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Question

Mathematics Question on mathematical reasoning

Let p,q,rp, q, r be three statements. Then ∼(p∨(q∧r))\sim (p \vee \left(q \wedge r\right)) is equal to

A

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

B

(∼p∨∼q)∧(∼p∨∼r)\left(\sim p \vee \sim q\right) \wedge \left(\sim p \vee \sim r\right)

C

(∼pΛ∼q)∨(∼pΛ∼r)\left(\sim p \Lambda \sim q\right) \vee \left(\sim p \Lambda \sim r\right)

D

(∼p∨∼q)∨(∼pΛ∼r)\left(\sim p \vee \sim q\right) \vee \left(\sim p \Lambda \sim r\right)

Answer

(∼pΛ∼q)∨(∼pΛ∼r)\left(\sim p \Lambda \sim q\right) \vee \left(\sim p \Lambda \sim r\right)

Explanation

Solution

∼(p∨(q∧r))=∼p∧∼(q∧r) \sim (p \vee(q \wedge r))=\sim p \wedge \sim(q \wedge r)
=∼p∧(∼q∨∼r)=\sim p \wedge(\sim q \vee \sim r)
=(∼p∧∼q)∨(∼p∧∼r)=(\sim p \wedge \sim q) \vee(\sim p \wedge \sim r)