Embibe Experts Solutions for Chapter: Mathematical Reasoning, Exercise 2: Level 2

If $p, q, r$ are simple propositions, then the truth value of $(~p\vee q)\wedge ~r\Rightarrow p$ is

Which of the following is always true?

Which of the following is true?

Which of the following is a tautology?

Which of the following is always true?

The statement $p\wedge \left(~q\vee r\right)\wedge \left(~r\vee q\right)$ is

Consider the following statements:
$S_1:$ Negation of $\left(~p\to q\right)$ is $\left[~\left(p\vee q\right)\right]\wedge \left[p\vee \left(~p\right)\right]$.
$S_2:$ Negation of $\left(p\leftrightarrow q\right)$ is $\left(p\wedge ~q\right)\vee \left(~p\wedge q\right)$.
$S_3:$ Negation of $\left(p\vee q\right)$ is $~p\wedge ~q$
$S_4: p\leftrightarrow q$ is equivalent to $\left(~p\vee q\right)\wedge \left(p\vee ~q\right)$.
State, in order, whether $S_1,{ S}_2,{ S}_3 \text{and} S_4$ are true or false.

Statement I :  $~\left(A\iff ~B\right)$ is equivalent to $A\iff B.$
Statement II :  $A\vee \left(~\left(A\wedge ~B\right)\right)$ a tautology.