JEE Main 2022 — Mathematical Reasoning Question with Solution

Let $x:(p\Delta q)\Rightarrow (p\Delta \sim q)\vee (\sim p\Delta q)$

Case-I

When $\Delta$ is same as $v$

Then $(p\Delta \sim q)\vee (\sim p\Delta q)$ becomes

$(p\vee \sim q)\vee (\sim p\vee q)$ which is always true, so $x$ becomes a tautology.

Case-II

When $\Delta$ is same as $\wedge$

Then $(p\wedge q)\Rightarrow (p\wedge \sim q)\vee (\sim p\wedge q)$

If $p\wedge q$ is $T$, then $(p\wedge \sim q)\vee (\sim p\wedge q)$ is $F$

so $x$ cannot be a tautology.

Case-III

When $\Delta$ is same as $\Rightarrow$

Then $(p\Rightarrow \sim q)\vee (\sim p\Rightarrow q)$ is same at $(\sim p\vee \sim q)\vee$ $(p\vee q)$, which is always true, so $x$ becomes a tautology.

Case-IV

When $\Delta$ is same as $\iff$

Then $(p\iff q)\Rightarrow (p\iff \sim q)\vee (\sim p\iff q)$

$p\iff q$ is true when $p$ and $q$ have same truth values, then $p\iff \sim q$ and $\sim p\iff q$ both are false. Hence $x$ cannot be a tautology.

So finally $x$ can be $\vee$ or $\Rightarrow$.