Embibe Experts Solutions for Chapter: Mathematical Reasoning, Exercise 1: JEE Main - 1 February 2023 Shift 2

If $p,q$ and $r$ are three propositions, then which of the following combination of truth values of $p, q$ and $\mathrm{r}$ makes the logical expression $\left\{\right(p\vee q)\wedge ((~p)\vee r\left)\right\}\to \left(\right(~q)\vee r)$ false ?

The statement $B\mathit{\Rightarrow }\left(\left(\mathit{~}A\right)\mathit{\vee }B\right)$ is not equivalent to :

Among the statements:

$\left(S1\right): \left(\left(p\vee q\right)\Rightarrow r\right)\iff \left(p\Rightarrow r\right)$

$\left(S2\right):\left(\left(p\vee q\right)\Rightarrow r\right)\iff \left(\left(p\Rightarrow r\right)\vee \left(q\Rightarrow r\right)\right)$

Consider the following statements:
$P$ : I have fever
$Q$ : I will not take medicine
$R$ : I will take rest
The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:

Consider:

$\left(\mathrm{S}1\right):\left(p\Rightarrow q\right)\vee \left(p\wedge \left(~q\right)\right)$ is a tautology.

$\left(\mathrm{S}2\right):\left(\left(~\mathrm{p}\right)\Rightarrow \left(~\mathrm{q}\right)\right)\wedge \left(\left(~\mathrm{p}\right)\vee \mathrm{q}\right)$ is a contradiction.

Then

The number of values of $r\in \left\{p,q,~p,~q\right\}$ for which $\left(\left(p\wedge q\right)\Rightarrow \left(r\vee q\right)\wedge (\left(p\wedge r\right)\Rightarrow q\right)$ is a tautology, is :

The negation of the expression $q\vee \left(\right(~q)\wedge p)$ is equivalent to

Which of the following statements is a tautology?