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To construct a truth table for \((r \vee q) \land (r \land p)\), we need to examine all possible truth values for the propositions \(p\), \(q\), and \(r\). Here’s a step-by-step solution for creating the truth table.

### Step-by-Step Truth Table Construction

1. **Identify Variables**: We have three variables \(p\), \(q\), and \(r\), which results in \(2^3 = 8\) possible combinations of truth values.
2. **Column Breakdown**:
   - **Columns for each variable**: \(p\), \(q\), \(r\)
   - **Intermediate expressions**: \(r \vee q\), \(r \land p\)
   - **Final expression**: \((r \vee q) \land (r \land p)\)

### Truth Table

| \(p\) | \(q\) | \(r\) | \(r \vee q\) | \(r \land p\) | \((r \vee q) \land (r \land p)\) |
|-------|-------|-------|--------------|---------------|----------------------------------|
| T     | T     | T     | T            | T             | T                                |
| T     | T     | F     | T            | F             | F                                |
| T     | F     | T     | T            | T             | T                                |
| T     | F     | F     | F            | F             | F                                |
| F     | T     | T     | T            | F             | F                                |
| F     | T     | F     | T            | F             | F                                |
| F     | F     | T     | T            | F             | F                                |
| F     | F     | F     | F            | F             | F                                |

### Explanation of Columns
- **\(r \vee q\)**: This is true if either \(r\) or \(q\) is true.
- **\(r \land p\)**: This is true only if both \(r\) and \(p\) are true.
- **\((r \vee q) \land (r \land p)\)**: This final expression is true if both \(r \vee q\) and \(r \land p\) are true.

Would you like further details or have any questions?

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### Related Questions
1. How do we create a truth table for a larger number of variables?
2. What is the difference between conjunction (\(\land\)) and disjunction (\(\vee\)) in logic?
3. Can a truth table be simplified using logical laws?
4. How do De Morgan’s laws apply to simplifying logical expressions?
5. What are the basic steps to converting a logical expression into a truth table?

### Tip
Remember that each row in a truth table represents a unique combination of truth values for each variable, covering all possibilities.

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

Learn how to construct a truth table for the logical expression (r ∨ q) ∧ (r ∧ p) with step-by-step guidance. This tutorial covers key concepts in logic, including conjunction and disjunction, and is suitable for high school students (Grades 9-12).

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