Mathematical Reasoning

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

(B) $$ $q\vee \left(-r\vee p\right)$

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

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

Using Venn diagram we get B as the correct option.

(A) $\left(\wedge ,\vee \right):\left(P\vee Q\right)\wedge \left(P\wedge \sim Q\right)$ $$ not a tautology

(B) $\left(\vee ,\vee \right):\left(P\vee Q\right)\vee \left(P\vee \sim Q\right)=\left(P\vee T\right)$ $$ in a tautology

$$  $\left(p\vee q\right)\vee \left(p\vee \sim q\right)=$ T using venn diagrams

 $S=\{\left(\begin{array}{cc}\hfill 1\hfill & \hfill a\hfill \\ \hfill 0\hfill & \hfill b\hfill \end{array}\right),a,b\in 1,2,3,.....100$

$\therefore A=\left(\begin{array}{cc}\hfill -1\hfill & \hfill a\hfill \\ \hfill 0\hfill & \hfill b\hfill \end{array}\right)$ then even power of A as A = $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$

If b = 1 & $a\in \left\{1,2,3.....100\right\}$ and n (n + 1) is always even

$\therefore T_1,T_2,T_3,........,T_n$ are all 1 for b = 1 and each value of a.

$\therefore {\displaystyle \underset{n=1}{\overset{1000}{\cap }}{T}_n=100}$

$?$ given statement is

$\left(A\wedge C\right)\to B$ then its negation is $\left\{\left(A\wedge C\right)\to B\right\}$

 $x_{1}y>0and{x}^3{y}^2=2^{15}$

$AM\ge GM$

$\frac{3x+2y}{5}\ge {\left(x^3{y}^2\right)}^{\frac{1}{5}}$

$\Rightarrow 3x+2y\ge 40$

pva ⇒$\left(\sim rvp\right)$

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

its negation as asked in question

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

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

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

$\left(p\vee q\right)\Rightarrow q$

= $\sim \left(p\vee q\right)\vee q$

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

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

= $\sim p\vee q$

now $\left(p\wedge q\right)\Delta \left(\sim p\vee q\right)istautolog\text{?}y$

pq $p\wedge q$ $\sim p\vee q$ $\left(p\wedge q\right)\Delta \left(\sim p\vee q\right)$

TTTTT

TFFFT

FTFTT

FFFTT

So,  $\Delta =\Rightarrow$

Let

P : Ramesh listens to music

Q : Ramesh is out of his village

R : It is Sunday

S : It is Saturday

Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday can be expressed as

$P\equiv$ Ramesh listens to music

$\sim q\equiv$ He is in village.

$r\vee s\equiv$ Saturday or Sunday

$P\Rightarrow \left(\left(\sim q\right)\wedge \left(r\vee s\right)\right)$

Let TV (r) denotes truth value of a statement $r$ .

$\text{Now, if}TV\left(p\right)=TV\left(q\right)=T$

$\Rightarrow \mathrm{T}\mathrm{V}\left({\mathrm{S}}_1\right)=\mathrm{F}$

Also, if $TV\left(p\right)=T$ and $TV\left(q\right)=F$

$\Rightarrow \mathrm{T}\mathrm{V}\left({\mathrm{S}}_2\right)=\mathrm{T}$