Key Concepts and Formulas

• Implication: $p \Rightarrow q \equiv \sim p \vee q$
• Commutative Law: $p \vee q \equiv q \vee p$ and $p \wedge q \equiv q \wedge p$
• Associative Law: $(p \vee q) \vee r \equiv p \vee (q \vee r)$ and $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$
• Distributive Law: $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$ and $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$
• Law of Contradiction: $p \wedge \sim p \equiv F$ (False)
• Identity Law: $p \vee F \equiv p$
• Idempotent Law: $p \vee p \equiv p$ and $p \wedge p \equiv p$

Step-by-Step Solution

Step 1: Convert the implications to their equivalent disjunction forms. We are given $(p \Rightarrow q) \wedge (q \Rightarrow \sim p)$. Using the implication rule $p \Rightarrow q \equiv \sim p \vee q$, we can rewrite the given expression as: $(\sim p \vee q) \wedge (\sim q \vee \sim p)$ This step is important because it transforms the implications into disjunctions, which are often easier to manipulate using logical equivalences.

Step 2: Apply the commutative law to the second term. We can rewrite the second term $(\sim q \vee \sim p)$ as $(\sim p \vee \sim q)$ using the commutative law for disjunction. This gives us: $(\sim p \vee q) \wedge (\sim p \vee \sim q)$ Rearranging terms allows us to potentially use the distributive law.

Step 3: Apply the distributive law. We can apply the distributive law in reverse, treating $\sim p$ as a common term. The expression now becomes: $\sim p \vee (q \wedge \sim q)$ This step simplifies the expression by isolating $\sim p$.

Step 4: Simplify using the law of contradiction. We know that $q \wedge \sim q$ is always false (F), according to the law of contradiction. So, we have: $\sim p \vee F$

Step 5: Simplify using the identity law. Using the identity law, $p \vee F \equiv p$, we get: $\sim p$

Now, we must go back and re-examine our steps because the correct answer given is 'q'. There must be an error. Let's start again.

Step 1: Convert the implications to their equivalent disjunction forms. We are given $(p \Rightarrow q) \wedge (q \Rightarrow \sim p)$. Using the implication rule $p \Rightarrow q \equiv \sim p \vee q$, we can rewrite the given expression as: $(\sim p \vee q) \wedge (\sim q \vee \sim p)$

Step 2: Apply the commutative law to the second term. We can rewrite the second term $(\sim q \vee \sim p)$ as $(\sim p \vee \sim q)$ using the commutative law for disjunction. This gives us: $(\sim p \vee q) \wedge (\sim q \vee \sim p)$

Step 3: Apply the distributive law. We can apply the distributive law: $(A \vee B) \wedge (A \vee C) \equiv A \vee (B \wedge C)$. $\sim p \vee (q \wedge \sim q)$

Step 4: Simplify using the law of contradiction. We know that $q \wedge \sim q$ is always false (F), according to the law of contradiction. So, we have: $\sim p \vee F$

Step 5: Simplify using the identity law. Using the identity law, $p \vee F \equiv p$, we get: $\sim p$

Since the correct answer is 'q', let's try a truth table:

Now, let's compare this to the truth table for 'q':

That doesn't match. Let's check "~p":

This matches! So the correct answer is ~p, not q. The provided "Correct Answer" is incorrect.

Common Mistakes & Tips

• Be careful when applying the distributive law, ensuring the correct terms are grouped together.
• Remember the key logical equivalences, especially the implication rule.
• Using a truth table can be a good way to verify your result.

Summary

We started by converting the implications into their disjunctive equivalents. Then, we applied the distributive law and simplified the expression using the law of contradiction and the identity law. This led us to the simplified form $\sim p$. However, after constructing a truth table, it became clear that the provided correct answer 'q' was incorrect. The truth table confirmed that the correct answer is indeed $\sim p$.

Final Answer

The final answer is \boxed{\sim p}, which corresponds to option (B).