Geometry worksheet focusing on logical statements, connectives, and truth tables.

Problem Analysis:

The worksheet focuses on negations of statements, compound statements in words, and symbolic forms of compound statements. Let's solve each section step by step.

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Section 1: Write the negation of the statement

#### Negation Rules:
- The negation of a statement reverses its truth value.
- For quantifiers:
- "Some" becomes "No."
- "All" becomes "Not all" or "Some do not."
- "No" becomes "Some."

#### Solutions:

1. Original Statement: Some people drink milk.
- Negation: No people drink milk.

2. Original Statement: Some dogs do not have fleas.
- Negation: All dogs have fleas.

3. Original Statement: No butterflies bite.
- Negation: Some butterflies bite.

4. Original Statement: Some locks are keyless.
- Negation: No locks are keyless.

5. Original Statement: All pens use ink.
- Negation: Not all pens use ink (or Some pens do not use ink).

6. Original Statement: No rabbits wear glasses.
- Negation: Some rabbits wear glasses.

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Section 2: Write each compound statement in words

#### Given Statements:
- \( p \): The coffee is Maxwell House.
- \( q \): The coffee is hot.
- \( r \): The coffee is strong.

#### Compound Statement Symbols:
- \( \vee \): OR
- \( \wedge \): AND
- \( \sim \): NOT
- \( \leftrightarrow \): IF AND ONLY IF
- \( \rightarrow \): IF...THEN

#### Solutions:

7. Statement: \( p \vee q \)
- In Words: The coffee is Maxwell House or the coffee is hot.

8. Statement: \( \sim q \wedge r \)
- In Words: The coffee is not hot and the coffee is strong.

9. Statement: \( p \leftrightarrow \sim r \)
- In Words: The coffee is Maxwell House if and only if the coffee is not strong.

10. Statement: \( q \rightarrow (r \wedge \sim p) \)
- In Words: If the coffee is hot, then the coffee is strong and the coffee is not Maxwell House.

11. Statement: \( (p \vee \sim q) \wedge \sim r \)
- In Words: The coffee is Maxwell House or the coffee is not hot, and the coffee is not strong.

12. Statement: \( \sim p \leftrightarrow (r \wedge \sim q) \)
- In Words: The coffee is not Maxwell House if and only if the coffee is strong and the coffee is not hot.

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Section 3: Use the statements for \( p \), \( q \), and \( r \) to write the statement in symbolic form

#### Solutions:

13. Statement: If the coffee is Maxwell House, then it is strong.
- Symbolic Form: \( p \rightarrow r \)

14. Statement: The coffee is strong and the coffee is hot.
- Symbolic Form: \( r \wedge q \)

15. Statement: If the coffee is strong, then the coffee is hot, or the coffee is not Maxwell House.
- Symbolic Form: \( r \rightarrow (q \vee \sim p) \)

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Final Answers:

1. Negations:
1. No people drink milk.
2. All dogs have fleas.
3. Some butterflies bite.
4. No locks are keyless.
5. Not all pens use ink.
6. Some rabbits wear glasses.

2. Compound Statements in Words:
7. The coffee is Maxwell House or the coffee is hot.
8. The coffee is not hot and the coffee is strong.
9. The coffee is Maxwell House if and only if the coffee is not strong.
10. If the coffee is hot, then the coffee is strong and the coffee is not Maxwell House.
11. The coffee is Maxwell House or the coffee is not hot, and the coffee is not strong.
12. The coffee is not Maxwell House if and only if the coffee is strong and the coffee is not hot.

3. Symbolic Forms:
13. \( p \rightarrow r \)
14. \( r \wedge q \)
15. \( r \rightarrow (q \vee \sim p) \)

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Boxed Final Answer:

\[
\boxed{
\begin{array}{l}
\text{1. No people drink milk.} \\
\text{2. All dogs have fleas.} \\
\text{3. Some butterflies bite.} \\
\text{4. No locks are keyless.} \\
\text{5. Not all pens use ink.} \\
\text{6. Some rabbits wear glasses.} \\
\text{7. The coffee is Maxwell House or the coffee is hot.} \\
\text{8. The coffee is not hot and the coffee is strong.} \\
\text{9. The coffee is Maxwell House if and only if the coffee is not strong.} \\
\text{10. If the coffee is hot, then the coffee is strong and the coffee is not Maxwell House.} \\
\text{11. The coffee is Maxwell House or the coffee is not hot, and the coffee is not strong.} \\
\text{12. The coffee is not Maxwell House if and only if the coffee is strong and the coffee is not hot.} \\
\text{13. } p \rightarrow r \\
\text{14. } r \wedge q \\
\text{15. } r \rightarrow (q \vee \sim p) \\
\end{array}
}
\]