Simplify $$$\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A \cdot \overline{B}$$$
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Simplify the boolean expression $$$\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A \cdot \overline{B}.$$$
Solution
Apply the commutative law:
$${\color{red}\left(\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A \cdot \overline{B}\right)} = {\color{red}\left(\overline{A} \cdot \overline{B} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A\right)}$$Apply the idempotent law $$$X \cdot X = X$$$ with $$$X = \overline{B}$$$:
$$\overline{A} \cdot {\color{red}\left(\overline{B} \cdot \overline{B}\right)} \cdot \overline{C} \cdot \overline{D} \cdot A = \overline{A} \cdot {\color{red}\left(\overline{B}\right)} \cdot \overline{C} \cdot \overline{D} \cdot A$$Apply the commutative law:
$${\color{red}\left(\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A\right)} = {\color{red}\left(A \cdot \overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D}\right)}$$Apply the complement law $$$X \cdot \overline{X} = 0$$$ with $$$X = A$$$:
$${\color{red}\left(A \cdot \overline{A}\right)} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} = {\color{red}\left(0\right)} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D}$$Apply the commutative law:
$${\color{red}\left(0 \cdot \overline{B} \cdot \overline{C} \cdot \overline{D}\right)} = {\color{red}\left(\overline{B} \cdot 0 \cdot \overline{C} \cdot \overline{D}\right)}$$Apply the dominant (null, annulment) law $$$X \cdot 0 = 0$$$ with $$$X = \overline{B}$$$:
$${\color{red}\left(\overline{B} \cdot 0\right)} \cdot \overline{C} \cdot \overline{D} = {\color{red}\left(0\right)} \cdot \overline{C} \cdot \overline{D}$$Apply the commutative law:
$${\color{red}\left(0 \cdot \overline{C} \cdot \overline{D}\right)} = {\color{red}\left(\overline{C} \cdot 0 \cdot \overline{D}\right)}$$Apply the dominant (null, annulment) law $$$X \cdot 0 = 0$$$ with $$$X = \overline{C}$$$:
$${\color{red}\left(\overline{C} \cdot 0\right)} \cdot \overline{D} = {\color{red}\left(0\right)} \cdot \overline{D}$$Apply the commutative law:
$${\color{red}\left(0 \cdot \overline{D}\right)} = {\color{red}\left(\overline{D} \cdot 0\right)}$$Apply the dominant (null, annulment) law $$$X \cdot 0 = 0$$$ with $$$X = \overline{D}$$$:
$${\color{red}\left(\overline{D} \cdot 0\right)} = {\color{red}\left(0\right)}$$Answer
$$$\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A \cdot \overline{B} = 0$$$