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Mathematical Reasoning
medium
If the Boolean expression $( p \wedge q ) \circledast( p \otimes q )$ is a tautology, then $\circledast$ and $\otimes$ are respectively given by
A
$\rightarrow, \rightarrow$
B
$\wedge, \vee$
C
$\vee, \rightarrow$
D
$\wedge, \rightarrow$
(JEE MAIN-2021)
Solution
Option $(1)$
$( p \wedge q ) \longrightarrow( p \rightarrow q )$
$=\sim(p \wedge q) \vee(\sim p \vee q)$
$=(\sim p \vee \sim q) \vee(\sim p \vee q)$
$=\sim p \vee(\sim q \vee q )$
$=\sim p \vee t$
$= t$
Option $(2)$
$( p \wedge q ) \wedge( p \vee q )=( p \wedge q )($ Not a tautology $)$
Option $(3)$
$(p \wedge q) \vee(p \rightarrow q)$
$=(p \wedge q) \vee(\sim p \vee q)$
$=\sim p \vee q$ $($ Not a tautology $)$
Option $(4)$
$=(p \wedge q) \wedge(\sim p \vee q)$
$= p \wedge q$ (Not a tautology)
Option $(1)$
Standard 11
Mathematics
