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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.
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