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Let's go through the logic table step by step.

### (a) Completing the table:

1. **Column `p ↔ q` (biconditional)**: 
   - The biconditional (↔) is true when both `p` and `q` have the same truth value.
   
   **Row 1**: p = T, q = T → p ↔ q = T (same values)  
   **Row 2**: p = T, q = F → p ↔ q = F (different values)  
   **Row 3**: p = F, q = T → p ↔ q = F (different values)  
   **Row 4**: p = F, q = F → p ↔ q = T (same values)  
   
   The column for `p ↔ q` is: T, F, F, T.

2. **Column `(p → q)` (implication)**:
   - The implication (→) is false only when `p` is true and `q` is false; otherwise, it is true.
   
   **Row 1**: p = T, q = T → p → q = T  
   **Row 2**: p = T, q = F → p → q = F (since p = T and q = F)  
   **Row 3**: p = F, q = T → p → q = T (since p = F, the implication is true)  
   **Row 4**: p = F, q = F → p → q = T (since p = F, the implication is true)  
   
   The column for `(p → q)` is: T, F, T, T.

### Full table (completed):

| p   | q   | p ↔ q | (p → q) | q   |
|-----|-----|-------|---------|-----|
| T   | T   | T     | T       | T   |
| T   | F   | F     | F       | F   |
| F   | T   | F     | T       | T   |
| F   | F   | T     | T       | F   |

### (b) Choose the correct answer:

This part presents logical statements with "Truthful Troll" and "freedom." The implication structure in answer choice **C** aligns with the logical implication truth table. The phrase "If the troll is Truthful Troll, then the eastern door leads to freedom" matches the structure of `(p → q)`, which is true in all cases except when `p = T` and `q = F`.

- **Correct answer: C** - "If the troll is Truthful Troll, then the eastern door leads to freedom."

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Would you like more detailed explanations or have any questions on this? Here are some follow-up questions to explore further:

1. What is the difference between implication and biconditional statements?
2. How would the truth table change if we added another variable?
3. Can you provide an example of a real-world scenario that uses logical biconditionals?
4. Why does implication hold true when the premise is false?
5. How does this logic structure relate to conditional statements in programming?

**Tip**: Always remember that in an implication, the statement is considered true if the premise is false, even if the conclusion is false!

Complete the truth table and choose the correct answer regarding logical statements involving a Truthful Troll and freedom.

Learn how to complete a truth table and solve logic problems involving implications and biconditionals with a Truthful Troll. This step-by-step guide covers key concepts in discrete mathematics and logical operators.

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