Geometry worksheet focusing on logical statements and truth tables, including negation, compound statements, and symbolic representation.

Let's solve each part of this Geometry(H) Worksheet: Statements, Logical Connectives, and Truth Tables step by step.

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Part 1: Write the negation of the statement.

Negation means forming a statement that is logically opposite to the original.

1. Some people drink milk.
→ "Some" means "at least one."
Negation: No people drink milk. or All people do not drink milk.

2. Some dogs do not have fleas.
→ "Some do not" = "Not all have fleas."
Negation: All dogs have fleas.

3. No butterflies bite.
→ "No" = "None" = "All do not."
Negation: Some butterflies bite. (At least one bites)

4. Some locks are keyless.
→ "Some are" = "At least one is."
Negation: No locks are keyless. or All locks require keys.

5. All pens use ink.
→ "All" = "Every pen uses ink."
Negation: Some pens do not use ink.

6. No rabbits wear glasses.
→ "No" = "None"
Negation: Some rabbits wear glasses.

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Part 2: Write each compound statement in words.

Given:
- $ p $: The coffee is Maxwell House
- $ q $: The coffee is hot
- $ r $: The coffee is strong

Logical connectives:
- $ \vee $: OR
- $ \wedge $: AND
- $ \neg $: NOT
- $ \rightarrow $: IF...THEN
- $ \leftrightarrow $: IF AND ONLY IF

7. $ p \vee q $
→ The coffee is Maxwell House or the coffee is hot.

8. $ \neg q \wedge r $
→ The coffee is not hot and the coffee is strong.

9. $ p \rightarrow \neg r $
→ If the coffee is Maxwell House, then the coffee is not strong.

10. $ q \rightarrow (r \wedge \neg p) $
→ If the coffee is hot, then the coffee is strong and it is not Maxwell House.

11. $ (p \vee \neg q) \wedge \neg r $
→ (The coffee is Maxwell House or the coffee is not hot) and the coffee is not strong.

12. $ \neg p \leftrightarrow (r \wedge \neg q) $
→ The coffee is not Maxwell House if and only if the coffee is strong and not hot.

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Part 3: Use the statements for p, q, and r to write the statement in symbolic form.

13. If the coffee is Maxwell House, then it is strong.
→ $ p \rightarrow r $

14. The coffee is strong and the coffee is hot.
→ $ r \wedge q $

15. If the coffee is strong then the coffee is hot, or the coffee is not Maxwell House.
→ This is: $ (r \rightarrow q) \vee \neg p $

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✔ Final Answers:

#### Negations (1–6):
1. No people drink milk.
2. All dogs have fleas.
3. Some butterflies bite.
4. No locks are keyless.
5. Some pens do not use ink.
6. Some rabbits wear glasses.

#### Compound Statements in Words (7–12):
7. The coffee is Maxwell House or the coffee is hot.
8. The coffee is not hot and the coffee is strong.
9. If the coffee is Maxwell House, then the coffee is not strong.
10. If the coffee is hot, then the coffee is strong and it 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 not hot.

#### Symbolic Form (13–15):
13. $ p \rightarrow r $
14. $ r \wedge q $
15. $ (r \rightarrow q) \vee \neg p $

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Let me know if you'd like a truth table for any of these!