We are tasked with proving that the two formulas are logically equivalent. The two formulas are:

1. $( \neg P \iff Q ) \implies R$
2. $( P \land Q ) \lor ( \neg P \land \neg Q ) \lor R$

To show that these two are equivalent, we will simplify both formulas step by step using logical rules.

Formula 1: $( \neg P \iff Q ) \implies R$

Step 1: Expand the biconditional (IFF)

The biconditional $\neg P \iff Q$ means $(\neg P \implies Q) \land (Q \implies \neg P)$. So, the first formula becomes:

$( (\neg P \implies Q) \land (Q \implies \neg P) ) \implies R$

Step 2: Express implications in terms of disjunction

Using the implication rule $(A \implies B) \equiv (\neg A \lor B)$, we rewrite both implications:

• $\neg P \implies Q \equiv P \lor Q$
• $Q \implies \neg P \equiv \neg Q \lor \neg P$

Thus, the formula becomes:

$((P \lor Q) \land (\neg Q \lor \neg P)) \implies R$

Step 3: Simplify the conjunction

Let's analyze the conjunction $(P \lor Q) \land (\neg Q \lor \neg P)$. This is a form of the exclusive OR ($P \oplus Q$), because it says that either $P$ is true and $Q$ is false,