[SOLVED] Binary Operations

Suppose that two binary operations, denoted by $\displaystyle \oplus$ and $\displaystyle \odot$, are defined on a nonempty set S, and that the following conditions are satisfied $\displaystyle \forall x,y,z\in S$.

(1) $\displaystyle x\oplus y$ and $\displaystyle x\odot y$ are in S

(2) $\displaystyle x\oplus (y\oplus z)=(x\oplus y) \oplus z$ and $\displaystyle x\odot (y\odot z)=(x\odot y) \odot z$

(3) $\displaystyle x\oplus y=y\oplus x$

Also, $\displaystyle \forall x\in S$ and $\displaystyle \forall n\in \mathbb{Z}^+$, the elements $\displaystyle nx$ and $\displaystyle x^n$ are defined recursively as follows:

$\displaystyle 1x=x^1=x$

if $\displaystyle kx$ and $\displaystyle x^k$ have been defined, then $\displaystyle (k+1)x=kx\oplus x$ and $\displaystyle x^{k+1}=x^k\odot x$

Which of the following must be true?

(i) $\displaystyle (x\odot y)^n=x^n\odot y^n$ $\displaystyle \forall x,y\in S$ and $\displaystyle \forall n\in\mathbb{Z}^+$

(ii) $\displaystyle n(x\oplus y)=nx\oplus ny$ $\displaystyle \forall x,y\in S$ and $\displaystyle \forall n\in\mathbb{Z}^+$

(ii) $\displaystyle x^m\odot x^n=x^{m+n}$ $\displaystyle \forall x\in S$ and $\displaystyle \forall m,n\in\mathbb{Z}^+$

This is one obviously true.

I am struggling with proving or disproving 1 and 2