JEE Main 2021 — Mathematical Reasoning Question with Solution

$\left(1\right) (p\Rightarrow q)\vee (~q\Rightarrow p)$

By applying $a\Rightarrow b\equiv ~a\vee b \& ~\left(~a\right)=a,$ we get

$=(~p\vee q)\vee (q\vee p)$

Since, the operation disjunction is commutative, hence, we have

$=(~p\vee q)\vee (p\vee q)$

Now, applying the distributive law i.e. $a\vee \left(b\vee c\right)\equiv \left(a\vee b\right)\vee \left(a\vee c\right),$ we get

$=(~p\vee p)\vee q$

Also, we know that $~a\vee a\equiv t \& t\vee a\equiv t,$ where $t$ represents tautology.

Thus, we have

$=t\vee q=t.$

Hence, the option $\left(A\right)$ is a tautology.

$\left(2\right) (q\Rightarrow p)\vee (~q\Rightarrow p)$

Again, using the same definitions, we get

$=(~q\vee p)\vee (q\vee p)$

$=(~q\vee q)\vee p$

$=t\vee p=t.$

Hence, the option $\left(B\right)$ is also a tautology.

$\left(3\right) (p\Rightarrow ~q)\vee (~q\Rightarrow p)$

$=(~p\vee ~q)\vee (q\vee p)$

$=(~p\vee p)\vee (~q\vee q)$

$=t\vee t=t.$

Hence, the option $\left(C\right)$ is also a tautology.

$\left(4\right) (~p\Rightarrow q)\vee (~q\Rightarrow p)$

$=(p\vee q)\vee (q\vee p)$

$=(p\vee q)\vee (p\vee q)$

$=p\vee q.$

Which is not a tautology.