JEE Main 2021 (Online) 27th August Evening ShiftQuestion 1
Let $\mathbb{Z}$ be the set of all integers,
$A=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}:(x-2)^{2}+y^{2}\le4\}$,
$B=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}:x^{2}+y^{2}\le4\}$ and
$C=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}:(x-2)^{2}+(y-2)^{2}\le4\}$.
If the total number of relations from $A\cap B$ to $A\cap C$ is $2^{p}$, then the value of $p$ is :
Sets, Relations and Function
Solution
$A\cap B$: points with $(x-2)^2+y^2\le4$ and $x^2+y^2\le4$: $(1,0),(1,1),(1,-1),(0,0),(2,0)$ checks: $(0,0)$ in both, $(2,0)$ in both, $(1,0),(1,\pm1)$ in both: 5 points. $A\cap C$ similarly has 5 points. Relations: $2^{5\cdot5}=2^{25}$, $p=25$. Option (C).