Let Z be the set of all integers, A = ( x , y ) ∈ Z × Z : ( x − 2 ) 2 + y 2 ≤ 4 B = ( x , y ) ∈ Z × Z : x 2 + y 2 ≤ 4 and C = ( x , y ) ∈ Z × Z : ( x − 2 ) 2 + ( y − 2 ) 2 ≤ 4 If the total number of relations from A ∩ B to A ∩ C is 2 p , then the value of p is:

Step-by-step solution

A = ( x , y ) ∈ Z × Z ; ( x − 2 ) 2 + y 2 ≤ 4 B = ( x , y ) ≤ Z × Z ; x 2 + y 2 ≤ 4 C = ( x , y ) ∈ Z × Z ; ( x − 2 ) 2 + ( y − 2 ) 2 ≤ 4 Here, we can see from the above diagram A ∩ B = ( 0 , 0 ) , ( 1 , 0 ) , ( 2 , 0 ) , ( 1 , 1 ) , ( 1 , - 1 ) ⇒ n ( A ∩ B ) = 5 Similarly n ( A ∩ C ) = 5 Relation from A ∩ B to A ∩ C = 2 5 × 5 = 2 p ⇒ p = 25