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Let's walk through completing the truth table for the statement \((\neg p \land (p \rightarrow q)) \rightarrow \neg q\).

### Steps to break down the expression:

1. **Negation of \(p\)**: \(\neg p\)
2. **Implication \(p \rightarrow q\)**: This is false only when \(p = T\) and \(q = F\), and true otherwise.
3. **Conjunction**: \(\neg p \land (p \rightarrow q)\) combines the results from steps 1 and 2.
4. **Negation of \(q\)**: \(\neg q\)
5. **Final Implication**: The overall expression \((\neg p \land (p \rightarrow q)) \rightarrow \neg q\) is evaluated by combining the conjunction from step 3 with the negation from step 4.

#### Let's fill in the missing values step by step:

### 1. Row 1 (\(p = T, q = T\)):
- \(\neg p = F\)
- \(p \rightarrow q = T\)
- \(\neg p \land (p \rightarrow q) = F \land T = F\)
- \(\neg q = F\)
- \((\neg p \land (p \rightarrow q)) \rightarrow \neg q = F \rightarrow F = T\)

### 2. Row 2 (\(p = T, q = F\)):
- \(\neg p = F\)
- \(p \rightarrow q = F\)
- \(\neg p \land (p \rightarrow q) = F \land F = F\)
- \(\neg q = T\)
- \((\neg p \land (p \rightarrow q)) \rightarrow \neg q = F \rightarrow T = T\)

### 3. Row 3 (\(p = F, q = T\)):
- \(\neg p = T\)
- \(p \rightarrow q = T\)
- \(\neg p \land (p \rightarrow q) = T \land T = T\)
- \(\neg q = F\)
- \((\neg p \land (p \rightarrow q)) \rightarrow \neg q = T \rightarrow F = F\)

### 4. Row 4 (\(p = F, q = F\)):
- \(\neg p = T\)
- \(p \rightarrow q = T\)
- \(\neg p \land (p \rightarrow q) = T \land T = T\)
- \(\neg q = T\)
- \((\neg p \land (p \rightarrow q)) \rightarrow \neg q = T \rightarrow T = T\)

### Filled Truth Table:
| \(p\) | \(q\) | \(\neg p\) | \(p \rightarrow q\) | \(\neg p \land (p \rightarrow q)\) | \(\neg q\) | \((\neg p \land (p \rightarrow q)) \rightarrow \neg q\) |
|:----:|:----:|:----------:|:------------------:|:----------------------------------:|:----------:|:-------------------------------------------------------:|
|  T   |  T   |     F      |         T          |                  F                 |     F      |                          T                              |
|  T   |  F   |     F      |         F          |                  F                 |     T      |                          T                              |
|  F   |  T   |     T      |         T          |                  T                 |     F      |                          F                              |
|  F   |  F   |     T      |         T          |                  T                 |     T      |                          T                              |

Now the table is complete.

Would you like further details or explanations on any part of this truth table?

### 5 Follow-up Questions:
1. What is the truth value of an implication when the premise is false?
2. How does negation affect the values of a logical statement?
3. Why does the conjunction in row 3 result in a false implication?
4. What would change if the expression was \((p \land (p \rightarrow q)) \rightarrow q\)?
5. How would you construct a truth table for a biconditional statement \(p \leftrightarrow q\)?

**Tip**: The implication \(p \rightarrow q\) is true unless \(p\) is true and \(q\) is false.

Complete the truth table for the logical statement (¬p ∧ (p → q)) → ¬q.

Learn how to complete the truth table for the logical expression (¬p ∧ (p → q)) → ¬q. This problem covers key concepts in logic, including implications, negations, and conjunctions. Suitable for high school students in grades 9-12.

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