Let Z be the set of all integers,
Question:

Let $\mathbb{Z}$ be the set of all integers,

$\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+\mathrm{y}^{2} \leq 4\right\}$

$\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}: \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\}$ and

$\mathrm{C}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+(\mathrm{y}-2)^{2} \leq 4\right\}$

If the total number of relation from $\mathrm{A} \cap \mathrm{B}$ to $\mathrm{A} \cap \mathrm{C}$ is $2^{\mathrm{p}}$, then the value of $\mathrm{p}$ is :

1. 16

2. 25

3. 49

4. 9

Correct Option: , 2

Solution:

$(x-2)^{2}+y^{2} \leq 4$

$x^{2}+y^{2} \leq 4$

No. of points common in $\mathrm{C}_{1} \& \mathrm{C}_{2}$ is 5 . $(0,0),(1,0),(2,0),(1,1),(1,-1)$

Similarly in $\mathrm{C}_{2} \& \mathrm{C}_{3}$ is 5 .

No. of relations $=2^{5 \times 5}=2^{25}$.