We are tasked with simplifying the logical statement $(q \rightarrow p) \land \neg(r \land q)$.

Step 1: Use Implication Law

The implication $q \rightarrow p$ can be rewritten using the Implication Law: $q \rightarrow p \equiv \neg q \lor p$ So, the original expression becomes: $(\neg q \lor p) \land \neg(r \land q)$

Step 2: Apply De Morgan's Law to Negation

Next, let's simplify $\neg(r \land q)$ using De Morgan's Law. This law states that: $\neg(r \land q) \equiv \neg r \lor \neg q$ So, the expression becomes: $(\neg q \lor p) \land (\neg r \lor \neg q)$

Step 3: Apply the Distributive Law

Now, distribute $(\neg q \lor p)$ over $(\neg r \lor \neg q)$ using the Distributive Law: $(\neg q \lor p) \land (\neg r \lor \neg q) \equiv (\neg q \land (\neg r \lor \neg q)) \lor (p \land (\neg r \lor \neg q))$ Simplify the two parts of the expression:

1. $\neg q \land (\neg r \lor \neg q)$: Since $\neg q$ appears in both terms, we simplify this part to just $\neg q$ (by Idempotent Law).

2. $p \land (\neg r \lor \neg q)$: Distribute $p$ over $\neg r \lor \neg q$: $p \land (\neg r \lor \neg q) \equiv (p \land \neg r) \lor (p \land \neg q)$

Step 4: Combine the Results

Now we combine the results from the two parts: $\neg q \lor ((p \land \neg r) \lor (p \land \neg q))$ We can simplify this further using the Associative Law and Distributive Law: $\neg q \lor (p \land \neg r) \lor (p \land \neg q)$

Final Simplified Expression

Thus, the simplified form of the original expression is: $\neg q \lor (p \land \neg r) \lor (p \land \neg q)$

Recap of Steps:

1. Applied Implication Law: $q \rightarrow p \equiv \neg q \lor p$.
2. Applied De Morgan's Law: $\neg(r \land q) \equiv \neg r \lor \neg q$.
3. Applied Distributive Law to distribute $\neg q \lor p$ over $\neg r \lor \neg q$.
4. Simplified the result using the Idempotent Law and Associative Law.

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Do you want more details on any of these steps or the logical laws?

Here are five related questions for further exploration:

1. How would you simplify $(p \lor q) \land \neg(p \land r)$?
2. What is the negation of $(q \rightarrow p) \land r$?
3. Can you simplify $p \rightarrow (q \rightarrow r)$ using logical equivalences?
4. How do De Morgan's Laws apply to three terms, e.g., $\neg(p \land q \land r)$?
5. What is the equivalent disjunctive normal form (DNF) of the simplified expression?

Tip: Always apply De Morgan’s laws when dealing with negations over conjunctions or disjunctions.