https://file.aiquickdraw.com/mathbot/file/1726767173679ggxqc62n.jpg

To construct a truth table for the logical expression \((p \lor q) \lor (\sim p \lor \sim r)\), let's break it down step by step:

1. **Variables**: We have three variables: \(p\), \(q\), and \(r\). 
2. **Operators**:
   - \(\lor\) stands for logical OR.
   - \(\sim\) stands for logical NOT.

### Truth Table Columns:
- First, list all possible truth values for \(p\), \(q\), and \(r\).
- Next, compute intermediate expressions:
  - \(p \lor q\) (logical OR of \(p\) and \(q\))
  - \(\sim p\) (negation of \(p\))
  - \(\sim r\) (negation of \(r\))
  - \(\sim p \lor \sim r\) (logical OR of \(\sim p\) and \(\sim r\))
- Finally, compute the full expression \((p \lor q) \lor (\sim p \lor \sim r)\).

### Step-by-Step Construction:

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

### Explanation:
- \(p \lor q\): This column shows the result of the logical OR between \(p\) and \(q\).
- \(\sim p\): This column negates the value of \(p\).
- \(\sim r\): This column negates the value of \(r\).
- \(\sim p \lor \sim r\): This column is the logical OR between \(\sim p\) and \(\sim r\).
- The final column, \((p \lor q) \lor (\sim p \lor \sim r)\), is the overall logical OR between the results from \(p \lor q\) and \(\sim p \lor \sim r\).

Would you like more details about any step? 

Here are 5 follow-up questions to deepen your understanding:
1. What is the truth table for a simple logical AND operation between two variables?
2. How does the negation of a proposition affect its truth value?
3. What happens to the truth table when more variables are added?
4. Can the truth table for this expression be simplified?
5. How can we verify logical equivalence between two expressions using truth tables?

**Tip**: When constructing a truth table, always double-check the logical operators and intermediate steps for accuracy!

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

Explore how to construct a truth table for the logical expression (p ∨ q) ∨ (∼ p ∨ ∼ r) with a detailed explanation of each step, including intermediate results for logical OR and NOT operations. Suitable for high school students learning propositional logic.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation