Consider the following statements: P: Suman is brilliant Q: Suman...

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

Options

A $~ Q ↔ ~ P ∨ R$
B $~ Q ↔ P ∨ ~ R$
C $~ Q ↔ P ∧ ~ R$
D $~ Q ↔ ~ P ∧ R$

Correct answer

C. $~ Q ↔ P ∧ ~ R$

Step-by-step solution

$P$ : Suman is brilliant

$~ P$ : Suman is not brilliant

$Q$ = Suman is rich

$~ Q$ : Suman is not rich

$R$ : Suman is honest

$~ R$ : Suman is not honest

Also, $~ \left(p ⇔ q\right) ≡ \left(~ p ⇔ q\right) ≡ \left(p ⇔ ~ q\right)$

Hence $~ \left[( P ∧ ∼ R ) ⟷ Q\right]is ∼ Q ⟷ \left(P ∧ ∼ R\right)$

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