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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example?
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Consider R² with the usual topology and A,B subset or equal R² then ch...
Proof:
To determine whether interior(A ∪ B) = (interior(A)) ∪ (interior(B)) holds for subsets A, B of R², let's consider the definitions of interior and union.
Definition of Interior:
The interior of a set A is the largest open set contained in A. Denoted as int(A).
Definition of Union:
The union of two sets A and B is the set containing all elements that are in A or in B or in both. Denoted as A ∪ B.
Counterexample:
Let's consider A = {(x, y) | x² + y² < 1}="" and="" b="{(x," y)="" |="" (x-2)²="" +="" y²="" />< />
The interior of A is the open disk of radius 1 centered at the origin, int(A) = {(x, y) | x² + y² < />
The interior of B is the open disk of radius 1 centered at (2, 0), int(B) = {(x, y) | (x-2)² + y² < />
The union A ∪ B consists of the two open disks without their boundaries, which is not an open set. Therefore, interior(A ∪ B) is not equal to (interior(A)) ∪ (interior(B)) in this case.
Thus, interior(A ∪ B) = (interior(A)) ∪ (interior(B)) does not hold for all subsets A, B of R².
To determine whether interior(A ∪ B) = (interior(A)) ∪ (interior(B)) holds for subsets A, B of R², let's consider the definitions of interior and union.
Definition of Interior:
The interior of a set A is the largest open set contained in A. Denoted as int(A).
Definition of Union:
The union of two sets A and B is the set containing all elements that are in A or in B or in both. Denoted as A ∪ B.
Counterexample:
Let's consider A = {(x, y) | x² + y² < 1}="" and="" b="{(x," y)="" |="" (x-2)²="" +="" y²="" />< />
The interior of A is the open disk of radius 1 centered at the origin, int(A) = {(x, y) | x² + y² < />
The interior of B is the open disk of radius 1 centered at (2, 0), int(B) = {(x, y) | (x-2)² + y² < />
The union A ∪ B consists of the two open disks without their boundaries, which is not an open set. Therefore, interior(A ∪ B) is not equal to (interior(A)) ∪ (interior(B)) in this case.
Thus, interior(A ∪ B) = (interior(A)) ∪ (interior(B)) does not hold for all subsets A, B of R².
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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example?
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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example?.
Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not, if not then give example?.
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