位运算的小 trick

\[\def\or{\operatorname{or}} \def\and{\operatorname{and}} \def\xor{\operatorname{xor}} \def\not{\operatorname{not}} \begin{aligned} &(x\or y)\and z=(x\and z)\or(y\and z)\\ &\xor (a_i\and v)=(\xor a_i)\and v\\ &\xor (a_i\or v)= \begin{cases} (\xor a_i)\and\not v & \text{if } n\text{ is even.}\\ (\xor a_i)\or v & \text{if }n\text{ is odd.} \end{cases} \end{aligned} \]