Let \( Z \) be the set of all integers,
\( A = \{(x, y) \in Z \times Z : (x - 2)^2 + y^2 \leq 4\} \),
\( B = \{(x, y) \in Z \times Z : x^2 + y^2 \leq 4\} \) and
\( C = \{(x, y) \in Z \times Z : (x - 2)^2 + (y - 2)^2 \leq 4\} \)
If the total number of relations from \( A \cap B \) to \( A \cap C \) is \( 2^p \), then the value of \( p \) is :
Show Hint
When finding integer points inside a circle, it's efficient to iterate through possible integer values of y and solve the resulting inequality for integer values of x. This systematic approach ensures no points are missed.
Step 1: Understanding the Question:
We are given three sets A, B, and C, which represent the integer points inside or on three different circles. We need to find the number of elements in the intersections A \(\cap\) B and A \(\cap\) C. Then, we use the formula for the total number of relations between two sets to find the value of p. Step 2: Key Formula or Approach:
The total number of relations from a set P to a set Q is given by \(2^{|P| \times |Q|}\), where $|P|$ and $|Q|$ are the number of elements in sets P and Q, respectively. Step 3: Finding the elements of the sets and their intersections: Set A: (x - 2)\(^2\) + y\(^2\) \(\le\) 4 represents a circle centered at (2, 0) with radius 2. We list the integer points (x, y) satisfying this:
- For y = 0, (x - 2)\(^2\) \(\le\) 4 \(\implies\) 0 \(\le\) x \(\le\) 4. Points: (0,0), (1,0), (2,0), (3,0), (4,0).
- For y = \(\pm\)1, (x - 2)\(^2\) \(\le\) 3 \(\implies\) 2 - \(\sqrt{3}\) \(\le\) x \(\le\) 2 + \(\sqrt{3}\). Integer x values are 1, 2, 3. Points: (1,1), (1,-1), (2,1), (2,-1), (3,1), (3,-1).
- For y = \(\pm\)2, (x - 2)\(^2\) \(\le\) 0 \(\implies\) x = 2. Points: (2,2), (2,-2).
A = {(0,0), (1,0), (2,0), (3,0), (4,0), (1,1), (1,-1), (2,1), (2,-1), (3,1), (3,-1), (2,2), (2,-2)}. Set B: x\(^2\) + y\(^2\) \(\le\) 4 represents a circle centered at (0, 0) with radius 2. The integer points are:
B = {(-2,0), (-1,0), (0,0), (1,0), (2,0), (-1,1), (0,1), (1,1), (-1,-1), (0,-1), (1,-1), (0,2), (0,-2)}. Set C: (x - 2)\(^2\) + (y - 2)\(^2\) \(\le\) 4 represents a circle centered at (2, 2) with radius 2. The integer points are:
C = {(0,2), (1,2), (2,2), (3,2), (4,2), (1,1), (2,1), (3,1), (1,3), (2,3), (3,3), (2,0), (2,4)}. Intersection A \(\cap\) B:
Comparing the elements of A and B, the common points are:
A \(\cap\) B = {(0,0), (1,0), (2,0), (1,1), (1,-1)}.
So, |A \(\cap\) B| = 5. Intersection A \(\cap\) C:
Comparing the elements of A and C, the common points are:
A \(\cap\) C = {(2,0), (1,1), (2,1), (3,1), (2,2)}.
So, |A \(\cap\) C| = 5. Step 4: Calculating p:
The total number of relations from A \(\cap\) B to A \(\cap\) C is given by \(2^{|A \cap B| \times |A \cap C|}\).
Number of relations = \(2^{5 \times 5} = 2^{25}\).
We are given that this number is 2\(^p\).
Therefore, p = 25.
