Create a truth table for (~p→q)˄p→~q

To create a truth table for the logical expression \((\sim p \to q) \land p \to \sim q\), we'll break it down step by step.

### Expression:
\[
(\sim p \to q) \land p \to \sim q
\]

This is a conditional statement where:
- \(\sim p\) means "not p."
- \(p \to q\) means "if p then q" (implication).
- \(\land\) means "and."
- The main conditional is the outer \(\to\).

### Steps for the Truth Table:
1. Calculate \(\sim p\) (negation of \(p\)).
2. Calculate \(\sim p \to q\) (implication).
3. Calculate \( (\sim p \to q) \land p \) (conjunction).
4. Calculate \(\sim q\) (negation of \(q\)).
5. Finally, calculate \((\sim p \to q) \land p \to \sim q\) (implication).

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

### Explanation:
- The third column is \(\sim p\), the negation of \(p\).
- The fourth column is the implication \(\sim p \to q\).
- The fifth column is the conjunction \((\sim p \to q) \land p\).
- The sixth column is \(\sim q\), the negation of \(q\).
- The final column is the implication \((\sim p \to q) \land p \to \sim q\), which gives us the truth value of the overall expression.

Would you like any clarifications on these steps?

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**Relative Questions:**
1. How do you evaluate implications in a truth table?
2. Why does the conjunction \((\sim p \to q) \land p\) affect the overall truth value?
3. What are common logical equivalences related to implications?
4. How can you use truth tables to determine tautologies or contradictions?
5. How does negating the whole statement affect the truth table?

**Tip:** When working with implications, remember that \(p \to q\) is false only when \(p\) is true and \(q\) is false.

Explore a detailed solution to create a truth table for the logical expression (~p→q)˄p→~q. This guide covers key logic concepts such as negation, implication, and conjunction, with step-by-step analysis and a comprehensive truth table. Suitable for Grades 9-12 students studying advanced logic.

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