TEST

$$ \begin{aligned} [f({\cal P})=f({\cal Q})]&=\sum_{\sf x}[P_{\sf x}=Q_{\sf x}] \\ &=\sum (P_{\sf x}\land Q_{\sf x})\lor(\overline{P_{\sf x}}\land \overline{Q_{\sf x}}) \\ &=\sum 1-\big(1-(P_{\sf x}\cdot Q_{\sf x})\big)\cdot\big(1-(1-P_{\sf x})\cdot (1-Q_{\sf x})\big) \\ &=\sum 2P_{\sf x}Q_{\sf x}-(P_{\sf x}+Q_{\sf x})+1 \\ &=\kern{-20px}\sum_{\substack{A\sube f({\cal P})\cap f({\cal Q})\\B\sube f({\cal P}),C\sube f({\cal Q})}}\kern{-18px}[A\cap B=\varnothing][A\cap C=\varnothing][B\cap C=\varnothing] 2^{|A|}(-1)^{|B|+|C|} \\ &=\kern{-19px}\sum_{S\sube f({\cal P}),T\sube f({\cal Q})}\kern{-17px}2^{|S\cap T|}(-1)^{|S|+|T|} \end{aligned} $$