Verify by method of contradiction, $p: \sqrt{11}$ is irrational.

Given the following two statements:

$\left(S_1\right):\left(q\vee p\right)\to \left(p\leftrightarrow ~q\right)$ is a tautology
$\left(S_2\right):~q\wedge \left(~p\leftrightarrow q\right)$ is a fallacy. Then :

Which of the following statement pattern is a tautology?

$S_1\equiv ~p\to \left(q\leftrightarrow p\right)$

$S_2\equiv ~p\vee ~q$

$S_3\equiv (p\to q)\wedge \left(q\to p\right)$

$S_4\equiv \left(q\to p\right)\vee \left(~p\leftrightarrow q\right)$

Which of the following statement pattern is a tautology?

$S_1\equiv \left(~q\wedge p\right)\wedge q$

$S_2\equiv \left[p\wedge \left(p\to q\right)\right]\to q$

$S_3\equiv \left(p\wedge q\right)\wedge \left(~p\vee ~q\right)$

$S_4\equiv \left(p\wedge q\right)\to r$

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

$\left(a\right) \left(~q\wedge \left(p\to q\right)\right)\to ~p$

$\left(b\right) \left(\left(p\vee q\right)\wedge ~p\right)\to q$

Then which of the following statements is correct?