Key Concepts and Formulas

• Tautology: A statement that is always true, regardless of the truth values of its components.
• Implication: $p \to q$ is equivalent to $\sim p \vee q$.
• De Morgan's Laws: $\sim (p \wedge q) \equiv \sim p \vee \sim q$ and $\sim (p \vee q) \equiv \sim p \wedge \sim q$.
• Contradiction: A statement that is always false (denoted by C or F). $p \wedge \sim p$ is a contradiction.
• Associative Law: $(p \vee q) \vee r \equiv p \vee (q \vee r)$
• Commutative Law: $p \vee q \equiv q \vee p$

Step-by-Step Solution

Step 1: Rewrite the given statement using the implication rule. We are given the statement $(\sim p) \wedge (p \vee q) \to q$. Using the implication rule, $p \to q \equiv \sim p \vee q$, we can rewrite the statement as:

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

Step 2: Apply De Morgan's Law. Using De Morgan's Law, $\sim (p \wedge q) \equiv \sim p \vee \sim q$, we have:

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

Step 3: Simplify the double negation and apply De Morgan's Law again. Since $\sim (\sim p) \equiv p$, and applying De Morgan's Law to $\sim(p \vee q) \equiv \sim p \wedge \sim q$, we get:

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

Step 4: Apply the distributive law. We can rewrite the expression using the distributive law $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$:

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

Step 5: Simplify using the fact that $p \vee \sim p$ is a tautology (T). Since $p \vee \sim p$ is always true (a tautology, which we denote by T), we have:

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

Step 6: Simplify using the identity $T \wedge p \equiv p$. Since $T \wedge (p \vee \sim q) \equiv (p \vee \sim q)$, we have:

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

Step 7: Apply the associative law. Using the associative law $(p \vee q) \vee r \equiv p \vee (q \vee r)$, we get:

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

Step 8: Simplify using the fact that $q \vee \sim q$ is a tautology (T). Since $q \vee \sim q$ is always true (a tautology, which we denote by T), we have:

$p \vee T$

Step 9: Simplify using the identity $p \vee T \equiv T$. Since $p \vee T$ is always true (a tautology), we have:

$T$

Therefore, the original statement is a tautology.

Common Mistakes & Tips

• Remember to apply De Morgan's Laws correctly. A common mistake is to only negate one term instead of both.
• When simplifying, use the basic logical equivalences such as $p \vee \sim p \equiv T$ and $p \wedge \sim p \equiv F$.
• When dealing with implications, remember that $p \to q \equiv \sim p \vee q$. This is crucial for simplifying expressions.

Summary

We started with the given expression $(\sim p) \wedge (p \vee q) \to q$. We then applied the implication rule, De Morgan's laws, the distributive law, and the associative law, along with the properties of tautologies and contradictions, to simplify the expression. Through these steps, we showed that the expression simplifies to $T$, which means it is a tautology. Therefore, option (A) is the correct answer.

Final Answer

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