does this operation hold?

Question

The answer tells me it does hold. But I just couldn't see it. Any help would be appreciated. Thank you!

Answer 1

It doesn't hold on $(\mathbb Z,+\times)$, right?! For instance $2^2+2^2\neq2^4$.

Answer 2

As noted, there is clearly a typo in the problem; it should be $\otimes$ and not $\oplus$. With that correction, the statement is true.

Let $n$ be given. We will show that $x^n\otimes x^m=x^{n+m}$ for all positive integers $m$ by induction.

First, $x^n\otimes x^1=x^{n+1}$ by your definition of the exponentiation notation.

Now, assume $x^n\otimes x^k=x^{n+k}$ for some positive integer $k$. Then

$$x^n\otimes x^{k+1}=x^n\otimes (x^k\otimes x)=(x^n\otimes x^k)\otimes x=x^{n+k}\otimes x=x^{n+(k+1)}$$ by the definition of the exponentiation, the associativity of $\otimes$, and the induction hypothesis. Therefore, by the principle of mathematical induction, the statement holds for all values of $m$ (and therefore also $n$, since it was arbitrarily chosen at the beginning).