Among the statements

$(S1)$: $(p \Rightarrow q) \vee((\sim p) \wedge q)$ is a tautology

$(S2)$: $(q \Rightarrow p) \Rightarrow((\sim p) \wedge q)$ is a contradiction

The negation of the Boolean expression $x \leftrightarrow \sim y$ is equivalent to 

Which of the following is true

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

The contrapositive of the statement "if I am not feeling well, then I will go to the doctor" is

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?