Question

Which of the following statements is a tautology?

• A. \(p \vee( p \wedge q )\)
• B. \(( p \wedge( p \rightarrow q )) \rightarrow \sim q\)
• C. \(( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )\)
• D. \(p \rightarrow( p \wedge( p \rightarrow q ))\)

Hint:

To check if a statement is a tautology, simplify the logical expression step-by-step using equivalence rules (e.g., distributive, associative, and De Morgan’s laws) and test it for all possible truth values of the variables.

Answer 1

The Correct Option is C

Step 1: Analyze Each Statement
(i) \( p \to (p \land (p \to q)) \):
\[p \to (p \land (p \to q)) \equiv \neg p \lor (p \land (p \to q)).\]
Using distributive property:
\[\neg p \lor (p \land (p \to q)) = (\neg p \lor p) \land (\neg p \lor (p \to q)).\]
This simplifies to:
\[\text{True} \land (\neg p \lor (\neg p \lor q)).\]
\[= \neg p \lor q.\]
This is not always true, so it is not a tautology.
(ii) \( (p \land q) \to \neg (p \to q) \):
\[(p \land q) \to \neg (p \to q) \equiv \neg (p \land q) \lor \neg (\neg p \lor q).\]
\[= (\neg p \lor \neg q) \lor (p \land \neg q).\]
This simplifies to:
\[\text{True for all cases}.\]
Hence, it is tautology
(iii) \( (p \land (p \to q)) \to \neg q \):
\[(p \land (p \to q)) \to \neg q \equiv \neg (p \land (p \to q)) \lor \neg q.\]
\[= (\neg p \lor \neg (p \to q)) \lor \neg q.\]
\[= (\neg p \lor (\neg p \land \neg q)) \lor \neg q.\]
\[= (\neg p \lor \neg q) \lor \neg q.\]
This is not always true, so it is not a tautology
(iv) \( p \lor (p \land q) \)
\[p \lor (p \land q) \equiv p.\]
This is not always true, so it is not a tautology.
Conclusion
The statement \((p \land q) \to \neg (p \to q)\)  is a tautology.