https://file.aiquickdraw.com/mathbot/file/1726506164474r94whfjt.jpg

The statement in the image requires constructing a truth table for the logical expression:

\[
(\neg p \land q) \rightarrow (p \land \neg q)
\]

To determine whether the statement is a tautology, we need to construct the truth table for all possible values of \(p\) and \(q\), evaluate the truth of the expression for each case, and then see if the statement is always true.

### Steps:
1. List all possible truth values for \(p\) and \(q\).
2. Evaluate \( \neg p \), \( \neg q \), \( \neg p \land q \), and \( p \land \neg q \).
3. Use the implication (\( \rightarrow \)) to find the truth value of the entire expression.

I'll begin by constructing the truth table now.Here is the completed truth table for the expression \( (\neg p \land q) \rightarrow (p \land \neg q) \):

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
p & q & \neg p & \neg q & \neg p \land q & p \land \neg q & (\neg p \land q) \rightarrow (p \land \neg q) \\
\hline
T & T & F & F & F & F & T \\
T & F & F & T & F & T & T \\
F & T & T & F & T & F & F \\
F & F & T & T & F & F & T \\
\hline
\end{array}
\]

### Conclusion:
The expression is **not** a tautology because in one case (when \(p = \text{False}\) and \(q = \text{True}\)), the statement is **false**. A tautology must be true in all cases.

Do you need further details or explanations?

Here are 5 questions to expand on this:

1. What is the difference between a tautology and a contradiction in logic?
2. How is the implication \(p \rightarrow q\) defined in logic?
3. Can a logical expression be simplified using logical equivalences?
4. How do truth tables help in validating logical arguments?
5. What are some common logical operators used in propositional logic?

**Tip**: When constructing truth tables, always start by evaluating individual components like negations and conjunctions before tackling implications.

Construct a truth table for the statement (~p ∧ q) → (p ∧ ~q). Then determine if the statement is a tautology.

Construct a truth table for the logical expression (~p ∧ q) → (p ∧ ~q) and determine if it is a tautology. Explore core concepts in propositional logic and how to evaluate statements using truth tables. Suitable for college-level discrete mathematics.

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