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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!

Construct a formal proof of validity for the argument:
1. (P => Q) ∧ (R => S)
2. ¬A => ¬Q
3. A => ¬B
4. B
Conclusion: ∴ ¬P ∨ ¬S

Learn how to construct a formal proof of validity for a logical argument involving implications and propositional logic. The problem includes key concepts such as contrapositive, disjunction, and modus ponens. Suitable for undergraduate students.

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