化简 $$$\left(\left(1 \cdot 0\right) + \overline{0}\right) \cdot \left(\overline{1} + 0 + 1\right)$$$

应用否定律 $$$\overline{0} = 1$$$：

$$\left(\left(1 \cdot 0\right) + {\color{red}\left(\overline{0}\right)}\right) \cdot \left(\overline{1} + 0 + 1\right) = \left(\left(1 \cdot 0\right) + {\color{red}\left(1\right)}\right) \cdot \left(\overline{1} + 0 + 1\right)$$

应用否定律 $$$\overline{1} = 0$$$：

$$\left(\left(1 \cdot 0\right) + 1\right) \cdot \left({\color{red}\left(\overline{1}\right)} + 0 + 1\right) = \left(\left(1 \cdot 0\right) + 1\right) \cdot \left({\color{red}\left(0\right)} + 0 + 1\right)$$

对 $$$x = 1 \cdot 0$$$ 应用支配（零化、湮灭）律 $$$x + 1 = 1$$$：

$${\color{red}\left(\left(1 \cdot 0\right) + 1\right)} \cdot \left(0 + 0 + 1\right) = {\color{red}\left(1\right)} \cdot \left(0 + 0 + 1\right)$$

对 $$$x = 0$$$ 应用支配（零化、湮灭）律 $$$x + 1 = 1$$$：

$$1 \cdot \left(0 + {\color{red}\left(0 + 1\right)}\right) = 1 \cdot \left(0 + {\color{red}\left(1\right)}\right)$$

对 $$$x = 0$$$ 应用支配（零化、湮灭）律 $$$x + 1 = 1$$$：

$$1 \cdot {\color{red}\left(0 + 1\right)} = 1 \cdot {\color{red}\left(1\right)}$$

对 $$$x = 1$$$ 应用恒等律 $$$x \cdot 1 = x$$$：

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