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

对 $$$x = A$$$ 和 $$$y = \overline{C}$$$ 应用德摩根定律 $$$\overline{x + y} = \overline{x} \cdot \overline{y}$$$：

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

对 $$$x = C$$$ 应用双重否定（对合）律 $$$\overline{\overline{x}} = x$$$：

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

改写:

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

应用交换律：

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

将冗余律 $$$x + \left(\overline{x} \cdot y\right) = x + y$$$ 应用于 $$$x = \overline{B}$$$ 和 $$$y = \overline{C}$$$：

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

改写:

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

应用交换律：

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

对 $$$x = C$$$ 应用补集律 $$$x + \overline{x} = 1$$$：

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

应用交换律：

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

对 $$$x = \overline{B}$$$ 应用支配（零化、湮灭）律 $$$x + 1 = 1$$$：

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

应用交换律：

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

对 $$$x = \overline{A}$$$ 应用恒等律 $$$x \cdot 1 = x$$$：

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