Consider the propositional formula ( NOT ( P ) IFF Q ) IMPLIES R and the formula ( P AND Q ) OR ( NOT ( P ) AND NOT ( Q ) ) OR R .Prove that the two formulas are equivalent, using logical rules

We are given two propositional formulas:

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

We need to prove that these two formulas are logically equivalent. To do this, we will simplify both formulas using logical rules and show that they reduce to the same form.

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

We'll begin by breaking down this formula step by step:

#### Step 1: Expand \( \neg P \iff Q \)
The biconditional \( \iff \) can be rewritten as two implications:

\[
\neg P \iff Q \equiv (\neg P \implies Q) \land (Q \implies \neg P)
\]

#### Step 2: Rewrite each implication
Using the implication identity \( A \implies B \equiv \neg A \lor B \), we rewrite both implications:

\[
\neg P \implies Q \equiv P \lor Q
\]
\[
Q \implies \neg P \equiv \neg Q \lor \neg P
\]

So, the biconditional becomes:

\[
\neg P \iff Q \equiv (P \lor Q) \land (\neg Q \lor \neg P)
\]

#### Step 3: Substitution into the original formula
Now we substitute this back into the original formula:

\[
((P \lor Q) \land (\neg Q \lor \neg P)) \implies R
\]

#### Step 4: Apply implication
Again, using the implication identity \( A \implies B \equiv \neg A \lor B \), we rewrite the entire formula as:

\[
\neg ((P \lor Q) \land (\neg Q \lor \neg P)) \lor R
\]

#### Step 5: Apply De Morgan’s Law
To simplify \( \neg ((P \lor Q) \land (\neg Q \lor \neg P)) \), we use De Morgan's Law:

\[
\neg ((P \lor Q) \land (\neg Q \lor \neg P)) \equiv \neg (P \lor Q) \lor \neg (\neg Q \lor \neg P)
\]

Using De Morgan's Law again on both parts:

\[
\neg (P \lor Q) \equiv \neg P \land \neg Q
\]
\[
\neg (\neg Q \lor \neg P) \equiv Q \land P
\]

Thus, the formula simplifies to:

\[
(\neg P \land \neg Q) \lor (P \land Q) \lor R
\]

### Formula 2: \( (P \land Q) \lor (\neg P \land \neg Q) \lor R \)



Consider the propositional formula ( NOT ( P ) IFF Q ) IMPLIES R and the formula ( P AND Q ) OR ( NOT ( P ) AND NOT ( Q ) ) OR R. Prove that the two formulas are equivalent, using logical rules.

This solution demonstrates how to prove the logical equivalence between two propositional formulas: (NOT (P) IFF Q) IMPLIES R and (P AND Q) OR (NOT (P) AND NOT (Q)) OR R. It covers key concepts in propositional logic, including biconditionals, implications, and De Morgan's Laws.

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