Dual and Self Dual of Boolean equation, Boolean Algebra in Digital Electronics

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$\begin{align*} A \cdot (A + B) &= (A \cdot A) + (A \cdot B) & \text{Distributivity of AND over OR}\\ &= A + (A \cdot B) & \text{Idempotence of AND}\\ &= (A \cdot 1) + (A \cdot B) & \text{Identity of AND}\\ &= A \cdot (1 + B) & \text{Distributivity of AND over OR} \\ &= A \cdot 1 & \text{Dominance of OR}\\ \therefore A \cdot (A + B) &= A & \text{Identity of AND} \end{align}$

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Dual and Self Dual of Boolean equation, Boolean Algebra in Digital Electronics