\[ F_S=\sum_{T \subseteq S}G_T \]
\[ G_S=\sum_{T \subseteq S}(-1)^{|S|-|T|}F_T \]
\[ F_{S_1,S_2...S_n}=\sum_{T_1 \subseteq S_1,T_2 \subseteq S_2 ... T_n \subseteq S_n}G_{T_1,T_2...T_n} \]
\[ G_{S_1,S_2...S_n}=\sum_{T_1 \subseteq S_1,T_2 \subseteq S_2 ... T_n \subseteq S_n}\prod_{i=1}^n (-1)^{|S_i|-|T_i|} F_{T_1,T_2...T_n} \]