Question:
Consider \( \mathbb{R}^2 \) with the usual metric. Let
\[
A = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\} \quad \text{and} \quad B = \{(x, y) \in \mathbb{R}^2 : (x - 2)^2 + y^2 \leq 1\}.
\]
Let \( M = A \cup B \) and \( N = \text{interior}(A) \cup \text{interior}(B) \).
Then, which of the following statements is TRUE?
Consider \( \mathbb{R}^2 \) with the usual metric. Let
\[
A = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\} \quad \text{and} \quad B = \{(x, y) \in \mathbb{R}^2 : (x - 2)^2 + y^2 \leq 1\}.
\]
Let \( M = A \cup B \) and \( N = \text{interior}(A) \cup \text{interior}(B) \).
Then, which of the following statements is TRUE?
Show Hint
A union of connected sets is connected if and only if the sets intersect. The interior of two disjoint sets results in a disconnected set.
Updated On: Dec 4, 2025
- \( M \) and \( N \) are connected
- Neither \( M \) nor \( N \) is connected
- \( M \) is connected and \( N \) is not connected
- \( M \) is not connected and \( N \) is connected
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The Correct Option is C
Solution and Explanation
Step 1: Understand the given sets.
- The set \( A \) represents the unit disk centered at the origin, and \( B \) represents the unit disk centered at \( (2, 0) \). - \( M = A \cup B \) is the union of these two disks, which are tangent to each other at the point \( (1, 0) \). Since the disks intersect at a single point, the union \( M \) is connected. Step 2: Analyze \( N \).
- \( N = \text{interior}(A) \cup \text{interior}(B) \), which consists of the interior of the two disks. Since the interiors of the two disks are disjoint, \( N \) is the union of two disconnected regions. Thus, \( M \) is connected, but \( N \) is not connected. Step 3: Conclusion.
The correct statement is (C): \( M \) is connected and \( N \) is not connected.
- The set \( A \) represents the unit disk centered at the origin, and \( B \) represents the unit disk centered at \( (2, 0) \). - \( M = A \cup B \) is the union of these two disks, which are tangent to each other at the point \( (1, 0) \). Since the disks intersect at a single point, the union \( M \) is connected. Step 2: Analyze \( N \).
- \( N = \text{interior}(A) \cup \text{interior}(B) \), which consists of the interior of the two disks. Since the interiors of the two disks are disjoint, \( N \) is the union of two disconnected regions. Thus, \( M \) is connected, but \( N \) is not connected. Step 3: Conclusion.
The correct statement is (C): \( M \) is connected and \( N \) is not connected.
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