Key Concepts and Formulas

• Tautology: A statement that is always true, regardless of the truth values of its components.
• Contradiction: A statement that is always false, regardless of the truth values of its components.
• Identity Laws:
  • $p \vee T \equiv T$ (where T is a tautology)
  • $p \wedge F \equiv F$ (where F is a contradiction)
• Complement Laws:
  • $p \vee \sim p \equiv T$
  • $p \wedge \sim p \equiv F$

Step-by-Step Solution

We need to determine which of the given options is a tautology. Let's analyze each option:

Option (A): $(( \sim q) \wedge p) \wedge q$

Step 1: Simplify the expression using the associative property.

• What: Rearrange the terms using the associative property of conjunction.
• Why: To group terms that might simplify using other logical equivalences.
• Math: $(( \sim q) \wedge p) \wedge q \equiv ( \sim q) \wedge (p \wedge q)$

Step 2: Rearrange the terms using the commutative property.

• What: Rearrange the terms using the commutative property of conjunction.
• Why: To group $q$ and $\sim q$ together, which might lead to a simplification.
• Math: $( \sim q) \wedge (p \wedge q) \equiv ( \sim q) \wedge (q \wedge p) \equiv (\sim q \wedge q) \wedge p$

Step 3: Apply the complement law.

• What: Simplify $\sim q \wedge q$.
• Why: To simplify the expression further.
• Math: $(\sim q \wedge q) \wedge p \equiv F \wedge p \equiv F$, where F is a contradiction.

Since option (A) simplifies to a contradiction, it is not a tautology.

Option (B): $(( \sim q) \wedge p) \wedge (p \wedge ( \sim p))$

Step 1: Simplify the expression within the second parenthesis using the complement law.

• What: Simplify $p \wedge \sim p$.
• Why: To simplify the expression further.
• Math: $(( \sim q) \wedge p) \wedge (p \wedge ( \sim p)) \equiv (( \sim q) \wedge p) \wedge F$, where F is a contradiction.

Step 2: Simplify using the identity law.

• What: Simplify the entire expression using the identity law for conjunction with a contradiction.
• Why: To determine if the expression is a tautology.
• Math: $(( \sim q) \wedge p) \wedge F \equiv F$

Since option (B) simplifies to a contradiction, it is not a tautology.

Option (C): $(( \sim q) \wedge p) \vee (p \vee ( \sim p))$

Step 1: Simplify the expression within the second parenthesis using the complement law.

• What: Simplify $p \vee \sim p$.
• Why: To simplify the expression further.
• Math: $(( \sim q) \wedge p) \vee (p \vee ( \sim p)) \equiv (( \sim q) \wedge p) \vee T$, where T is a tautology.

Step 2: Simplify using the identity law.

• What: Simplify the entire expression using the identity law for disjunction with a tautology.
• Why: To determine if the expression is a tautology.
• Math: $(( \sim q) \wedge p) \vee T \equiv T$

Since option (C) simplifies to a tautology, it is the correct answer.

Option (D): $(p \wedge q) \wedge ( \sim p \wedge q)$

Step 1: Rearrange terms using the associative and commutative properties.

• What: Rearrange the terms.
• Why: To group $p$ and $\sim p$ together.
• Math: $(p \wedge q) \wedge ( \sim p \wedge q) \equiv p \wedge q \wedge \sim p \wedge q \equiv p \wedge \sim p \wedge q \wedge q$

Step 2: Apply the complement law.

• What: Simplify $p \wedge \sim p$.
• Why: To simplify the expression further.
• Math: $p \wedge \sim p \wedge q \wedge q \equiv F \wedge q \wedge q \equiv F \wedge q$

Step 3: Apply the identity law.

• What: Simplify $F \wedge q$.
• Why: To determine if the expression is a tautology.
• Math: $F \wedge q \equiv F$

Since option (D) simplifies to a contradiction, it is not a tautology.

Common Mistakes & Tips

• Remember the difference between conjunction ($\wedge$) and disjunction ($\vee$).
• Be careful when applying De Morgan's Laws.
• Recognize tautologies and contradictions quickly to simplify expressions.

Summary

We analyzed each option to determine if it simplifies to a tautology (a statement that is always true). Options (A), (B), and (D) simplified to contradictions, while option (C) simplified to a tautology. Therefore, option (C) is the correct answer.

Final Answer

The final answer is \boxed{(( \sim q) \wedge p) \vee (p \vee ( \sim p))}, which corresponds to option (C).