The image you provided contains a symbolic logic problem. Let's break down the two statements and conclusion step by step.

Given:

1. $P \rightarrow (Q \vee R)$ (If $P$ is true, then $Q$ or $R$ must be true.)
2. $\neg Q \wedge \neg R$ (Both $Q$ and $R$ are false.)

Conclusion:

$\therefore \neg P \vee \neg R$ (The goal is to prove that either $P$ is false or $R$ is false.)

Steps:

1. From statement (2), we have both $\neg Q$ (not $Q$) and $\neg R$ (not $R$). So neither $Q$ nor $R$ is true.

2. Using Modus Tollens on statement (1):

   • $P \rightarrow (Q \vee R)$ means that if $P$ is true, then $Q$ or $R$ must be true.
   • However, from statement (2), we know $Q$ is false and $R$ is false, i.e., $Q \vee R$ is false.
   • By Modus Tollens, if $P \rightarrow (Q \vee R)$ is true and $Q \vee R$ is false, then $P$ must be false.
   • Hence, $\neg P$ (not $P$) is true.

3. Therefore, since we know $\neg P$ is true from step (2) and $\neg R$ is given in statement (2), we can conclude $\neg P \vee \neg R$, which matches the desired conclusion.

Thus, the conclusion $\neg P \vee \neg R$ is valid.

Would you like more details or have any questions on this?

Here are 5 related questions for practice:

1. What is the difference between Modus Ponens and Modus Tollens?
2. How do you use De Morgan's laws in symbolic logic?
3. Can you derive $P$ given that both $Q$ and $R$ are true?
4. How would the conclusion change if $Q$ were true but $R$ were false?
5. How does truth table analysis help in verifying logical conclusions?

Tip: When working with logical proofs, always remember to check if you can apply inference rules like Modus Ponens or Modus Tollens to simplify your statements.