The statement ( p ∧ ( ~ q ) ) ∨ ( ( ~ p ) ∧ q ) ∨ ( ( ~ p ) ∧ ( ~ q...

The statement

( p ∧ ( ~ q ) ) ∨ ( ( ~ p ) ∧ q ) ∨ ( ( ~ p ) ∧ ( ~ q ) )

is equivalent to

_____

Options

A $\left(~ p\right) ∨ q$
B $\left(~ p\right) ∨ \left(~ q\right)$
C $p ∨ \left(~ q\right)$
D $p ∨ q$

Correct answer

B. $\left(~ p\right) ∨ \left(~ q\right)$

Step-by-step solution

Given,
$( p ∧ ( ~ q ) ) ∨ ( ( ~ p ) ∧ q ) ∨ ( ( ~ p ) ∧ ( ~ q ) )$
$= ( ( ~ p ) ∧ q ) ∨ ( ( ~ p ) ∧ ( ~ q ) ) ∨ ( p ∧ ( ~ q ) )$
$= \left(~ p ∧ \left(q ∨ ~ q\right)\right) ∨ \left(p ∧ ~ q\right)$
We know that $~ A ∨ A = T$ , where $T$ is tautology,
$= \left(~ p ∧ T\right) ∨ \left(p ∧ ~ q\right)$
$= ~ p ∨ \left(p ∧ ~ q\right)$
$= \left(~ p ∨ p\right) ∧ \left(~ p ∨ ~ q\right)$
$= T ∧ \left(~ p ∨ ~ q\right)$
$= ~ p ∨ ~ q$

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