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### I. Write each symbolic statement in words (for p, q, r, s, t, u).

1. **\( p \land \neg q \)**  
   - The tour goes to Italy, and the tour does not go to Spain.

2. **\( r \to \neg s \)**  
   - If we go to Venice, then we do not go to Florence.

3. **\( s \leftrightarrow \neg r \)**  
   - We go to Florence if and only if we do not go to Venice.

4. **\( r \lor s \)**  
   - We go to Venice, or we go to Florence.

5. **\( \neg t \lor u \)**  
   - The hotel fees are not included, or the meals are not included.

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### II. Write each symbolic statement as an English sentence (for p, q, r, s, t).

6. **\( (p \lor r) \land \neg q \)**  
   - Taylor Swift is a singer or Taylor Swift is an actress, and Taylor Swift is not a songwriter.

7. **\( p \to (q \land \neg r) \)**  
   - If Taylor Swift is a singer, then Taylor Swift is not a songwriter, and Taylor Swift is not an actress.

8. **\( (s \land \neg q) \to \neg t \)**  
   - If Taylor Swift plays the piano and is not a songwriter, then Taylor Swift does not play the guitar.

9. **\( t \leftrightarrow (\neg r \land \neg p) \)**  
   - Taylor Swift does not play the guitar if and only if Taylor Swift is not an actress and Taylor Swift is not a singer.

10. **\( \neg s \to (p \land \neg q) \)**  
    - If Taylor Swift does not play the piano, then Taylor Swift is a singer, and Taylor Swift is not a songwriter.

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### III. Write the negation of each quantified statement.

11. **Some lions are playful.**  
    - No lions are playful.

12. **Some dogs are not friendly.**  
    - All dogs are friendly.

13. **All classic movies were first produced in black and white.**  
    - Some classic movies were not first produced in black and white.

14. **Everybody enjoyed the dinner.**  
    - Somebody did not enjoy the dinner.

15. **No even numbers are odd numbers.**  
    - Some even numbers are odd numbers.

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### IV. Use truth tables to show that two statements are equivalent.

- **16.** \( \neg[p \lor (q \land r)] \equiv \neg p \land (\neg q \lor \neg r) \)
- **17.** \( [(\neg p \lor \neg q) \land r] \lor [(p \land \neg q) \land \neg r] \equiv [p \land (q \land r)] \lor [(\neg p \land \neg q) \land r] \)

For these two problems, we will need to construct truth tables to verify the logical equivalence step by step.

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### V. Construct a truth table for the given compound statement.

- **18.** \( S \land [\neg(\neg r \lor q) \lor \neg p] \)

This will also require a truth table to evaluate each possible combination of truth values for \( S \), \( r \), \( q \), and \( p \).

Would you like me to construct truth tables for the problems in parts IV and V?

Please answer this problem involving symbolic logic, including tasks such as writing symbolic statements in words, negating quantified statements, and constructing truth tables.

Learn how to solve symbolic logic problems by converting logical statements into English, negating quantified statements, and using truth tables. This tutorial covers key logical concepts like logical AND, OR, NOT, implications, and biconditional equivalences. Suitable for undergraduate students.

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