Let $\Sigma - \{0, 1\}$. Let $A, \: B$ be arbitrary subsets of $\Sigma^*$. We define the following operatins on such sets:
$$ A+B := \{ w \in \Sigma^* \mid w \in A \text{ or } w \in B \}$$
$$A \cdot B := \{ uv \in \Sigma^* \mid u \in A \text{ and } v \in B \} $$
$$ 2A := \{ ww \in \Sigma^* \mid w \in A \}$$
Is it true that $(A+B) \cdot (A+B) = A \cdot A + B \cdot B +2(A \cdot B)$ for all choices of $A$ and $B$? If yes, give a proof. If not, provide suitable $A$ and $B$ for which this equation fails.