Simplify $$$\overline{\overline{A \cdot B} + \left(\overline{D} \cdot A\right)}$$$

The calculator will simplify the boolean expression $$$\overline{\overline{A \cdot B} + \left(\overline{D} \cdot A\right)}$$$, with steps shown.

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Your Input

Simplify the boolean expression $$$\overline{\overline{A \cdot B} + \left(\overline{D} \cdot A\right)}$$$.

Solution

Apply de Morgan's theorem $$$\overline{X + Y} = \overline{X} \cdot \overline{Y}$$$ with $$$X = \overline{A \cdot B}$$$ and $$$Y = \overline{D} \cdot A$$$:

$${\color{red}\left(\overline{\overline{A \cdot B} + \left(\overline{D} \cdot A\right)}\right)} = {\color{red}\left(\overline{\overline{A \cdot B}} \cdot \overline{\overline{D} \cdot A}\right)}$$

Apply the double negation (involution) law $$$\overline{\overline{X}} = X$$$ with $$$X = A \cdot B$$$:

$${\color{red}\left(\overline{\overline{A \cdot B}}\right)} \cdot \overline{\overline{D} \cdot A} = {\color{red}\left(A \cdot B\right)} \cdot \overline{\overline{D} \cdot A}$$

Apply de Morgan's theorem $$$\overline{X \cdot Y} = \overline{X} + \overline{Y}$$$ with $$$X = \overline{D}$$$ and $$$Y = A$$$:

$$A \cdot B \cdot {\color{red}\left(\overline{\overline{D} \cdot A}\right)} = A \cdot B \cdot {\color{red}\left(\overline{\overline{D}} + \overline{A}\right)}$$

Apply the double negation (involution) law $$$\overline{\overline{X}} = X$$$ with $$$X = D$$$:

$$A \cdot B \cdot \left({\color{red}\left(\overline{\overline{D}}\right)} + \overline{A}\right) = A \cdot B \cdot \left({\color{red}\left(D\right)} + \overline{A}\right)$$

Apply the commutative law:

$${\color{red}\left(A \cdot B \cdot \left(D + \overline{A}\right)\right)} = {\color{red}\left(A \cdot \left(D + \overline{A}\right) \cdot B\right)}$$

Apply the commutative law:

$$A \cdot {\color{red}\left(D + \overline{A}\right)} \cdot B = A \cdot {\color{red}\left(\overline{A} + D\right)} \cdot B$$

Apply the redundancy law $$$X \cdot \left(\overline{X} + Y\right) = X \cdot Y$$$ with $$$X = A$$$ and $$$Y = D$$$:

$${\color{red}\left(A \cdot \left(\overline{A} + D\right)\right)} \cdot B = {\color{red}\left(A \cdot D\right)} \cdot B$$

Answer

$$$\overline{\overline{A \cdot B} + \left(\overline{D} \cdot A\right)} = A \cdot D \cdot B$$$