(~p V q) ^ ~ r make this to a truth table

To create a truth table for the expression \((\sim p \vee q) \land \sim r\), we'll follow these steps:

### Variables:
- \( p \)
- \( q \)
- \( r \)

### Logical Operations:
- \( \sim p \): Negation of \( p \)
- \( \sim r \): Negation of \( r \)
- \( \sim p \vee q \): Logical OR between \( \sim p \) and \( q \)
- \( (\sim p \vee q) \land \sim r \): Logical AND between the result of \( \sim p \vee q \) and \( \sim r \)

### Truth Table Layout:

We will calculate each of the sub-expressions step by step.

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

### Explanation:
- For each row in the truth table, we evaluate the basic variables \( p \), \( q \), and \( r \).
- Next, we compute the intermediate steps \( \sim p \) and \( \sim r \).
- Then, we compute \( \sim p \vee q \).
- Finally, we calculate the result of \( (\sim p \vee q) \land \sim r \).

Would you like a more detailed breakdown of any of the steps?

### Relative Questions:
1. How do negation and conjunction interact in logic?
2. What does the logical OR operation represent in truth tables?
3. How would the truth table change if we replaced OR with AND in the original expression?
4. Can you construct a truth table for \( p \rightarrow q \)?
5. How do De Morgan's laws apply to the expression above?

### Tip:
When constructing truth tables, always work systematically by calculating intermediate logical operations step by step.

Construct a truth table for the logical expression (~p V q) ^ ~r with a detailed breakdown of negation, logical OR, and logical AND. This exercise covers key concepts in propositional logic, suitable for students in Grades 9-12.

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