The map $\phi: \mathbb{Q} \to \mathbb{Q}$ defined by $\phi (x) = 3x-1$ for $x \in \mathbb{Q}$ is one to one and onto $\mathbb{Q}$, Give the definition of a binary operation * on $\mathbb{Z}$ such that $\phi$ is an isomorphism mapping
a) <$\mathbb{Q}$,+> onto <$\mathbb{Q}$, * >
b) <$\mathbb{Q}$,*> onto <$\mathbb{Q}$, + >
The solution said
Question
For (a), I tried doing it like this
$\phi(x+y) = 3(x+y) -1=3x + 3y - 1$
$\phi(x) *\phi(y)=(3x-1)*(3y-1)$
Setting them equal to each other, I get $(3x-1)*(3y-1)= 3x + 3y - 1 \iff (3x)*(3y) = 3x + 3y + 1 \iff x * y = x + y + 1/3$. It doesn't match up...
The method worked for (b). Why doesn't it work on (a)?