The logical statement $[\sim(\sim p\vee q)\vee (p\wedge r)]\wedge (\sim q \wedge r)$is equivalent to :Option 1)  $(\sim p \wedge \sim q)\wedge r$Option 2)  $(p\wedge \sim q) \vee r$Option 3)  $\sim p \vee r$Option 4)  $(p \wedge r) \wedge \sim q$

Truth value of "And" Conjuction -

The statement $p\wedge q$ has the truth value T whenever both p and q have the truth value T.

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Truth Value of Disjunction "OR" -

The statement p$\vee$q has the truth value F if both p and q have the truth value F.

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Negation of a Statement -

Negation is a connective although it doesn't combine two or more statements

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Given,

$[\sim (\sim p\vee q)\vee (p\wedge r) \wedge (\sim q\wedge r))] \\ = [ (p\wedge \sim q)\vee (p\wedge r)] \wedge (\sim q\wedge r)) \\ = [ (p\wedge (\sim q\vee r)] \wedge (\sim q\wedge r)) \\= p\wedge (\sim q\wedge r) \\ \equiv (p\wedge r)\wedge \sim q$

Option 1)

$(\sim p \wedge \sim q)\wedge r$

Option 2)

$(p\wedge \sim q) \vee r$

Option 3)

$\sim p \vee r$

Option 4)

$(p \wedge r) \wedge \sim q$

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