6.5.1 Predicate calculus

Problem: Reduce the predicate logic formula to prenex normal form. Is the formula on the set \( M=\{1,2\} \) : 1) satisfiable, 2) refutable, 3) generally valid, 4) unsatisfiable? Calculate the truth value of the formula on the set \( M \) with the following predicates: \begin{tabular}{|c|c|c|} \hline\( x \) & 1 & 2 \\ \hline\( P(x) \) & 1 & 0 \\ \hline\( R(x) \) & 0 & 1 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline\( Q(x, y) \) & 1 & 2 \\ \hline 1 & 1 & 0 \\ \hline 2 & 0 & 0 \\ \hline \end{tabular} \[ \forall x P(x) \rightarrow(R(x) \rightarrow \exists y Q(x, y)) \text {. } \]