Simplify $\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A \cdot \overline{B}$

The calculator will simplify the boolean expression $\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A \cdot \overline{B}$, with steps shown.

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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 $X = \overline{D}$:

$${\color{red}\left(\overline{D} \cdot 0\right)} = {\color{red}\left(0\right)}$$

$\overline{A} \cdot \overline{B} \cdot \overline{C} \cdot \overline{D} \cdot A \cdot \overline{B} = 0$