How to prove the following distributive property?
$\prod\limits_{i}^{n} (A_i \oplus B_i) = (\prod\limits_{i}^{n} A_i) \oplus (\prod\limits_{i}^{n} B_i)$
where $\prod$ means Cartesian Product, $\oplus$ means Minkowski sum, and A, B are sets.
I tried a simple example in a 2-d plane and the equality seems like correct. But I did not find a distribution law like this.
Thanks in advance!
Let $x$ such that for all $i$ there are $a_i\in A_i$ and $b_i\in B_i$ such that $x_i=a_i+b_i$. Then $a=(a_1,\cdots, a_n)\in \prod_{i=1}^nA_i$ and $b=(b_1,\cdots, b_n)\in \prod_{i=1}^nB_i$ satsify $a+b=x$. On the other hand, let $a+b=x$, with $a_i\in A_i$ and $b_i\in B_i$ for all $i$. Then, $x_i=a_i+b_i\in A_i\oplus B_i$ for all $i$. So $x\in \prod_{i=1}^n (A_i\oplus B_i)$.