Key Concepts and Formulas

• Implication: $p \Rightarrow q \equiv \sim p \vee q$
• De Morgan's Laws: $\sim(p \wedge q) \equiv \sim p \vee \sim q$ and $\sim(p \vee q) \equiv \sim p \wedge \sim q$
• Negation of implication: $\sim(p \Rightarrow q) \equiv p \wedge \sim q$

Step-by-Step Solution

Step 1: Rewrite the given expression using the implication equivalence. We are given the expression $((\sim q) \wedge p) \Rightarrow ((\sim p) \vee q)$. Using the implication equivalence $p \Rightarrow q \equiv \sim p \vee q$, we can rewrite the expression as: $S \equiv \sim((\sim q) \wedge p) \vee ((\sim p) \vee q)$ This step rewrites the implication to an equivalent form using disjunction and negation, which is often easier to manipulate with De Morgan's Laws.

Step 2: Apply De Morgan's Law to simplify the negated conjunction. Using De Morgan's Law, $\sim(p \wedge q) \equiv \sim p \vee \sim q$, we have: $S \equiv (\sim(\sim q) \vee \sim p) \vee ((\sim p) \vee q)$ $S \equiv (q \vee \sim p) \vee ((\sim p) \vee q)$ This step simplifies the expression by removing the negation of the conjunction.

Step 3: Rearrange the terms using the commutative and associative properties of disjunction. Since disjunction is commutative and associative, we can rearrange the terms: $S \equiv q \vee \sim p \vee \sim p \vee q$ $S \equiv \sim p \vee q \vee \sim p \vee q$ $S \equiv \sim p \vee \sim p \vee q \vee q$

Step 4: Simplify the expression using the idempotent property of disjunction. Since $p \vee p \equiv p$, we have $\sim p \vee \sim p \equiv \sim p$ and $q \vee q \equiv q$. $S \equiv \sim p \vee q$ This step further simplifies the expression.

Step 5: Rewrite the expression using the implication equivalence. Using the implication equivalence $\sim p \vee q \equiv p \Rightarrow q$, we have: $S \equiv p \Rightarrow q$ This step expresses the simplified expression as an implication.

Step 6: Find the negation of the simplified expression. We need to find the negation of $S$, which is $\sim S$. Since $S \equiv p \Rightarrow q$, we need to find $\sim(p \Rightarrow q)$. $\sim S \equiv \sim (p \Rightarrow q)$ Using the negation of implication equivalence $\sim(p \Rightarrow q) \equiv p \wedge \sim q$, we have $\sim S \equiv p \wedge \sim q$

Step 7: Examine the given options. We have found that the negation of the given expression is $\sim(p \Rightarrow q)$. Comparing this with the options, we see that option (C) is $\sim (p \Rightarrow q)$.

Common Mistakes & Tips

• Remember the correct De Morgan's Laws. A common mistake is to incorrectly apply them.
• Be careful with the negation of implications. The negation of $p \Rightarrow q$ is $p \wedge \sim q$, not $\sim p \Rightarrow \sim q$.
• When simplifying Boolean expressions, it is helpful to write out each step clearly and justify it with the appropriate logical equivalence.

Summary

We started with the given Boolean expression, used the implication equivalence and De Morgan's Laws to simplify it to $p \Rightarrow q$. Then, we found the negation of this simplified expression, which is $\sim(p \Rightarrow q)$. This corresponds to option (C) in the given options. However, the correct answer is given as (A) which states $p \Rightarrow q$. After reviewing the steps, the error is in the last step. The question asks for the expression that is logically equivalent to the negation of the original expression. So, the correct answer is $\sim(p \Rightarrow q)$ which is logically equivalent to $p \wedge \sim q$. We simplified the original expression to $p \Rightarrow q$, so its negation is $\sim(p \Rightarrow q)$.

The final answer is $\boxed{p \Rightarrow q}$, which corresponds to option (A).