Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Show that if $A$ and $B$ are two subsets of $\mathbb{R}^n$ and $\gamma \in \mathbb{Z}^n$ we have
$$E_{A \cup B} + E_{A \cap B} = E_A + E_B \quad \text{and} \quad E_{\gamma + A} = x^\gamma E_A.$$