(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