Key Concepts and Formulas

• Tautology: A statement that is always true, regardless of the truth values of its components.
• Associative Law: $(p \vee q) \vee r \equiv p \vee (q \vee r)$ and $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$
• Commutative Law: $p \vee q \equiv q \vee p$ and $p \wedge q \equiv q \wedge p$
• Identity Law: $p \vee F \equiv p$ and $p \wedge T \equiv p$
• Complement Law: $p \vee \sim p \equiv T$ and $p \wedge \sim p \equiv F$
• Idempotent Law: $p \vee p \equiv p$ and $p \wedge p \equiv p$
• Distributive Law: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$ and $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
• De Morgan's Laws: $\sim(p \vee q) \equiv \sim p \wedge \sim q$ and $\sim(p \wedge q) \equiv \sim p \vee \sim q$

Step-by-Step Solution

We need to determine which of the given Boolean expressions is a tautology. A tautology is always true, regardless of the truth values of $p$ and $q$. Let's analyze each option:

Option (A): $(p \vee q) \wedge (\sim p \vee \sim q)$ Step 1: Analyze the expression. We want to determine if $(p \vee q) \wedge (\sim p \vee \sim q)$ is a tautology.

Step 2: Apply De Morgan's Law. Rewrite $(\sim p \vee \sim q)$ as $\sim(p \wedge q)$ using De Morgan's Law. The expression becomes $(p \vee q) \wedge \sim(p \wedge q)$.

Step 3: Expand the expression. Consider the truth values of $p$ and $q$. If $p$ and $q$ are both true, $(p \vee q)$ is true, but $\sim(p \wedge q)$ is false (since $p \wedge q$ is true), so the entire expression is false. Thus, (A) is not a tautology.

Option (B): $(p \vee q) \vee (p \vee \sim q)$ Step 1: Analyze the expression. We want to determine if $(p \vee q) \vee (p \vee \sim q)$ is a tautology.

Step 2: Apply the associative law. Rearrange the terms using the associative law: $p \vee (q \vee p) \vee \sim q$. Then use the commutative law: $p \vee p \vee (q \vee \sim q)$.

Step 3: Apply the idempotent and complement laws. Since $p \vee p \equiv p$ and $q \vee \sim q \equiv T$, we get $p \vee T$.

Step 4: Apply the identity law. Finally, $p \vee T \equiv T$. Therefore, the expression is a tautology.

Option (C): $(p \wedge q) \vee (p \wedge \sim q)$ Step 1: Analyze the expression. We want to determine if $(p \wedge q) \vee (p \wedge \sim q)$ is a tautology.

Step 2: Apply the distributive law. Apply the distributive law: $p \wedge (q \vee \sim q)$.

Step 3: Apply the complement law. Since $q \vee \sim q \equiv T$, we have $p \wedge T$.

Step 4: Apply the identity law. Finally, $p \wedge T \equiv p$. Since $p$ is not always true, this is not a tautology.

Option (D): $(p \vee q) \wedge (p \vee \sim q)$ Step 1: Analyze the expression. We want to determine if $(p \vee q) \wedge (p \vee \sim q)$ is a tautology.

Step 2: Apply the distributive law. Apply the distributive law: $p \vee (q \wedge \sim q)$.

Step 3: Apply the complement law. Since $q \wedge \sim q \equiv F$, we have $p \vee F$.

Step 4: Apply the identity law. Finally, $p \vee F \equiv p$. Since $p$ is not always true, this is not a tautology.

The correct answer according to the ground truth is (A), but our derivations show that (B) is the tautology. There must be an error in the answer key. We will proceed with the assumption that the answer key is incorrect and the true answer is (B).

Common Mistakes & Tips

• Remember De Morgan's Laws correctly. It's a common source of errors.
• When simplifying Boolean expressions, it's helpful to use truth tables to check your work, especially for more complex expressions.
• Be careful when applying the distributive law. Make sure you are distributing correctly.

Summary

We analyzed each of the given Boolean expressions to determine which one is a tautology. By applying the associative, commutative, idempotent, complement, distributive, and identity laws, as well as De Morgan's Laws, we simplified each expression. Option (B), $(p \vee q) \vee (p \vee \sim q)$, simplifies to $T$, which means it is a tautology. The provided solution states that (A) is the correct answer, but our derivation proves that (B) is the correct answer.

Final Answer

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