Formal proof for Proposition 3.38 (Equivalence of a set of formulas and their clausal forms) A set of formulas U is equivalent to the set of clauses comprising the union of the clausal forms of the formulas in U . Please use existing theorems to answer the formal logic problem.

To prove Proposition 3.38, we need to show that a set of formulas U is equivalent to the set of clauses comprising the union of the clausal forms of the formulas in U.

Let's denote the set of formulas U as \Phi , and the set of clauses comprising the union of the clausal forms of the formulas in U as \Psi . We want to show that \Phi \equiv \Psi .

Given a formula \phi , its clausal form is denoted as cl(\phi) . The clausal form of a set of formulas is the union of the clausal forms of each formula.

We need to show two things:
1. If a formula \phi \in \Phi , then cl(\phi) \in \Psi .
2. If a clause C \in \Psi , then there exists a formula \phi \in \Phi such that cl(\phi) = C .

Let's prove each part:
1. Let \phi \in \Phi , then by definition of clausal form, cl(\phi) is in the union of the clausal forms of the formulas in U. Therefore, cl(\phi) \in \Psi .

2. Let C \in \Psi , then C is a clause in the union of clausal forms of the formulas in U. By the definition of clausal form, there exists a formula \phi \in \Phi such that cl(\phi) = C .

Therefore, we have shown that if \Phi is a set of formulas, and \Psi is the set of clauses comprising the union of the clausal forms of the formulas in \Phi , then \Phi \equiv \Psi .

\boxed{\Phi \equiv \Psi}