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If the Boolean expression $\left( {p \oplus q} \right) \wedge \left( { \sim p\,\Theta\, q} \right)$ is equivalent to $p \wedge q$, where $ \oplus $ , $\Theta \in \left\{ { \wedge , \vee } \right\}$ , ,then the ordered pair $\left( { \oplus ,\Theta } \right)$ is
$\left( { \vee , \wedge } \right)$
$\left( { \vee , \vee } \right)$
$\left( { \wedge , \vee } \right)$
$\left( { \wedge , \wedge } \right)$
Solution
check all option repeatedly
$\left( i \right)\left( {A \wedge B} \right) \wedge \left( { \sim A \vee B} \right) \equiv A \wedge \left( {B \wedge \left( { \sim A \vee B} \right)} \right)$
$ \equiv A \wedge \left( B \right) \equiv A \wedge B$
$ \Rightarrow \left( i \right)$ is correct
$\left( {ii} \right)\left( {A \wedge B} \right) \wedge \left( { \sim A \vee B} \right) \equiv \left( {A \wedge \sim A} \right) \wedge B$
$ \equiv f \wedge B \equiv f$
$\left( {iii} \right)\left( {A \vee B} \right) \wedge \left( { \sim A \vee B} \right) \equiv B$
$\left( {iv} \right)\left( {A \vee B} \right)\left( { \sim A \vee B} \right)$
$ \equiv B \vee \left( {A \wedge \sim A} \right) \equiv B \vee f \equiv f$
$ \Rightarrow $ only $(1)$ is correct
