Question

If \(A=\{(x,y)\mid x^2+y^2\le 4;\; x,y\in \mathbb{R}\}\) and \(B=\{(x,y)\mid x^2+y^2\ge 9;\; x,y\in \mathbb{R}\}\), then

• A. \(A-B=\phi\)
• B. \(B-A=\phi\)
• C. \(A\cap B\neq \phi\)
• D. \(A\cap B=\phi\)

Hint:

For sets defined by \(x^2+y^2\):
\(x^2+y^2\le r^2\) represents a disc of radius \(r\)
\(x^2+y^2\ge r^2\) represents the exterior of a circle
Discs with non-overlapping radii have empty intersection

Answer 1

The Correct Option is D

Step 1: Describe the set \(A\). \[ A=\{(x,y):x^2+y^2\le 4\} \] This represents the closed disc of radius \(2\) centered at the origin.
Step 2: Describe the set \(B\). \[ B=\{(x,y):x^2+y^2\ge 9\} \] This represents the region outside or on the circle of radius \(3\) centered at the origin.
Step 3: Compare the two regions. \[ x^2+y^2\le 4 \quad \text{and} \quad x^2+y^2\ge 9 \] There is no real number which is simultaneously \(\le 4\) and \(\ge 9\).
Step 4: Hence, no point can satisfy both conditions at the same time. \[ \Rightarrow A\cap B=\phi \]