Help to prove this combinatorial lemma

Question

Let $V_1,V_2 \subseteq \{1, \dots , n\}.$ Consider the function $$F(V_1,V_2)= \sum_{\substack{ U \subseteq\{1, \dots,n\} \\ V_1 \cup V_2 \subseteq U}} (-\iota )^{card(U - V_1)+card(U-V_2)}$$ Then $F(V_1,V_2)=0$ provided $V_1 \cup V_2 \ne \{1, \dots , n\}.$ This lemma is in a research paper. Can anyone help in proving the lemma? Thanks in advanced.

Answer 1

We write $U=(V_1 \cup V_2)\cup \hat U$ with $\hat U \cap (V_1 \cup V_2)= \emptyset.$ This determine $\hat U$ uniquely. Now

$$card(U-V_1)+card(U-V_2)=card(((V_1 \cup V_2)-V_1)\cup \hat U)+card(((V_1 \cup V_2)-V_2)\cup \hat U).$$ Since $ (V_1 \cup V_2)-V_i$ and $\hat U$ are disjoint for $i=1,2$ we get

$$card(U-V_1)+card(U-V_2)=card((V_1 \cup V_2)-V_1)+card((V_1 \cup V_2)-V_2) + 2 card(\hat U).$$ In the last line, the first two summands do not depend on $\hat U$ so we get

$$ \sum_{\substack{ U \\ V_1 \cup V_2 \subseteq U}} (-\iota )^{card(U - V_1)+card(U-V_2)}=(-\iota )^{card((V_1 \cup V_2)-V_1)+card((V_1 \cup V_2)-V_2)}.\sum_{\substack{\hat U\\ (V_1 \cup V_2) \cap \hat U \ne \emptyset}}(-\iota)^{2.card (\hat U)}$$ $$ =(-\iota )^{card((V_1 \cup V_2)-V_1)+card((V_1 \cup V_2)-V_2)}.\sum_{\substack{\hat U \subseteq \{1, \dots ,n\}-(V_1 \cup V_2)}}(-1)^{card (\hat U)}$$

This is not the complete proof but can someone tell me up till now is it right or not?, I have doubt in the 2nd last step.

Thank you!