# The Boolean expression $\left ( \left ( p\wedge q \right )\vee \left ( p\: \vee \sim q \right ) \right )\wedge \left ( \sim p\: \wedge \sim q \right )$ is equivalent to : Option 1)$\left ( \sim p \right )\wedge \left ( \sim q \right )$      Option 2)$p\wedge q$Option 3)$p\vee \left ( \sim q \right )$Option 4)$p\wedge \left ( \sim q \right )$

Negation of Conditional Statement -

$\sim (p\Rightarrow q)\equiv p\wedge\sim q$

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Construction of truth table -

We prepare table of rows and columns. We write variables denoting sub-statements and we write the truth value of sub statement to get compound statement.

- wherein

$\left \{ \left [\left (p\wedge q\right ) \vee p \right ] \vee \left [\left(p\wedge q \right )\vee \sim q \right ] \right \}\wedge \sim(p\vee q) \\\\\Rightarrow\left \{ p\vee \left [\left (p\vee \sim q\right ) \wedge (q\vee \sim q) \right ] \right \}\wedge \sim(p\vee q) \\\\\Rightarrow \left \{ p\vee \left [p\vee \sim q\right ] \right \}\wedge \sim(p\vee q) \\\\\Rightarrow \left (p\vee \sim q\right )\wedge \sim(p\vee q) \Rightarrow \sim (p \vee q) \\\\\Rightarrow \sim p\;\wedge \sim q$

Option 1)

$\left ( \sim p \right )\wedge \left ( \sim q \right )$

Option 2)

$p\wedge q$

Option 3)

$p\vee \left ( \sim q \right )$

Option 4)

$p\wedge \left ( \sim q \right )$

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