Question

The statements p, q and r have truth values True, False and False respectively. The truth values of a logical statement \([\sim (p∧\sim q)∨(q∨\sim r)]\) and its dual are, respectively.....

• A. True, True
• B. True, False
• C. False, True
• D. False, False

Hint:

Evaluate each connective step by step for the dual too.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The dual of a statement is formed by swapping \(\wedge\) with \(\vee\) (and T with F), keeping \(\sim\) as is. We evaluate the statement and its dual with the given truth values.

Step 2: Given values:
\(p=T\), \(q=F\), \(r=F\). So \(\sim q=T\) and \(\sim r=T\).

Step 3: Evaluate the statement:
\(S=\sim(p\wedge\sim q)\vee(q\vee\sim r)\).
\(p\wedge\sim q=T\wedge T=T\), so \(\sim(\ldots)=F\).
\(q\vee\sim r=F\vee T=T\).
So \(S=F\vee T=T\).

Step 4: Dual:
Swap the connectives: \(S^d=\sim(p\vee\sim q)\wedge(q\wedge\sim r)\).
\(p\vee\sim q=T\vee T=T\), so \(\sim(\ldots)=F\).
\(q\wedge\sim r=F\wedge T=F\).
So \(S^d=F\wedge F=F\).

Step 5: Choose:
True and False, option (B).

Final Answer:
The statement is True and its dual is False. \[ \boxed{\text{True, False (B)}} \]