Question

The dual of the statement pattern \((p∧\sim q)⟶(q∧\sim p)\) is equivalent to

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

Hint:

Rewrite the implication with or, take the dual by swapping and and or, then match.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The dual of a statement made with \(\wedge\), \(\vee\), \(\sim\) is found by swapping \(\wedge\) with \(\vee\) (and t with f). An implication must first be written using these symbols.

Step 2: Rewrite
\[ (p \wedge \sim q) \to (q \wedge \sim p) \equiv \sim(p \wedge \sim q) \vee (q \wedge \sim p) \equiv (\sim p \vee q) \vee (q \wedge \sim p) \]

Step 3: Take the dual
Swap \(\vee\) and \(\wedge\):
\[ (\sim p \wedge q) \wedge (q \vee \sim p) \equiv \sim p \wedge q \]

Step 4: Match with the options
Option (B): \((p \to q) \wedge \sim(q \to p) = (\sim p \vee q) \wedge (q \wedge \sim p) = q \wedge \sim p\). This equals the dual.
Option (A): \(\sim(p \to q) \wedge (q \to p) = (p \wedge \sim q) \wedge (p \vee \sim q) = p \wedge \sim q\), which is different.

Final Answer:
The dual is equivalent to \(\sim p \wedge q\), which matches option (B). \[ \boxed{(p\to q)\wedge \sim(q\to p)} \]