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Which of the following statement(s) is(are) TRUE?
- a)There exists a connected set in R which is not compact
- b)Arbitrary union of closed intervals in R need not be compact
- c)Arbitrary union of closed intervals in R is always closed
- d)Every bounded infinite subset R of R has a limit point in V itself
Correct answer is option 'A,B'. Can you explain this answer?
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Statement A: There exists a connected set in R which is not compact
To prove this statement, we can consider the set of all real numbers, R. R is a connected set because it contains all the real numbers and there are no gaps or breaks in between. However, R is not compact because it is unbounded. We can always find an open cover for R that does not have a finite subcover. For example, we can consider the open cover consisting of the open intervals (-n, n) for all positive integers n. This cover covers the entire set R, but no finite number of intervals can cover R completely. Therefore, statement A is true.
Statement B: Arbitrary union of closed intervals in R need not be compact
To prove this statement, we can consider the union of closed intervals [n, n+1] for all positive integers n. Each individual closed interval [n, n+1] is compact because it is a closed and bounded set in R. However, the union of these intervals is not compact. We can construct an open cover for this union by considering open intervals (n-1, n+2) for all positive integers n. This cover covers the entire union, but no finite number of intervals can cover the union completely. Therefore, statement B is true.
Statement C: Arbitrary union of closed intervals in R is always closed
This statement is false. The arbitrary union of closed intervals in R may not always be closed. For example, consider the union of closed intervals [0, 1/n] for all positive integers n. Each individual closed interval [0, 1/n] is closed, but the union of these intervals is the half-open interval [0, 1) which is not closed. Therefore, statement C is false.
Statement D: Every bounded infinite subset R of R has a limit point in itself
This statement is true. By the Bolzano-Weierstrass theorem, every bounded sequence in R has a convergent subsequence. Therefore, every bounded infinite subset R of R has at least one limit point in itself. Therefore, statement D is true.
To prove this statement, we can consider the set of all real numbers, R. R is a connected set because it contains all the real numbers and there are no gaps or breaks in between. However, R is not compact because it is unbounded. We can always find an open cover for R that does not have a finite subcover. For example, we can consider the open cover consisting of the open intervals (-n, n) for all positive integers n. This cover covers the entire set R, but no finite number of intervals can cover R completely. Therefore, statement A is true.
Statement B: Arbitrary union of closed intervals in R need not be compact
To prove this statement, we can consider the union of closed intervals [n, n+1] for all positive integers n. Each individual closed interval [n, n+1] is compact because it is a closed and bounded set in R. However, the union of these intervals is not compact. We can construct an open cover for this union by considering open intervals (n-1, n+2) for all positive integers n. This cover covers the entire union, but no finite number of intervals can cover the union completely. Therefore, statement B is true.
Statement C: Arbitrary union of closed intervals in R is always closed
This statement is false. The arbitrary union of closed intervals in R may not always be closed. For example, consider the union of closed intervals [0, 1/n] for all positive integers n. Each individual closed interval [0, 1/n] is closed, but the union of these intervals is the half-open interval [0, 1) which is not closed. Therefore, statement C is false.
Statement D: Every bounded infinite subset R of R has a limit point in itself
This statement is true. By the Bolzano-Weierstrass theorem, every bounded sequence in R has a convergent subsequence. Therefore, every bounded infinite subset R of R has at least one limit point in itself. Therefore, statement D is true.
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Which of the following statement(s) is(are) TRUE? a)There exists a connected set in R which is not compactb)Arbitrary union of closed intervals in R need not be compactc)Arbitrary union of closed intervals in R is always closedd)Every bounded infinite subset R of R has a limit point in VitselfCorrect answer is option 'A,B'. Can you explain this answer?
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Which of the following statement(s) is(are) TRUE? a)There exists a connected set in R which is not compactb)Arbitrary union of closed intervals in R need not be compactc)Arbitrary union of closed intervals in R is always closedd)Every bounded infinite subset R of R has a limit point in VitselfCorrect answer is option 'A,B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which of the following statement(s) is(are) TRUE? a)There exists a connected set in R which is not compactb)Arbitrary union of closed intervals in R need not be compactc)Arbitrary union of closed intervals in R is always closedd)Every bounded infinite subset R of R has a limit point in VitselfCorrect answer is option 'A,B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statement(s) is(are) TRUE? a)There exists a connected set in R which is not compactb)Arbitrary union of closed intervals in R need not be compactc)Arbitrary union of closed intervals in R is always closedd)Every bounded infinite subset R of R has a limit point in VitselfCorrect answer is option 'A,B'. Can you explain this answer?.
Which of the following statement(s) is(are) TRUE? a)There exists a connected set in R which is not compactb)Arbitrary union of closed intervals in R need not be compactc)Arbitrary union of closed intervals in R is always closedd)Every bounded infinite subset R of R has a limit point in VitselfCorrect answer is option 'A,B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which of the following statement(s) is(are) TRUE? a)There exists a connected set in R which is not compactb)Arbitrary union of closed intervals in R need not be compactc)Arbitrary union of closed intervals in R is always closedd)Every bounded infinite subset R of R has a limit point in VitselfCorrect answer is option 'A,B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statement(s) is(are) TRUE? a)There exists a connected set in R which is not compactb)Arbitrary union of closed intervals in R need not be compactc)Arbitrary union of closed intervals in R is always closedd)Every bounded infinite subset R of R has a limit point in VitselfCorrect answer is option 'A,B'. Can you explain this answer?.
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