化簡 $$$\left(\overline{A} \cdot \overline{B}\right) + \left(\overline{A} \cdot B \cdot \overline{C}\right) + \overline{A + \overline{C}}$$$

將德摩根定律 $$$\overline{x + y} = \overline{x} \cdot \overline{y}$$$ 應用於 $$$x = A$$$ 與 $$$y = \overline{C}$$$：

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

以 $$$x = C$$$ 套用雙重否定（對合）律 $$$\overline{\overline{x}} = x$$$：

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

改寫：

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

應用交換律：

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

將冗餘律$$$x + \left(\overline{x} \cdot y\right) = x + y$$$應用於$$$x = \overline{B}$$$與$$$y = \overline{C}$$$：

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

改寫：

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

應用交換律：

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

將補集法則 $$$x + \overline{x} = 1$$$ 應用於 $$$x = C$$$：

$$\left({\color{red}\left(C + \overline{C}\right)} + \overline{B}\right) \cdot \overline{A} = \left({\color{red}\left(1\right)} + \overline{B}\right) \cdot \overline{A}$$

應用交換律：

$${\color{red}\left(1 + \overline{B}\right)} \cdot \overline{A} = {\color{red}\left(\overline{B} + 1\right)} \cdot \overline{A}$$

對 $$$x = \overline{B}$$$ 應用支配（零、歸零）律 $$$x + 1 = 1$$$：

$${\color{red}\left(\overline{B} + 1\right)} \cdot \overline{A} = {\color{red}\left(1\right)} \cdot \overline{A}$$

應用交換律：

$${\color{red}\left(1 \cdot \overline{A}\right)} = {\color{red}\left(\overline{A} \cdot 1\right)}$$

將恆等律 $$$x \cdot 1 = x$$$ 應用於 $$$x = \overline{A}$$$：

$${\color{red}\left(\overline{A} \cdot 1\right)} = {\color{red}\left(\overline{A}\right)}$$