Key Concepts and Formulas

• Implication: $p \Rightarrow q$ is equivalent to $\sim p \vee q$.
• Tautology: A statement that is always true, regardless of the truth values of its components.
• Truth Table: A table that shows all possible truth values of a statement based on the truth values of its components.

Step-by-Step Solution

Step 1: Rewrite the given statement using the implication equivalence. We are given the statement $(p \wedge \sim q) \Rightarrow (p \Rightarrow \sim q)$. We want to determine if this is a tautology, contradiction, or equivalent to some other statement. Using the implication rule, we can rewrite the implication $p \Rightarrow \sim q$ as $\sim p \vee \sim q$. Thus, the given statement becomes $(p \wedge \sim q) \Rightarrow (\sim p \vee \sim q)$

Step 2: Rewrite the implication again. Applying the implication rule $A \Rightarrow B \equiv \sim A \vee B$ to the above, where $A = (p \wedge \sim q)$ and $B = (\sim p \vee \sim q)$, we get: $\sim (p \wedge \sim q) \vee (\sim p \vee \sim q)$

Step 3: Apply De Morgan's Law. De Morgan's Law states that $\sim (p \wedge q) \equiv \sim p \vee \sim q$. Applying this to $\sim (p \wedge \sim q)$, we get $\sim p \vee \sim (\sim q)$, which simplifies to $\sim p \vee q$. Substituting this back into our expression, we have: $(\sim p \vee q) \vee (\sim p \vee \sim q)$

Step 4: Simplify the expression. Since the expression involves only disjunctions (OR operations), we can rearrange the terms using the associative property: $\sim p \vee q \vee \sim p \vee \sim q$ $\sim p \vee \sim p \vee q \vee \sim q$ Since $p \vee p \equiv p$, we have $\sim p \vee \sim p \equiv \sim p$. Also, $q \vee \sim q \equiv T$ (True). Therefore, $\sim p \vee (q \vee \sim q)$ $\sim p \vee T$

Step 5: Evaluate the final expression. Since anything ORed with True is True, $\sim p \vee T \equiv T$. Therefore, the given statement is equivalent to True.

Step 6: Construct a Truth Table (Alternative Approach). Let $E = (p \wedge \sim q) \Rightarrow (p \Rightarrow \sim q)$. We can construct a truth table to verify that $E$ is always true:

Since the last column is always T, the statement is a tautology.

Common Mistakes & Tips

• Remember the equivalence $p \Rightarrow q \equiv \sim p \vee q$. This is crucial for simplifying implications.
• Be careful with De Morgan's Laws: $\sim (p \wedge q) \equiv \sim p \vee \sim q$ and $\sim (p \vee q) \equiv \sim p \wedge \sim q$.
• When simplifying, look for opportunities to use the identities $p \vee \sim p \equiv T$ and $p \wedge \sim p \equiv F$.

Summary

We started with the statement $(p \wedge \sim q) \Rightarrow (p \Rightarrow \sim q)$ and used logical equivalences to simplify it. We applied the implication rule and De Morgan's Law to arrive at the expression $\sim p \vee T$, which is always true. Therefore, the given statement is a tautology. We also verified this result by constructing a truth table.

Final Answer The final answer is \boxed{a tautology}, which corresponds to option (A).