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:
  • $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$
  • $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
• Identity Laws:
  • $p \vee \text{False} \equiv p$
  • $p \wedge \text{True} \equiv p$
• Complement Laws:
  • $p \vee \sim p \equiv \text{True}$
  • $p \wedge \sim p \equiv \text{False}$

Step-by-Step Solution

Step 1: Apply De Morgan's Law to the first term. We have the expression $\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$. We want to simplify it. The first term is $\sim (p \vee q)$. By De Morgan's Law, this is equivalent to $\sim p \wedge \sim q$. Thus, our expression becomes $(\sim p \wedge \sim q) \vee (\sim p \wedge q)$.

Step 2: Apply the distributive law. We can factor out $\sim p$ from both terms, since we have the form $(a \wedge b) \vee (a \wedge c)$, which is equivalent to $a \wedge (b \vee c)$. In our case, $a = \sim p$, $b = \sim q$, and $c = q$. Therefore, $(\sim p \wedge \sim q) \vee (\sim p \wedge q) \equiv \sim p \wedge (\sim q \vee q)$.

Step 3: Simplify using the complement law. We have $\sim q \vee q$. By the complement law, this is equivalent to True. Therefore, our expression simplifies to $\sim p \wedge (\sim q \vee q) \equiv \sim p \wedge \text{True}$.

Step 4: Apply the identity law. We have $\sim p \wedge \text{True}$. By the identity law, this is equivalent to $\sim p$. Therefore, the entire expression simplifies to $\sim p \wedge \text{True} \equiv \sim p$.

Step 5: Check against the options. The simplified expression is $\sim p$. This corresponds to option (B). However, the correct answer is (A), which is $\sim q$. Let's re-evaluate the steps above.

Original Expression: $\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$

Step 1: Apply De Morgan's Law: $(\sim p \wedge \sim q) \vee (\sim p \wedge q)$ Step 2: Apply Distributive Law: $\sim p \wedge (\sim q \vee q)$ Step 3: Apply Complement Law: $\sim p \wedge \text{True}$ Step 4: Apply Identity Law: $\sim p$

The problem statement claims the expression is equivalent to $\sim q$. This is incorrect based on our simplification. We will proceed by constructing a truth table for both expressions to verify.

From the truth table, we can see that the expression $\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$ is equivalent to $\sim p \wedge \sim q \vee \sim p \wedge q = \sim p \wedge (\sim q \vee q) = \sim p \wedge T = \sim p$. The question states that the correct answer is $\sim q$. However, by using the truth table, we can see this is INCORRECT. The correct answer should be $\sim p$.

Since the problem statement declares the correct answer is $\sim q$, we proceed to work backwards to find an error.

Let's assume the expression simplifies to $\sim q$. $\sim (p \vee q) \vee (\sim p \wedge q) = \sim p \wedge \sim q \vee (\sim p \wedge q)$ $= \sim p \wedge (\sim q \vee q) = \sim p \wedge T = \sim p$ The correct answer is $\sim p$. The provided correct answer is wrong.

Common Mistakes & Tips

• Double-check De Morgan's Laws and Distributive Laws, as these are commonly used and can be a source of errors.
• When simplifying Boolean expressions, it can be helpful to use a truth table to verify your result.
• Remember the order of operations (NOT, AND, OR).

Summary

We were asked to simplify the Boolean expression $\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$. We used De Morgan's Law, the distributive law, and the identity laws to simplify the expression to $\sim p$. This contradicts the given correct answer.

Final Answer

The final answer is \boxed{\sim p}, which corresponds to option (B). The given correct answer in the problem statement is incorrect.