Answer :
Sure, I'll provide detailed, step-by-step solutions for translating the sentences between English and Boolean notation:
1. [tex]\( P \& Q \)[/tex]:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( P \& Q \)[/tex]: Pia is guilty and Quinn is guilty.
2. [tex]\( Q \vee(\sim P \& \sim R) \)[/tex]:
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( \sim P \)[/tex]: Pia is not guilty.
- [tex]\( \sim R \)[/tex]: Raquel is not guilty.
- [tex]\( \sim P \& \sim R \)[/tex]: Pia is not guilty and Raquel is not guilty.
- [tex]\( Q \vee(\sim P \& \sim R) \)[/tex]: Quinn is guilty or (Pia is not guilty and Raquel is not guilty).
3. [tex]\( (P \vee R) \& \sim Q \)[/tex]:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( \sim Q \)[/tex]: Quinn is not guilty.
- [tex]\( P \vee R \)[/tex]: Pia is guilty or Raquel is guilty.
- [tex]\( (P \vee R) \& \sim Q \)[/tex]: (Pia is guilty or Raquel is guilty) and Quinn is not guilty.
4. [tex]\( \sim P \vee \sim Q \)[/tex]:
- [tex]\( \sim P \)[/tex]: Pia is not guilty.
- [tex]\( \sim Q \)[/tex]: Quinn is not guilty.
- [tex]\( \sim P \vee \sim Q \)[/tex]: Pia is not guilty or Quinn is not guilty.
5. Either Pia is guilty or Quinn and Raquel:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( Q \& R \)[/tex]: Quinn and Raquel are guilty.
- [tex]\( P \vee (Q \& R) \)[/tex]: Either Pia is guilty or (Quinn and Raquel are guilty).
6. Neither Quinn nor Raquel are guilty:
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( \sim Q \)[/tex]: Quinn is not guilty.
- [tex]\( \sim R \)[/tex]: Raquel is not guilty.
- [tex]\( \sim Q \& \sim R \)[/tex]: Quinn is not guilty and Raquel is not guilty.
7. Quinn is guilty but Pia is innocent:
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( \sim P \)[/tex]: Pia is not guilty.
- [tex]\( Q \& \sim P \)[/tex]: Quinn is guilty and Pia is not guilty.
8. Pia and Quinn are guilty but Raquel and Stan are innocent:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( S \)[/tex]: Stan is guilty.
- [tex]\( \sim R \)[/tex]: Raquel is not guilty.
- [tex]\( \sim S \)[/tex]: Stan is not guilty.
- [tex]\( P \& Q \)[/tex]: Pia and Quinn are guilty.
- [tex]\( \sim R \& \sim S \)[/tex]: Raquel and Stan are not guilty.
- [tex]\( P \& Q \& \sim R \& \sim S \)[/tex]: Pia and Quinn are guilty and Raquel and Stan are not guilty.
1. [tex]\( P \& Q \)[/tex]:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( P \& Q \)[/tex]: Pia is guilty and Quinn is guilty.
2. [tex]\( Q \vee(\sim P \& \sim R) \)[/tex]:
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( \sim P \)[/tex]: Pia is not guilty.
- [tex]\( \sim R \)[/tex]: Raquel is not guilty.
- [tex]\( \sim P \& \sim R \)[/tex]: Pia is not guilty and Raquel is not guilty.
- [tex]\( Q \vee(\sim P \& \sim R) \)[/tex]: Quinn is guilty or (Pia is not guilty and Raquel is not guilty).
3. [tex]\( (P \vee R) \& \sim Q \)[/tex]:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( \sim Q \)[/tex]: Quinn is not guilty.
- [tex]\( P \vee R \)[/tex]: Pia is guilty or Raquel is guilty.
- [tex]\( (P \vee R) \& \sim Q \)[/tex]: (Pia is guilty or Raquel is guilty) and Quinn is not guilty.
4. [tex]\( \sim P \vee \sim Q \)[/tex]:
- [tex]\( \sim P \)[/tex]: Pia is not guilty.
- [tex]\( \sim Q \)[/tex]: Quinn is not guilty.
- [tex]\( \sim P \vee \sim Q \)[/tex]: Pia is not guilty or Quinn is not guilty.
5. Either Pia is guilty or Quinn and Raquel:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( Q \& R \)[/tex]: Quinn and Raquel are guilty.
- [tex]\( P \vee (Q \& R) \)[/tex]: Either Pia is guilty or (Quinn and Raquel are guilty).
6. Neither Quinn nor Raquel are guilty:
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( \sim Q \)[/tex]: Quinn is not guilty.
- [tex]\( \sim R \)[/tex]: Raquel is not guilty.
- [tex]\( \sim Q \& \sim R \)[/tex]: Quinn is not guilty and Raquel is not guilty.
7. Quinn is guilty but Pia is innocent:
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( \sim P \)[/tex]: Pia is not guilty.
- [tex]\( Q \& \sim P \)[/tex]: Quinn is guilty and Pia is not guilty.
8. Pia and Quinn are guilty but Raquel and Stan are innocent:
- [tex]\( P \)[/tex]: Pia is guilty.
- [tex]\( Q \)[/tex]: Quinn is guilty.
- [tex]\( R \)[/tex]: Raquel is guilty.
- [tex]\( S \)[/tex]: Stan is guilty.
- [tex]\( \sim R \)[/tex]: Raquel is not guilty.
- [tex]\( \sim S \)[/tex]: Stan is not guilty.
- [tex]\( P \& Q \)[/tex]: Pia and Quinn are guilty.
- [tex]\( \sim R \& \sim S \)[/tex]: Raquel and Stan are not guilty.
- [tex]\( P \& Q \& \sim R \& \sim S \)[/tex]: Pia and Quinn are guilty and Raquel and Stan are not guilty.