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The logical statement $[ \sim \,( \sim \,P\, \vee \,q)\, \vee \,\left( {p\, \wedge \,r} \right)\, \wedge \,( \sim \,q\, \wedge \,r)]$ is equivalent to
$\left( {p\, \wedge \,r} \right)\, \wedge \, \sim \,q$
$( \sim \,p\,\, \wedge \sim \,q)\, \wedge \,r$
$ \sim \,p\,\, \vee {\kern 1pt} \,r$
$\left( {p\, \wedge \sim q} \right) \wedge \,r\,$
Solution
$\left[ { \sim \left( { \sim p \vee q} \right) \wedge \left( {p \wedge r} \right)} \right] \cap \left( { \sim q \wedge r} \right)$
$ \equiv \left[ {\left( {p \wedge \sim q} \right) \vee \left( {p \wedge r} \right)} \right] \wedge \left( { \sim q \wedge r} \right)$
$ \equiv \left[ {p \wedge \left( { \sim q \vee r} \right)} \right] \wedge \left( { \sim q \wedge r} \right)$
$ \equiv p \wedge \left( { \sim q \wedge r} \right)$
$ \equiv \left( {p \wedge r} \right) \sim q$
