Negation of Boolean expression p Leftrightarrow( q ⇒ p ) is.

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Mathematical Reasoning

hard

Negation of the Boolean expression $p \Leftrightarrow( q \Rightarrow p )$ is.

$(\sim p ) \wedge q$

$p \wedge(\sim q )$

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

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

(JEE MAIN-2022)

Solution

$\sim( p \leftrightarrow( q \rightarrow p ))$

$\sim( p \leftrightarrow q )=( p \wedge \sim q ) \vee( q \wedge \sim p )$

$\sim( p \leftrightarrow( q \rightarrow p ))=( p \wedge \sim( q \rightarrow p )) \vee(( q \rightarrow p ) \wedge \sim p )$

$( p \wedge \sim( q \rightarrow p ))= p \wedge( q \wedge \sim p )=( p \wedge \sim p ) \wedge q = c$

$( q \rightarrow p ) \wedge \sim p =(\sim q \vee p ) \wedge \sim p =\sim p \wedge(\sim q \vee p )$ $=(\sim p \wedge \sim q ) \vee(\sim p \wedge p )=\sim p \wedge \sim q$

$\sim( p \leftrightarrow( q \rightarrow p ))= c \vee(\sim p \wedge \sim q )=\sim p \wedge \sim q$

Standard 11

Mathematics

Which of the following is a statement

easy

If $p$ and $q$ are simple propositions, then $p \Leftrightarrow \sim \,q$ is true when

medium

$(\sim (\sim p)) \wedge q$ is equal to ………

easy

The contrapositive of the statement "If I reach the station in time, then I will catch the train" is

medium

(JEE MAIN-2020)

The statement $p \to ( q \to p)$ is equivalent to