# Intersection of decision problems?

Say we have two problems $$\Pi_1\in NP$$ and $$\Pi_2\in coNP$$. Where does $$\Pi_1\cap\Pi_2$$ live?

• What do you mean by $\Pi_1 \land \Pi_2$? $\{ (x_1, x_2) \mid x_1 \in \Pi_1 \land x_2 \in \Pi_2 \}$? – dkaeae Jul 12 '19 at 15:50
• I'm not sure I understand. Do you mean then $\Pi_1 \cap \Pi_2$ (i.e., set intersection)? – dkaeae Jul 12 '19 at 16:07
• Given two formulas $\phi_1$ and $\phi_2$ we have $\exists x:\phi_1(x)=1$ is in $NP$ and $\forall y:\phi_2(y)=0$ is in $coNP$. Where is $\{\exists x:\phi_1(x)=1\}\wedge\{\forall y:\phi_2(y)=0\}$? – T.... Jul 12 '19 at 16:09
• No. As @dkaeae has explained, problems are sets of strings. – David Richerby Jul 12 '19 at 19:00
• I think Ariel meant second level. – Yuval Filmus Jul 13 '19 at 14:06