Question

Let $Z$ be the set of all integers.
$A=\left\{\right(x,y)\in Z\times Z:(x-2{)}^2+y^2\le 4\}$,
$B=\left\{\right(x,y)\in Z\times Z:x^2+y^2\le 4\}$ and
$C=\left\{\right(x,y)\in Z\times Z:(x-2{)}^2+(y^2-2{)}^2\le 4\}$
If the total number of relations from $A\cap B$ to $A\cap C$ is $2^p$, then the value of $p$ is

• A. $16$
• B. $49$
• C. $25$
• D. $9$

Answer 1

The correct option is C $25$

The set $A$ and $B$ are represented as :


$\therefore A\cap B=\left\{\right(0,0),(1,0),(2,0),(1,1),(1,-1\left)\right\}$
The set $A$ and set $C$ are represented as:


$\therefore A\cap C=\left\{\right(1,1),(2,0),(2,1),(2,2),(3,2\left)\right\}$
Total number relations from $A\cap B$ to $A\cap C=2^{5\times 5}.$
$\therefore p=25$