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Mathematics Question on mathematical reasoning

The logical statement [∼(∼p∨q)∨(p∧r)∧(∼q∧r)][\sim (\sim p \vee q) \vee (p \wedge r) \wedge (\sim q \wedge r)] is equivalent to:

A

(p∧r)∧∼q(p \wedge r) \wedge \sim q

B

(∼p∧∼q)∧r(\sim p \wedge \sim q) \wedge r

C

∼p∨r\sim p \vee r

D

(p∧∼q)∨r(p \wedge \sim q) \vee r

Answer

(p∧r)∧∼q(p \wedge r) \wedge \sim q

Explanation

Solution

s[∼(∼p∨q)∧(p∧r)]∩(∼q∧r)s [\sim (\sim p \vee q) \wedge (p \wedge r )] \cap ( \sim q \wedge r)
≡[(p∧∼q)∨(p∧r)]∧(∼q∧r)\equiv [(p \wedge \sim q) \vee ( p \wedge r)] \wedge (\sim q \wedge r)
≡[p∧(∼q∨r)]∧(∼q∧r)\equiv [ p \wedge (\sim q \vee r)] \wedge ( \sim q \wedge r )
≡  p∧(∼q∧r)\equiv \; p \wedge (\sim q \wedge r )
≡(p∧r)∼q\equiv (p \wedge r) \sim q