x \in \mathbb{R}$.
I don't understand what isomorphism means. Can anyone explain it? I tried reading the wiki but didn't understand it.
Best Answer
A map is an isomorphism if it is a homomorphism (i.e. a map that preserves the operations given in the structure) and is bijective (i.e. has an inverse).
Now $\phi:\Bbb R\to\Bbb R$ has an inverse (namely $x\mapsto \sqrt[5]x$ which is well defined on whole $\Bbb R$ as $5$ is odd), but, it does not preserve $+$, meaning that in general $$\phi(x+y) \ne\phi(x)+\phi(y)\,.$$