what are the truth values of this statement   ~pconditional_transp.gif(pand_0.gif~q

To determine the truth values of the given logical statement, let's first understand the symbols and the structure of the expression:

### Symbols and Structure

1. **~ (tilde):** Represents logical NOT (negation).
2. **p, q:** Represent propositional variables (statements that can be either true (T) or false (F)).
3. **→ (conditional):** Represents logical implication (if... then...).
4. **∧ (and):** Represents logical AND.

### Statement Breakdown

The given statement is:  
\[
\sim p \to (p \land \sim q)
\]

This can be read as: "NOT p implies (p AND NOT q)".

### Evaluating the Truth Table

To evaluate the truth values, we need to construct a truth table considering all possible truth values of \( p \) and \( q \). Let's evaluate the truth table step-by-step.

| p   | q   | ~p  | ~q  | p ∧ ~q | ~p → (p ∧ ~q) |
|-----|-----|-----|-----|--------|---------------|
| T   | T   | F   | F   | F      | T             |
| T   | F   | F   | T   | T      | T             |
| F   | T   | T   | F   | F      | F             |
| F   | F   | T   | T   | F      | F             |

### Explanation of Each Column:

1. **p, q:** All possible combinations of truth values for \( p \) and \( q \) (True or False).
2. **~p:** The negation of \( p \). If \( p \) is True, \( \sim p \) is False, and vice versa.
3. **~q:** The negation of \( q \). If \( q \) is True, \( \sim q \) is False, and vice versa.
4. **p ∧ ~q:** Logical AND between \( p \) and \( \sim q \). This is True only if both \( p \) is True and \( \sim q \) is True.
5. **~p → (p ∧ ~q):** The conditional statement. The implication \( \sim p \to (p \land \sim q) \) is False only when \( \sim p \) is True, and \( p \land \sim q \) is False; otherwise, it is True.

### Conclusion:

The truth values of the statement \( \sim p \to (p \land \sim q) \) are **True, True, False, False** for the combinations of \( (p, q) \) as follows:  
1. \( (T, T) \): True  
2. \( (T, F) \): True  
3. \( (F, T) \): False  
4. \( (F, F) \): False  

Would you like more details on the explanation, or do you have any questions?

### Related Questions:

1. What is the truth table for a biconditional statement?
2. How do you determine the truth values of complex logical statements?
3. What is the difference between a tautology and a contradiction in logic?
4. Can you provide an example of De Morgan's Laws in a truth table?
5. How do logical connectives interact in compound statements?

**Tip:** When evaluating a truth table, always ensure to consider every possible combination of truth values for all variables involved.

Explore the truth values of the logical statement ~p → (p ∧ ~q) using a detailed truth table analysis. This problem covers propositional logic, logical connectives, and truth table evaluations. Suitable for advanced high school students and anyone interested in logic.

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