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):

(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."

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!