Question

If the Boolean expression (p ⊕ q) ∧ (~pΘq) is equivalent to p ∧ q , where ⊕ , Θ ∈ {∧, ∨}, then the ordered pair (⊕, Θ) is : (A) (∨, ∧) (B) (∨, ∨)
(C) (∧, ∨) (D) (∧, ∧)
[JEE Main 2019]

Answer 1

then the ordered pair (⊕, Θ) is (  ∧ , ∨  )

Lets check each option.

( A )  ( ∨ , ∧ ) =

• ( p ∨ q ) ∧ ( ~ p ∧ q ) = ( ~p ∧ q )

( B ) ( ∨, ∨ ) =

• ( p ∨ q ) ∧ ( ~p ∨ q )  = ( ( p ∨ q ) ∧ ~p ) ∨ ( ( p ∨ q ) ∧ q )
• = null set ∨ q = q

( C ) ( ∧ , ∨ ) =

• ( p ∧ q ) ∧ ( ~p ∨ q ) = {  ( p ∧ q ) ∧ ~p } ∨  { ( p ∧ q ) ∧ q }= null ∨ ( p ∧ q )
• =  p ∧ q

( D ) ( ∧ , ∧ ) =

• ( p ∧ q ) ∧ ( ~p ∧ q ) = ( ( p ∧ q ) ∧ ~p ) ∧ ( ( p ∧ q ) ∧ q ) =
• = null set ∧ ( p ∧ q ) = null set