UPSC Exam > UPSC Questions > Consider R² with the usual topology and A,B s...
Start Learning for Free
Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)?
Most Upvoted Answer
Consider R² with the usual topology and A,B subset or equal R² then ch...
Intuition:
The interior of a set represents all the points inside the set that do not touch the boundary. In the case of the union of two sets, the interior will include points that are in the interior of either set.
Verification:
Claim:
int(A ∪ B) = int(A) ∪ int(B)
Proof:
1. Inclusion of int(A ∪ B) in int(A) ∪ int(B):
- Let x be an element of int(A ∪ B). This means that there exists an open ball around x contained entirely within A ∪ B.
- Since x is in A ∪ B, it must be in either A or B. Without loss of generality, assume x is in A.
- Then, the open ball around x is entirely contained within A, making x an interior point of A. Thus, x is in int(A).
- Therefore, int(A ∪ B) is a subset of int(A).
2. Inclusion of int(A) ∪ int(B) in int(A ∪ B):
- Let x be an element of int(A) ∪ int(B). This means that x is either in int(A) or int(B).
- If x is in int(A), there exists an open ball around x contained entirely within A. Since A is a subset of A ∪ B, this open ball is also contained in A ∪ B.
- Similarly, if x is in int(B), there exists an open ball around x contained entirely within B, and hence in A ∪ B.
- Therefore, x is in int(A ∪ B), implying int(A) ∪ int(B) is a subset of int(A ∪ B).
Conclusion:
- Combining both inclusions, we have int(A ∪ B) = int(A) ∪ int(B).
The interior of a set represents all the points inside the set that do not touch the boundary. In the case of the union of two sets, the interior will include points that are in the interior of either set.
Verification:
Claim:
int(A ∪ B) = int(A) ∪ int(B)
Proof:
1. Inclusion of int(A ∪ B) in int(A) ∪ int(B):
- Let x be an element of int(A ∪ B). This means that there exists an open ball around x contained entirely within A ∪ B.
- Since x is in A ∪ B, it must be in either A or B. Without loss of generality, assume x is in A.
- Then, the open ball around x is entirely contained within A, making x an interior point of A. Thus, x is in int(A).
- Therefore, int(A ∪ B) is a subset of int(A).
2. Inclusion of int(A) ∪ int(B) in int(A ∪ B):
- Let x be an element of int(A) ∪ int(B). This means that x is either in int(A) or int(B).
- If x is in int(A), there exists an open ball around x contained entirely within A. Since A is a subset of A ∪ B, this open ball is also contained in A ∪ B.
- Similarly, if x is in int(B), there exists an open ball around x contained entirely within B, and hence in A ∪ B.
- Therefore, x is in int(A ∪ B), implying int(A) ∪ int(B) is a subset of int(A ∪ B).
Conclusion:
- Combining both inclusions, we have int(A ∪ B) = int(A) ∪ int(B).
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Top Courses for UPSCView all
Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)?
Question Description
Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)? 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 int(A union B)=int(A) union int(B)? 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 int(A union B)=int(A) union int(B)?.
Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)? 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 int(A union B)=int(A) union int(B)? 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 int(A union B)=int(A) union int(B)?.
Solutions for Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)? defined & explained in the simplest way possible. Besides giving the explanation of
Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)?, a detailed solution for Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)? has been provided alongside types of Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)? theory, EduRev gives you an
ample number of questions to practice Consider R² with the usual topology and A,B subset or equal R² then check int(A union B)=int(A) union int(B)? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup