To create a truth table for the expression \( \sim q \to (p \vee \sim r) \), we first rewrite the implication \( A \to B \) using its equivalent form: \( \sim A \vee B \). Thus, the given expression can be rewritten as:

\[
\sim q \to (p \vee \sim r) \equiv \sim (\sim q) \vee (p \vee \sim r) \equiv q \vee (p \vee \sim r).
\]

This means we need to evaluate the expression \( q \vee (p \vee \sim r) \). Let's construct the truth table step by step:

### Variables
- \( p \)
- \( q \)
- \( r \)

### Derived Columns
1. \( \sim q \)
2. \( \sim r \)
3. \( p \vee \sim r \)
4. \( \sim q \to (p \vee \sim r) \)

#### Truth Table

| \( p \) | \( q \) | \( r \) | \( \sim q \) | \( \sim r \) | \( p \vee \sim r \) | \( \sim q \to (p \vee \sim r) \) |
|--------|--------|--------|--------------|--------------|---------------------|--------------------------------|
|   T    |   T    |   T    |      F       |      F       |         T           |               T                |
|   T    |   T    |   F    |      F       |      T       |         T           |               T                |
|   T    |   F    |   T    |      T       |      F       |         T           |               T                |
|   T    |   F    |   F    |      T       |      T       |         T           |               T                |
|   F    |   T    |   T    |      F       |      F       |         F           |               T                |
|   F    |   T    |   F    |      F       |      T       |         T           |               T                |
|   F    |   F    |   T    |      T       |      F       |         F           |               F                |
|   F    |   F    |   F    |      T       |      T       |         T           |               T                |

### Explanation
1. **Column \( \sim q \):** The negation of \( q \).
2. **Column \( \sim r \):** The negation of \( r \).
3. **Column \( p \vee \sim r \):** The logical OR between \( p \) and \( \sim r \).
4. **Column \( \sim q \to (p \vee \sim r) \):** Evaluated as \( \sim q \vee (p \vee \sim r) \), combining the results.

Let me know if you'd like more detailed steps for each column! 😊

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### 5 Related Questions
1. What is the difference between logical implication (\( \to \)) and equivalence (\( \leftrightarrow \))?
2. How do truth tables help verify logical equivalences?
3. Can we derive \( p \vee q \) from \( \sim q \to p \)? Why or why not?
4. What is the purpose of using De Morgan's Laws in logical expressions?
5. How does \( \sim (\sim p) \) simplify in logical expressions?

### Tip
When constructing a truth table, ensure each variable has \( 2^n \) rows, where \( n \) is the number of distinct variables. This ensures all combinations are covered!

Learn how to construct a truth table for the logical expression ~q → (p ∨ ~r). This detailed guide covers key concepts in logic, including implication equivalence, De Morgan's Laws, and truth table construction. Ideal for students in Grades 10-12.

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