To solve the problems presented using direct proof, let's break down each argument logically, employing only rules of inference and rules of replacement.

Argument 1:

Premises:

1. $(N \land A) \vee (K \land A)$
2. $\neg K \rightarrow N$

Conclusion:

• $\neg K \rightarrow N$

Since premise 2 already provides the conclusion directly, we don't need further manipulation. The argument is valid by the rule of implication (if $\neg K \rightarrow N$ is given, it holds as the conclusion).

Argument 2:

Premises:

1. $(\neg R \land W) \rightarrow A$
2. $\neg R$
3. $S \vee W$

Conclusion:

• $(A \land S) \vee \neg W$

Step-by-Step Breakdown:

• From premise (2) $\neg R$, we can use this to simplify premise (1). Since $\neg R$ is true, the first part of premise (1) $\neg R \land W$ becomes true if $W$ is true, which leads to $A$.

• Now, premise (3) $S \vee W$ is available. Using disjunction elimination (since one of the parts of the disjunction must be true), if $W$ is true, then $A$ follows from premise (1).

• If $A$ is true, then $(A \land S) \vee \neg W$ is true because we can satisfy the left-hand side of the disjunction.

Thus, the argument holds by direct proof.

Argument 3:

Premises:

1. $\neg (A \vee N)$
2. $G \rightarrow F$
3. $F \rightarrow N$

Conclusion:

• $\neg F \rightarrow \neg N$

Step-by-Step Breakdown:

• From premise (1) $\neg (A \vee N)$, using De Morgan’s law, we get $\neg A \land \neg N$. This means $\neg N$ is true.

• Premise (2) and (3) give us the implications $G \rightarrow F$ and $F \rightarrow N$.

• Since we know $\neg N$, we can work backward using the contrapositive of $F \rightarrow N$, which is $\neg N \rightarrow \neg F$.

Thus, the argument holds by direct proof.

Would you like further clarification on these steps or any part of the reasoning? Here are 5 related questions to help you deepen your understanding:

1. How can De Morgan’s law help simplify logical statements in proofs?
2. Can you describe other rules of inference apart from the contrapositive and disjunction elimination?
3. What is the difference between direct proof and proof by contradiction?
4. How does the use of implication and its contrapositive affect logical arguments?
5. What are the essential elements of a valid argument in logic?

Tip: Practice writing down logical steps systematically to ensure clarity when handling more complex proofs.