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

To determine which statement is a tautology, we need to analyze each of the given logical expressions. A tautology is a statement that is true in every possible interpretation. Let's evaluate each option:

1. \(p \vee( p \wedge q )\)
2. \(( p \wedge( p \rightarrow q )) \rightarrow \sim q\)
3. \(( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )\)
4. \(p \rightarrow ( p \wedge ( p \rightarrow q ))\)

After evaluating all options, we conclude that the correct tautology is \(( p \wedge q ) \rightarrow(\sim( p ) \rightarrow q )\).