The proposition $\left( { \sim p} \right) \vee \left( {p\, \wedge  \sim q} \right)$

Consider

Statement $-1 :$$\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \wedge q} \right)$ is a fallacy.

Statement $-2 :$$(p \rightarrow q) \leftrightarrow ( \sim q \rightarrow   \sim  p )$  is a tautology.

The logical statement $(p \Rightarrow q){\wedge}(q \Rightarrow \sim p)$ is equivalent to

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

Let $p , q , r$ be three logical statements. Consider the compound statements $S _{1}:((\sim p ) \vee q ) \vee((\sim p ) \vee r ) \text { and }$ and $S _{2}: p \rightarrow( q \vee r )$ Then, which of the following is NOT true$?$

If $q$ is false and $p\, \wedge \,q\, \leftrightarrow \,r$ is true, then which one of the following statements is a tautology?