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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
$\; \sim \left( {{\rm{Q}} \leftrightarrow \left( {{\rm{P}} \wedge {\rm{\;}} \sim {\rm{R}}} \right)} \right)$
$ \sim {\rm{Q}} \leftrightarrow {\rm{\;}} \sim {\rm{P}} \wedge {\rm{R}}$
${\rm{\;}} \sim \left( {{\rm{P}} \wedge {\rm{\;}} \sim {\rm{R}}} \right) \leftrightarrow Q$
$\; \sim P \wedge \left( {{\rm{Q\;}} \leftrightarrow \sim {\rm{R}}} \right)$
Solution
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
$\sim(Q \leftrightarrow(P \wedge \sim R)$
