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

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Sets Relations and Functions

Let

Z

, be the set of all integers,

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

B = {(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4}

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

If the total number of relations from

A \cap B

A \cap C

2^{p}

, then the value of

p

is:

Answer: (D)

The integer points of

A

lying in

B

are

(0, 0), (1, - 1), (1, 0), (1, 1), (2, 0)

, so

∣ A \cap B ∣ = 5

. The integer points of

A

lying in

C

are

(1, 1), (2, 0), (2, 1), (2, 2), (3, 1)

, so

∣ A \cap C ∣ = 5

. The number of relations from a set of size

m

to a set of size

n

2^{mn}

, here

2^{5 \times 5} = 2^{25}

, so

p = 25

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Single correct option