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 equivalently expressed as :

 

 

  • Option 1)

    \sim Q\leftrightarrow \sim P\wedge R\; \; \;

  • Option 2)

    \sim Q\leftrightarrow \sim P\vee R\; \; \;

  • Option 3)

    \sim Q\leftrightarrow P\: \: \vee\sim R\; \; \;

  • Option 4)

    \sim Q\leftrightarrow P\, \wedge \sim R\; \; \;

 

Answers (1)

As we learnt in

Negation of Biconditional Statement -

Negation of p\Leftrightarrow q is disjunction of negation of implication p\Rightarrow q and the negation of implication q\Rightarrow p

-

 

 Given statement is

(P\wedge \sim R) \leftrightarrow Q

which is same as

Q \leftrightarrow (P\wedge \sim R)

Since it is a biconditional statement

Hence negation is 

\sim (Q \leftrightarrow (P\wedge \sim R) 

 

 


Option 1)

\sim Q\leftrightarrow \sim P\wedge R\; \; \;

This option is incorrect.

Option 2)

\sim Q\leftrightarrow \sim P\vee R\; \; \;

This option is incorrect.

Option 3)

\sim Q\leftrightarrow P\: \: \vee\sim R\; \; \;

This option is incorrect.

Option 4)

\sim Q\leftrightarrow P\, \wedge \sim R\; \; \;

This option is correct.

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