The problem provided asks to find the Conjunctive Normal Form (CNF) of the given logical statement:

$(p \iff q) \implies (\neg p \land r)$

Steps to Convert to CNF:

1. Rewrite the logical operators: We need to express the bi-conditional (\iff) and implication (\implies) in terms of logical conjunctions (\land), disjunctions (\lor), and negations (\neg).

   • The bi-conditional (p \iff q) is equivalent to: $(p \implies q) \land (q \implies p)$ Which simplifies further as: $(\neg p \lor q) \land (\neg q \lor p)$

   • The implication (A \implies B) is equivalent to: $\neg A \lor B$ Hence, $(p \iff q) \implies (\neg p \land r)$ becomes: $\neg((p \iff q)) \lor (\neg p \land r)$ Now substitute the expression for p \iff q: $\neg((\neg p \lor q) \land (\neg q \lor p)) \lor (\neg p \land r)$

2. Distribute and simplify: Apply De Morgan's law and distribute the negations.

I will complete the calculation steps to find the CNF based on this. Let me work this out fully.The Conjunctive Normal Form (CNF) of the given logical expression $(p \iff q) \implies (\neg p \land r)$ is:

$(p \lor q \lor r) \land (\neg p \lor \neg q)$

This means the expression is broken down into two clauses, both in disjunctive form, which are combined using a conjunction (\land).

Would you like further clarification on the CNF, or a step-by-step breakdown of how this was derived?

Here are 5 related questions:

1. How does the CNF format help in simplifying logical statements?
2. What are the key steps involved in converting a logical expression into CNF?
3. Can every logical expression be converted into CNF?
4. How does the truth table method validate the CNF form?
5. What are De Morgan's laws and how do they apply in CNF conversion?

Tip: When converting a logical expression to CNF, always break down the logical operators (\iff, \implies) into simpler forms first before applying distributive and De Morgan's laws.