Statement $-1 :$ $\sim (p \leftrightarrow \sim q)$ is equivalent to $p\leftrightarrow q $

Statement $-2 :$ $\sim (p \leftrightarrow \sim q)$ s a tautology

For the statements $p$ and $q$, consider the following compound statements :

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

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

Then which of the following statements is correct?

Which of the following Boolean expressions is not a tautology ?

Given the following two statements :

$\left( S _{1}\right):( q \vee p ) \rightarrow( p \leftrightarrow \sim q )$ is a tautology.

$\left( S _{2}\right): \sim q \wedge(\sim p \leftrightarrow q )$ is a fallacy.

Then

Consider the following statements 

$P :$ Suman is brilliant

$Q :$ Suman is rich

$R :$ Suman is honest

The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as 

The statement $(p \wedge(\sim q) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q))$ is equivalent to