化简 $$$\left(X \cdot Y\right) + \overline{Y \cdot Z} + \left(X \cdot \overline{Y} \cdot Z \cdot \left(\left(X \cdot Y\right) + Z\right)\right)$$$

应用交换律：

$$\left(X \cdot Y\right) + \overline{Y \cdot Z} + \left(X \cdot \overline{Y} \cdot Z \cdot {\color{red}\left(\left(X \cdot Y\right) + Z\right)}\right) = \left(X \cdot Y\right) + \overline{Y \cdot Z} + \left(X \cdot \overline{Y} \cdot Z \cdot {\color{red}\left(Z + \left(X \cdot Y\right)\right)}\right)$$

将吸收律 $$$x \cdot \left(x + y\right) = x$$$ 应用于 $$$x = Z$$$ 和 $$$y = X \cdot Y$$$：

$$\left(X \cdot Y\right) + \overline{Y \cdot Z} + \left(X \cdot \overline{Y} \cdot {\color{red}\left(Z \cdot \left(Z + \left(X \cdot Y\right)\right)\right)}\right) = \left(X \cdot Y\right) + \overline{Y \cdot Z} + \left(X \cdot \overline{Y} \cdot {\color{red}\left(Z\right)}\right)$$

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

$$\left(X \cdot Y\right) + {\color{red}\left(\overline{Y \cdot Z}\right)} + \left(X \cdot \overline{Y} \cdot Z\right) = \left(X \cdot Y\right) + {\color{red}\left(\overline{Y} + \overline{Z}\right)} + \left(X \cdot \overline{Y} \cdot Z\right)$$

应用交换律：

$${\color{red}\left(\left(X \cdot Y\right) + \overline{Y} + \overline{Z} + \left(X \cdot \overline{Y} \cdot Z\right)\right)} = {\color{red}\left(\left(X \cdot Y\right) + \overline{Y} + \left(X \cdot \overline{Y} \cdot Z\right) + \overline{Z}\right)}$$

应用交换律：

$$\left(X \cdot Y\right) + \overline{Y} + {\color{red}\left(X \cdot \overline{Y} \cdot Z\right)} + \overline{Z} = \left(X \cdot Y\right) + \overline{Y} + {\color{red}\left(\overline{Y} \cdot X \cdot Z\right)} + \overline{Z}$$

将吸收律 $$$x + \left(x \cdot y\right) = x$$$ 应用于 $$$x = \overline{Y}$$$ 和 $$$y = X \cdot Z$$$：

$$\left(X \cdot Y\right) + {\color{red}\left(\overline{Y} + \left(\overline{Y} \cdot X \cdot Z\right)\right)} + \overline{Z} = \left(X \cdot Y\right) + {\color{red}\left(\overline{Y}\right)} + \overline{Z}$$

应用交换律：

$${\color{red}\left(\left(X \cdot Y\right) + \overline{Y} + \overline{Z}\right)} = {\color{red}\left(\overline{Y} + \left(X \cdot Y\right) + \overline{Z}\right)}$$

应用交换律：

$$\overline{Y} + {\color{red}\left(X \cdot Y\right)} + \overline{Z} = \overline{Y} + {\color{red}\left(Y \cdot X\right)} + \overline{Z}$$

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

$${\color{red}\left(\overline{Y} + \left(Y \cdot X\right)\right)} + \overline{Z} = {\color{red}\left(\overline{Y} + X\right)} + \overline{Z}$$