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.