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To construct a truth table for the expression \( p \land (q \lor \sim r) \), let's break down the steps and create a table that examines all possible truth values of \( p \), \( q \), and \( r \).

### Variables and Operators
- **\( p \):** First variable, which can be either True (T) or False (F).
- **\( q \):** Second variable, also True or False.
- **\( r \):** Third variable, also True or False.
- **\( \sim r \):** The negation of \( r \).
- **\( q \lor \sim r \):** The logical "OR" of \( q \) and \( \sim r \).
- **\( p \land (q \lor \sim r) \):** The logical "AND" of \( p \) and \( (q \lor \sim r) \).

### Truth Table
Here’s the step-by-step truth table calculation for each logical operation.

| \( p \) | \( q \) | \( r \) | \( \sim r \) | \( q \lor \sim r \) | \( p \land (q \lor \sim r) \) |
|---------|---------|---------|--------------|----------------------|-------------------------------|
| T       | T       | T       | F            | T                    | T                             |
| T       | T       | F       | T            | T                    | T                             |
| T       | F       | T       | F            | F                    | F                             |
| T       | F       | F       | T            | T                    | T                             |
| F       | T       | T       | F            | T                    | F                             |
| F       | T       | F       | T            | T                    | F                             |
| F       | F       | T       | F            | F                    | F                             |
| F       | F       | F       | T            | T                    | F                             |

### Explanation of Columns
1. **\( p \), \( q \), \( r \)**: These columns show each possible combination of truth values for the variables \( p \), \( q \), and \( r \).
2. **\( \sim r \)**: This column is the negation of \( r \).
3. **\( q \lor \sim r \)**: This column shows the result of the "OR" operation between \( q \) and \( \sim r \).
4. **\( p \land (q \lor \sim r) \)**: This is the final column, showing the result of the "AND" operation between \( p \) and \( (q \lor \sim r) \).

Would you like further details on any part of the process or have any questions?

### Additional Questions
1. How is the negation operator \( \sim \) applied in logical statements?
2. What is the difference between "AND" (\( \land \)) and "OR" (\( \lor \)) in logic?
3. How would the truth table change if the expression was \( p \lor (q \land \sim r) \)?
4. Can truth tables be used to verify logical equivalences?
5. What are common applications of truth tables in computer science and logic?

#### Tip:
In truth tables, make sure each possible combination of variables is represented; for three variables, there are always \( 2^3 = 8 \) rows.

Construct a truth table for p ∧ (q ∨ ∼ r). Use T for true and F for false.

Construct a complete truth table for the logical expression p ∧ (q ∨ ∼ r) using the values T for true and F for false. This problem covers key concepts in propositional logic such as conjunction, disjunction, and negation. Suitable for students in Grades 9-12.

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