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Translate the following sentences either from English into BOOL or from BOOL into English. Use the standard atomic sentences from the book (i.e. [tex]\(P\)[/tex] = "Pia is guilty", [tex]\(Q\)[/tex] = "Quinn is guilty", [tex]\(R\)[/tex] = "Raquel is guilty", etc.).

1. [tex]\(P \& Q\)[/tex]
2. [tex]\(Q \vee (\sim P \& \sim R)\)[/tex]
3. [tex]\((P \vee R) \& \sim Q\)[/tex]
4. [tex]\(\sim P \vee \sim Q\)[/tex]
5. Either Pia is guilty or Quinn and Raquel are guilty.
6. Neither Quinn nor Raquel are guilty.
7. Quinn is guilty but Pia is innocent.
8. Pia and Quinn are guilty but Raquel and Stan are innocent.



Answer :

GinnyAnswer
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.

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