Question

The negation of the contrapositive of the statement \((p∨\sim q)\rightarrow (p∧\sim q)\) is

• A. \((p∧\sim q)∨(\sim p∧\sim q)\)
• B. \((\sim p∧q)∨(p∧\sim q)\)
• C. \((\sim p∨\sim q)∧(p∨q)\)
• D. \((\sim p∨q)∧(p∨\sim q)\)

Hint:

The negation of X implies Y is X and not Y.

Answer 1

The Correct Option is D

Step 1: Shortcut
The negation of the contrapositive of $A\rightarrow B$ equals the negation of $A\rightarrow B$ itself (they are logically equivalent). So we only need $\sim(A\rightarrow B) = A\wedge\sim B$.

Step 2: Substitute
$A = p\vee\sim q$, $B = p\wedge\sim q$, so $\sim B = \sim p\vee q$.

Step 3: Result
$(p\vee\sim q)\wedge(\sim p\vee q)$, which is option (D).

Final Answer:
Option D. \[ \boxed{\text{(D)}\ (\sim p\vee q)\wedge(p\vee\sim q)} \]