Question

The statement $(p \wedge(\sim q)) \Rightarrow(p \Rightarrow(\sim q))$ is

• A. equivalent to $p \vee q$
• B. equivalent to $(-p) \vee(-q)$
• C. a contradiction
• D. a tautology

Answer 1

The Correct Option is D

$(p\wedge \sim q)\to (p\to \sim q)$
$\equiv (\sim (p\wedge \sim q))\vee (\sim p\vee \sim q)$
$\equiv (\sim p\vee q)\vee (\sim p\vee \sim q)$
$\equiv \sim p\vee t\equiv t$

Answer 2

1. The statement can be simplified as follows: \[ (p \land (\sim q)) \Rightarrow (p \Rightarrow (\sim q)) \] Using the implication equivalence \( A \Rightarrow B = (\sim A) \lor B \): \[ = (\sim (p \land (\sim q))) \lor ((\sim p) \lor (\sim q)). \] 2. Simplify further: \[ = ((\sim p) \lor q) \lor ((\sim p) \lor (\sim q)). \] 3. Combine terms: \[ = (\sim p) \lor q \lor (\sim q). \] 4. Since \(q \lor (\sim q) = \text{True}\), the entire expression becomes: \[ = \text{True}. \] Thus, the statement is a tautology. A tautology is a logical statement that is always true, regardless of the truth values of its components.