Given that $(p \land Q) \Rightarrow (\neg S \lor R)$ is false, we can determine the truth values of the statements $S \Rightarrow R$, $\neg P \lor \neg S$, $(P \land \neg Q) \Leftrightarrow R$, and $P \land R$ as follows:

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

A conditional statement $A \Rightarrow B$ is false only when $A$ is true and $B$ is false.

So, for $(p \land Q) \Rightarrow (\neg S \lor R)$ to be false:

• $p \land Q$ must be true (i.e., $p$ is true and $Q$ is true).
• $\neg S \lor R$ must be false.

Step 2: Analyze $\neg S \lor R$

The disjunction $\neg S \lor R$ is false only when both $\neg S$ and $R$ are false.

• $\neg S$ is false implies $S$ is true.
• $R$ is false.

So, we have:

• $S$ is true.
• $R$ is false.
• $p$ is true.
• $Q$ is true.

Step 3: Determine the truth values for each statement

a) $S \Rightarrow R$:

$S \Rightarrow R$ is true if $S$ is false or $R$ is true. Since $S$ is true and $R$ is false, the statement $S \Rightarrow R$ is false.

b) $\neg P \lor \neg S$:

$\neg P \lor \neg S$ is true if either $\neg P$ or $\neg S$ is true. Given $P$ is true and $S$ is true, $\neg P$ is false and $\neg S$ is false. Therefore, $\neg P \lor \neg S$ is false.

c) $(P \land \neg Q) \Leftrightarrow R$:

$P \land \neg Q$ would be false since $Q$ is true. $R$ is false. The equivalence $(P \land \neg Q) \Leftrightarrow R$ will be true since both sides are false.

d) $P \land R$:

$P \land R$ is true if both $P$ and $R$ are true. Since $P$ is true and $R$ is false, $P \land R$ is false.

Summary of Truth Values:

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

Would you like more details or have any questions?

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1. What is the truth table for a conditional statement?
2. How does the truth value of a conjunction work?
3. Can you explain the negation of a disjunction?
4. What is the biconditional operation in logic?
5. How does one determine the truth value of complex logical statements?

Tip: When dealing with logical statements, it's essential to carefully consider the conditions under which compound statements are true or false.