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### **Q1. Determine if the Expression is a Tautology**
**Expression:** ¬q → (¬q ∧ p) → ¬q

To determine whether the expression is a tautology, we must check if it is true for all possible truth values of `q` and `p`.

1. **Construct the truth table:**

| q   | p   | ¬q  | ¬q ∧ p | (¬q ∧ p) → ¬q |
|-----|-----|-----|--------|---------------|
| T   | T   | F   | F      | T             |
| T   | F   | F   | F      | T             |
| F   | T   | T   | T      | T             |
| F   | F   | T   | F      | T             |

2. **Analyze the expression ¬q → ((¬q ∧ p) → ¬q):**

- Since the expression `(¬q ∧ p) → ¬q` is true for all combinations of `q` and `p`, ¬q → (¬q ∧ p) → ¬q is also always true.

**Conclusion:** The expression is a tautology.

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Here are some related questions to consider:
1. What is a tautology in logic?
2. How do you construct a truth table?
3. What does the inverse of a logical statement mean?
4. What is the converse of a logical statement?
5. What is the contrapositive of a logical statement?
6. How do you prove that a set is a subset of another set?
7. What is the significance of the universal set in set theory?
8. How do you find the binary expansion of a hexadecimal number?

**Tip:** When determining if a logical statement is a tautology, the key is to check if the statement is true for all possible truth values of its components.

Learn how to determine if the logical expression ¬q → (¬q ∧ p) → ¬q is a tautology using truth tables and logical reasoning. This problem explores fundamental concepts in logic, including tautologies and truth values. Suitable for advanced learners and those studying formal logic.

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