To determine the truth values of the given statements based on the condition that $(P \land Q) \Rightarrow (\neg S \lor R)$ is false, let's analyze each part step by step.

Given:

$(P \land Q) \Rightarrow (\neg S \lor R)$ This statement is false.

For an implication $A \Rightarrow B$ to be false, $A$ must be true and $B$ must be false.

So, $P \land Q$ must be true, and $\neg S \lor R$ must be false.

Analysis:

1. $P \land Q$ is true:

   • This implies $P$ is true and $Q$ is true.

2. $\neg S \lor R$ is false:

   • For $\neg S \lor R$ to be false, both $\neg S$ and $R$ must be false.
   • If $\neg S$ is false, $S$ must be true.
   • If $R$ is false, $R$ is false.

Now, using this information, let's determine the truth values for the given statements.

a) $S \Rightarrow R$

• $S$ is true, and $R$ is false.
• The implication $S \Rightarrow R$ is false when $S$ is true and $R$ is false.

Truth Value: False

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

• $P$ is true, so $\neg P$ is false.
• $S$ is true, so $\neg S$ is false.
• Since both $\neg P$ and $\neg S$ are false, the disjunction $\neg P \lor \neg S$ is false.

Truth Value: False

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

• $P$ is true, and $Q$ is true, so $P \land \neg Q$ is false.
• $R$ is false.
• The biconditional $(P \land \neg Q) \Leftrightarrow R$ is true when both sides have the same truth value (both true or both false).
• Since both $P \land \neg Q$ and $R$ are false, the biconditional is true.

Truth Value: True

d) $P \land R$

• $P$ is true, and $R$ is false.
• The conjunction $P \land R$ is true only if both $P$ and $R$ are true.
• Since $R$ is false, $P \land R$ is false.

Truth Value: False

Summary of Truth Values:

a) $S \Rightarrow R$: False
b) $\neg P \lor \neg S$: False
c) $(P \land \neg Q) \Leftrightarrow R$: True
d) $P \land R$: False

Would you like details on any of these steps or have any other questions?

Related Questions:

1. How do truth tables help in determining the truth value of compound statements?
2. What conditions make an implication statement true or false?
3. How does the truth value of a biconditional differ from that of an implication?
4. Can you explain the logical relationship between disjunction and negation in compound statements?
5. How does knowing the truth value of individual propositions help in evaluating a compound logical statement?

Tip:

When analyzing logical statements, breaking them down into simpler components (like individual propositions) makes it easier to determine the truth values of more complex expressions.