https://file.aiquickdraw.com/mathbot/file/17255888912191n3vv7gk.jpg

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.

P → (Q ∨ R), ¬Q ∧ ¬R, therefore ¬P ∨ ¬R.

Solve a propositional logic problem using Modus Tollens and disjunction. This problem involves logical inference with implications, conjunctions, and disjunctions. Suitable for students studying logic and discrete mathematics at the undergraduate level.

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