Key Concepts and Formulas

• De Morgan's Laws:
  • $\sim (p \vee q) \equiv \sim p \wedge \sim q$
  • $\sim (p \wedge q) \equiv \sim p \vee \sim q$
• Distributive Law: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$
• Negation of Negation: $\sim (\sim p) \equiv p$

Step-by-Step Solution

Step 1: Simplify the given Boolean expression using the distributive law.

We are given the expression $p \vee (\sim p \wedge q)$. We want to simplify this expression. Using the distributive law, we can rewrite this as:

$p \vee (\sim p \wedge q) \equiv (p \vee \sim p) \wedge (p \vee q)$

Step 2: Simplify further using the property $p \vee \sim p \equiv T$ (Tautology).

Since $p \vee \sim p$ is always true, we can replace it with T.

$(p \vee \sim p) \wedge (p \vee q) \equiv T \wedge (p \vee q)$

Step 3: Simplify using the property $T \wedge p \equiv p$.

Since $T \wedge (p \vee q)$ is equivalent to $p \vee q$, the expression becomes:

$T \wedge (p \vee q) \equiv p \vee q$

Therefore, the simplified expression is $p \vee q$.

Step 4: Find the negation of the simplified expression.

We need to find the negation of $p \vee q$, which is $\sim (p \vee q)$. Using De Morgan's Law, we have:

$\sim (p \vee q) \equiv \sim p \wedge \sim q$

Step 5: Consider an alternative simplification method.

The problem asks for the negation of $p \vee (\sim p \wedge q)$. Let's directly negate the original expression:

$\sim [p \vee (\sim p \wedge q)]$

Applying De Morgan's Law:

$\sim p \wedge \sim (\sim p \wedge q)$

Applying De Morgan's Law again:

$\sim p \wedge (\sim (\sim p) \vee \sim q)$

Simplifying:

$\sim p \wedge (p \vee \sim q)$

Applying the distributive law:

$(\sim p \wedge p) \vee (\sim p \wedge \sim q)$

Since $\sim p \wedge p \equiv F$ (False):

$F \vee (\sim p \wedge \sim q)$

Which simplifies to:

$\sim p \wedge \sim q$

Step 6: Examine the options to see if any are equivalent to $p \wedge \sim q$.

We are given that the correct answer is $p \wedge \sim q$. The derived solution is $\sim p \wedge \sim q$. Let's re-examine our initial simplification to see if there was an error. The negation of the Boolean expression $p \vee (\sim p \wedge q)$ is equivalent to $p \wedge \sim q$. This means that $p \vee (\sim p \wedge q)$ must be equivalent to $\sim (p \wedge \sim q) = \sim p \vee q$. Let's check this.

$p \vee (\sim p \wedge q) = (p \vee \sim p) \wedge (p \vee q) = T \wedge (p \vee q) = p \vee q$.

We want this to be equivalent to $\sim p \vee q$.

$p \vee q = \sim p \vee q$ is NOT generally true.

Therefore, the problem statement or the given correct answer is incorrect. The derived negation, $\sim p \wedge \sim q$ is correct. Let us re-examine the problem statement and options for potential errors.

The original expression is $p \vee (\sim p \wedge q)$. Its negation is $\sim [p \vee (\sim p \wedge q)]$. Using De Morgan's Law, this is $\sim p \wedge \sim (\sim p \wedge q)$. Using De Morgan's Law again, this becomes $\sim p \wedge (\sim (\sim p) \vee \sim q) = \sim p \wedge (p \vee \sim q)$. Using the distributive law, this becomes $(\sim p \wedge p) \vee (\sim p \wedge \sim q) = F \vee (\sim p \wedge \sim q) = \sim p \wedge \sim q$.

The correct negation is $\sim p \wedge \sim q$.

However, the provided correct answer is $p \wedge \sim q$. There must be an error in the problem statement or the provided correct answer.

Let's assume the correct answer IS $p \wedge \sim q$. Then, the original expression MUST be equivalent to $\sim (p \wedge \sim q) = \sim p \vee q$. Then, $p \vee (\sim p \wedge q)$ would have to be equivalent to $\sim p \vee q$. But we already showed that $p \vee (\sim p \wedge q) = p \vee q$. This leads to $p \vee q = \sim p \vee q$. This is NOT generally true. Therefore, the problem statement or the provided answer is incorrect.

The correct negation is $\sim p \wedge \sim q$.

Common Mistakes & Tips

• Carefully apply De Morgan's Laws. It's easy to make a mistake in distributing the negation.
• Remember the basic identities like $p \vee \sim p \equiv T$ and $p \wedge \sim p \equiv F$.
• When negating a complex expression, it can be helpful to break it down into smaller steps.

Summary

We started by simplifying the given Boolean expression using the distributive law and logical identities. We then found the negation of the simplified expression using De Morgan's Law. The negation of $p \vee (\sim p \wedge q)$ simplifies to $\sim p \wedge \sim q$. However, the correct answer given is $p \wedge \sim q$. This is a contradiction. The correct negation is $\sim p \wedge \sim q$.

Final Answer

The final answer is $\boxed{\sim p \wedge \sim q}$, which corresponds to option (D).