The converse of the statement “If sun is not shining, then sky is filled with clouds” is

(a) If sky is filled with clouds, then the sun is not shining.

(b) If sun is shining, then sky is filled with clouds.

(c) If sky is clear, then sun is shining.

(d) If sun is not shining, then sky is not filled with clouds.

Case – I $\Delta \equiv \nabla \equiv \wedge$  

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

              it can be false if r is false,

              so not a tautology

   &n

pva $\left(\sim rvp\right)$ $$

$\sim \left(p\vee q\right)\vee \left(\sim rvp\right)$

its negation as asked in question

$\left(p\vee q\right)\wedge \left(\sim p\wedge r\right)$

= $\left(p\wedge \sim p\wedge r\right)\vee \left(q\wedge r\wedge \sim p\right)$

= $\left(p\wedge r\wedge \sim p\right)\left[asp\wedge \sim pisfalse\right]$

$2021\equiv -2$ mod (7)

$\Rightarrow {\left(2021\right)}^{2023}\equiv {\left(-2\right)}^{2023}mod\left(7\right)$   …… (i)

Now,   ${\left(-2\right)}^3\equiv -1mod\left(7\right)$  

$\Rightarrow {\left(-2\right)}^{2023}\equiv \left(-2\right)mod\left(7\right)\equiv 5mod\left(7\right)$       ……. (ii)

(i) & (ii)

$\Rightarrow {\left(2021\right)}^{2023}\equiv 5mod\left(7\right)$  

$\therefore$  Remainder = 5

kindly consider the following Image 

$\sim \left(p\iff \sim q\right)\wedge q=\left(p\iff \sim q\right)\wedge qis:$

$\therefore \left(\sim \left(P\iff \sim Q\right)\right)\wedge$ q is equivalent to $\left(p\Rightarrow q\right)\wedge$