Given that $(\neg p \lor Q) \Rightarrow (R \land \neg R)$ is true, we need to determine the truth values of the statements $R \Rightarrow Q$, $P \land \neg R$, $(Q \Rightarrow R) \land R$, and $(Q \land R) \Rightarrow P$.

Step 1: Analyze the statement $(\neg p \lor Q) \Rightarrow (R \land \neg R)$

First, notice that the expression $R \land \neg R$ represents a contradiction because $R$ and $\neg R$ cannot both be true simultaneously. A contradiction is always false. Thus, $R \land \neg R$ is false.

For the conditional statement $(\neg p \lor Q) \Rightarrow (R \land \neg R)$ to be true, the antecedent $\neg p \lor Q$ must be false. This is because a conditional statement with a false consequent (which we have here) is true only when the antecedent is false.

Step 2: Analyze $\neg p \lor Q$

The disjunction $\neg p \lor Q$ is false only when both $\neg p$ and $Q$ are false.

• $\neg p$ is false implies $P$ is true.
• $Q$ is false.

So, we have:

• $P$ is true.
• $Q$ is false.
• $R$ can be either true or false (since $R \land \neg R$ forces no specific value on $R$).

Step 3: Determine the truth values for each statement

a) $R \Rightarrow Q$:

$R \Rightarrow Q$ is true if $R$ is false or $Q$ is true. Since $Q$ is false, the truth value of $R \Rightarrow Q$ depends on $R$:

• If $R$ is true, $R \Rightarrow Q$ is false.
• If $R$ is false, $R \Rightarrow Q$ is true.

b) $P \land \neg R$:

$P \land \neg R$ is true if both $P$ is true and $\neg R$ is true (i.e., $R$ is false). Since $P$ is true:

• If $R$ is true, $P \land \neg R$ is false.
• If $R$ is false, $P \land \neg R$ is true.

c) $(Q \Rightarrow R) \land R$:

$Q \Rightarrow R$ is true if $Q$ is false (which it is) or $R$ is true. Since $Q$ is false, $Q \Rightarrow R$ is true regardless of $R$.

• If $R$ is true, $(Q \Rightarrow R) \land R$ is true.
• If $R$ is false, $(Q \Rightarrow R) \land R$ is false.

d) $(Q \land R) \Rightarrow P$:

$Q \land R$ is true only if both $Q$ and $R$ are true. Since $Q$ is false, $Q \land R$ is false, and the conditional $(Q \land R) \Rightarrow P$ is true regardless of $P$.

Summary of Truth Values:

The truth values depend on whether $R$ is true or false:

• If $R$ is true:

  • a) $R \Rightarrow Q$ is false.
  • b) $P \land \neg R$ is false.
  • c) $(Q \Rightarrow R) \land R$ is true.
  • d) $(Q \land R) \Rightarrow P$ is true.

• If $R$ is false:

  • a) $R \Rightarrow Q$ is true.
  • b) $P \land \neg R$ is true.
  • c) $(Q \Rightarrow R) \land R$ is false.
  • d) $(Q \land R) \Rightarrow P$ is true.

Would you like further clarification or have any questions?

---

1. What does it mean when a conditional statement has a contradiction as a consequent?
2. How does the truth value of a disjunction work?
3. Can you explain why a conditional statement with a false antecedent is always true?
4. What are the conditions under which a conjunction is true?
5. How do the truth values of implications depend on the antecedent and consequent?

Tip: When solving logic problems, carefully determine the truth values of individual components before combining them into larger expressions.