The image you've shared asks for a formal proof of validity for the given argument. Let's break down the argument and work through a proof step by step.

The Argument:

1. $(P \Rightarrow Q) \land (R \Rightarrow S)$ (Conjunction of implications)
2. $\neg A \Rightarrow \neg Q$ (Implication)
3. $A \Rightarrow \neg B$ (Implication)
4. $B$ (Statement)

Conclusion to Prove:

$\therefore \neg P \lor \neg S$

Step-by-Step Proof:

Step 1: Break down the conjunction from premise (1)

From the first premise, $(P \Rightarrow Q) \land (R \Rightarrow S)$, we can split it into two separate implications:

$P \Rightarrow Q \quad \text{(1a)}$ $R \Rightarrow S \quad \text{(1b)}$

Step 2: Derive intermediate conclusions from premises

• From premise (4) $B$, and premise (3) $A \Rightarrow \neg B$, we can conclude that $A$ must be false. If $A$ were true, it would lead to $\neg B$, which contradicts $B$. Therefore:

$\neg A \quad \text{(From (3) and (4))}$

• Now, use $\neg A$ in premise (2) $\neg A \Rightarrow \neg Q$. Since $\neg A$ is true, we can conclude:

\neg Q \quad \text{(From (2) and \neg A)}

Step 3: Use the contrapositive

• We have $\neg Q$. Using the contrapositive of $P \Rightarrow Q$ from (1a), we know that if $Q$ is false, then $P$ must also be false:

\neg P \quad \text{(Contrapositive of (1a) using \neg Q)}

Step 4: Consider $R \Rightarrow S$

• To derive $\neg S$, we turn to the second part of premise (1b), $R \Rightarrow S$. We need to argue that either $R$ is false or $S$ is false.

• We know that the goal is to prove $\neg P \lor \neg S$. Since we have already concluded $\neg P$, we do not need to further investigate whether $R$ or $S$ are true or false.

Conclusion:

We have established that $\neg P$. Therefore, the disjunction $\neg P \lor \neg S$ holds because one of the disjuncts is true.

Thus, the conclusion $\neg P \lor \neg S$ follows from the premises, and the argument is valid.

Would you like more details on any part of this proof? Here are 5 related questions to consider:

1. Why do we need the contrapositive to conclude $\neg P$ from $\neg Q$?
2. How do conjunctions like $(P \Rightarrow Q) \land (R \Rightarrow S)$ influence formal proofs?
3. Can the disjunction $\neg P \lor \neg S$ still hold if $\neg S$ were true instead of $\neg P$?
4. What role does premise (3) $A \Rightarrow \neg B$ play in leading to the conclusion?
5. How would the proof change if the premise $B$ were different?

Tip: In logical proofs, applying the contrapositive is often a powerful tool when dealing with implications!