Question

Which of the following statement is a tautology?

• A. \((\sim q \land p) \land q\)
• B. \((\sim q \land p) \land (p \land \sim p)\)
• C. \((\sim q \land p) \lor (p \lor \sim p)\)
• D. \((p \land q) \land (\sim (p \land q)\)

Answer 1

The Correct Option is C

To determine which statement is a tautology, we need to analyze each given option and identify the logical structure that is always true, irrespective of the truth values of the individual propositions.

Let's evaluate each option:

1. (\sim q \land p) \land q

   This expression can never be a tautology because if q is true, then \sim q is false, which makes (\sim q \land p) false, leading to the entire conjunction being false.

2. (\sim q \land p) \land (p \land \sim p)

   Notice that (p \land \sim p) is a contradiction since a proposition p cannot be both true and false at the same time. Therefore, the whole statement is a contradiction, not a tautology.

3. (\sim q \land p) \lor (p \lor \sim p)

   The sub-expression (p \lor \sim p) is a tautology because any proposition p is either true or false (law of excluded middle). Therefore, the entire expression evaluates to true regardless of the truth value of p or q, making it a tautology.

4. (p \land q) \land (\sim (p \land q))

   This expression is a direct contradiction because (p \land q) cannot simultaneously be true and false. Thus, it cannot be a tautology.

Upon this analysis, only option three (\sim q \land p) \lor (p \lor \sim p) results in a tautology because of the presence of the component (p \lor \sim p), which is always true according to the law of excluded middle.