JEE Main 2023 — Mathematical Reasoning Question with Solution

Given,

$\left(\right(A\wedge (B\vee C))\Rightarrow (A\vee B\left)\right)\Rightarrow A$

We know that, $X\Rightarrow Y\equiv ~X\vee Y$,

Now using the above formula we get,

$\equiv \left(~(A\wedge (B\vee C\left)\right)\vee (A\vee B)\right)\Rightarrow A$

$\equiv ~(~(A\wedge (B\vee C))\vee (A\vee B\left)\right)\vee A$

$\equiv (A\wedge (B\vee C\left)\right)\wedge ~(A\vee B)\vee A$

Now to solve $(A\wedge (B\vee C\left)\right)\wedge ~(A\vee B)$ let us assume a universal set $\left\{1,2,3,4,5,6,7,8\right\}$, set $A=\left\{1,2,3\right\}$, set $B=\left\{3,4,5\right\}$ and set $C=\left\{5,6,7\right\}$

So, $A\wedge \left(B\vee C\right)$ will be $\left\{3\right\}$ and $~(A\vee B)$ will be $\left\{6,7,8\right\}$, 

So, $(A\wedge (B\vee C\left)\right)\wedge ~(A\vee B)=\varnothing$

$\equiv F\vee A$

$\equiv A$

$\therefore$ Negation of statement$=~A$