对称差的结合律

对于集合 $A,B$，有
$$\begin{array}{rl}(A\oplus B)\oplus C& =((A-B)\cup (B-A))\oplus C\\ & =((A\cup B)\cap (\bar{A}\cup \bar{B}))\oplus C\\ & =((A\cup B)\cap (\bar{A}\cup \bar{B})\cup C)\cap ((\bar{A}\cap \bar{B})\cup (A\cap B)\cup \bar{C})\\ & =((A\cup B\cup C)\cap (\bar{A}\cup \bar{B}\cup C))\cap (((\bar{A}\cup B)\cap (A\cup \bar{B}))\cup \bar{C})\\ & =((A\cup B\cup C)\cap (\bar{A}\cup \bar{B}\cup C))\\ & \cap ((\bar{A}\cup B\cup \bar{C})\cap (A\cup \bar{B}\cup \bar{C}))\\ & =(A\cup B\cup C)\cap (\bar{A}\cup \bar{B}\cup C)\cap (\bar{A}\cup B\cup \bar{C})\cap (A\cup \bar{B}\cup \bar{C})\end{array}$$

对于结合后的对称差，有

$$\begin{array}{rl}A\oplus (B\oplus C)& =(B\oplus C)\oplus A=((B-C)\cup (C-B))\oplus A\\ & =((B\cup C)\cap (\bar{B}\cup \bar{C}))\oplus A\\ & =((B\cup C)\cap (\bar{B}\cup \bar{C})\cup A)\cap ((\bar{B}\cap \bar{C})\cup (B\cap C)\cup \bar{A})\\ & =((B\cup C\cup A)\cap (\bar{B}\cup \bar{C}\cup A))\cap (((\bar{B}\cup C)\cap (B\cup \bar{C}))\cup \bar{A})\\ & =((B\cup C\cup A)\cap (\bar{B}\cup \bar{C}\cup A))\\ & \cap ((\bar{B}\cup C\cup \bar{A})\cap (B\cup \bar{C}\cup \bar{A}))\\ & =(B\cup C\cup A)\cap (\bar{B}\cup \bar{C}\cup A)\cap (\bar{B}\cup C\cup \bar{A})\cap (B\cup \bar{C}\cup \bar{A})\\ & =(A\oplus B)\oplus C\end{array}$$

证毕。