Question

Consider the following statements:
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest
The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as

• A. $\sim P\wedge (Q\leftrightarrow \sim R)$
• B. $\sim (Q\leftrightarrow (P\wedge \sim R\left)\right)$
• C. $\sim Q\leftrightarrow \sim (P\wedge R)$
• D. $\sim (P\wedge \sim R)\leftrightarrow Q$

Answer 1

The correct option is B $\sim (Q\leftrightarrow (P\wedge \sim R\left)\right)$

Here $\sim \equiv Negation$ and $\leftrightarrow \equiv ifandonlyif,\wedge \equiv AND$

Suman is brilliant $\equiv P$

Suman is dishonest $\equiv \sim R$

Suman is rich $\equiv Q$

Thus the negation of statement $S$:"Suman is brilliant and dishonest if and only if Suman is rich"

$\equiv \sim \left(\right(P\wedge \sim R)\leftrightarrow Q)$