IIT JAM Exam > IIT JAM Questions > Let G and H be nonempty subsets of R , where ...
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Let G and H be nonempty subsets of R , where G is connected and G U H is not connected.
Which one of the following statements is true for all such G and H ?
Which one of the following statements is true for all such G and H ?
- a)If G ∩ H = Ø, then H is connected
- b)If G ∩ H = Ø, then H is not connected
- c)If G ∩ H ≠ Ø, then H is connected
- d)If G ∩ H ≠ Ø, then H is not connected
Correct answer is option 'D'. Can you explain this answer?
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Let G and H be nonempty subsets of R , where G is connected and G U H ...
To prove that the correct answer is option 'D', let's consider the implications of each statement and analyze them one by one.
Assumptions:
- G and H are nonempty subsets of R (the set of real numbers).
- G is connected.
- G U H is not connected.
Statement a) If G ∩ H = ∅, then H is connected.
- This statement suggests that if the intersection of G and H is empty, then H must be connected.
- However, this statement does not hold in general. Consider the following counterexample:
- Let G = [0, 1] and H = (1, 2).
- G and H are nonempty subsets of R.
- G is connected (since it is a closed interval).
- G U H = [0, 1] U (1, 2) = [0, 2], which is not connected.
- G ∩ H = {1}, which is not an empty set.
- Therefore, statement a) is not true in general.
Statement b) If G ∪ H = R, then H is not connected.
- This statement suggests that if the union of G and H is the entire set of real numbers, then H cannot be connected.
- This statement does not hold in general. Consider the following counterexample:
- Let G = [0, 1] and H = (2, 3).
- G and H are nonempty subsets of R.
- G is connected (since it is a closed interval).
- G ∪ H = [0, 1] ∪ (2, 3) = [0, 1] ∪ [2, 3] = [0, 1] ∪ [2, 3], which is connected.
- Therefore, statement b) is not true in general.
Statement c) If G ∩ H ≠ ∅, then H is connected.
- This statement suggests that if the intersection of G and H is not empty, then H must be connected.
- This statement does not hold in general. Consider the following counterexample:
- Let G = [0, 1] and H = (0, 1).
- G and H are nonempty subsets of R.
- G is connected (since it is a closed interval).
- G ∩ H = (0, 1), which is not an empty set.
- Therefore, statement c) is not true in general.
Statement d) If G ∩ H = ∅, then H is not connected.
- This statement suggests that if the intersection of G and H is empty, then H cannot be connected.
- This statement holds true in general. Consider the following proof by contradiction:
- Assume G ∩ H = ∅ and H is connected.
- Let A and B be two nonempty disjoint open sets in R such that H = A ∪ B.
- Since G is connected, it must be contained in either A or B (but not both).
- Without loss of generality, assume G ⊆ A.
- Since G U H is not connected, there must exist a point x in G and a point y in H such that there is no connected subset of R containing both x and y.
- However, since G ⊆ A
Assumptions:
- G and H are nonempty subsets of R (the set of real numbers).
- G is connected.
- G U H is not connected.
Statement a) If G ∩ H = ∅, then H is connected.
- This statement suggests that if the intersection of G and H is empty, then H must be connected.
- However, this statement does not hold in general. Consider the following counterexample:
- Let G = [0, 1] and H = (1, 2).
- G and H are nonempty subsets of R.
- G is connected (since it is a closed interval).
- G U H = [0, 1] U (1, 2) = [0, 2], which is not connected.
- G ∩ H = {1}, which is not an empty set.
- Therefore, statement a) is not true in general.
Statement b) If G ∪ H = R, then H is not connected.
- This statement suggests that if the union of G and H is the entire set of real numbers, then H cannot be connected.
- This statement does not hold in general. Consider the following counterexample:
- Let G = [0, 1] and H = (2, 3).
- G and H are nonempty subsets of R.
- G is connected (since it is a closed interval).
- G ∪ H = [0, 1] ∪ (2, 3) = [0, 1] ∪ [2, 3] = [0, 1] ∪ [2, 3], which is connected.
- Therefore, statement b) is not true in general.
Statement c) If G ∩ H ≠ ∅, then H is connected.
- This statement suggests that if the intersection of G and H is not empty, then H must be connected.
- This statement does not hold in general. Consider the following counterexample:
- Let G = [0, 1] and H = (0, 1).
- G and H are nonempty subsets of R.
- G is connected (since it is a closed interval).
- G ∩ H = (0, 1), which is not an empty set.
- Therefore, statement c) is not true in general.
Statement d) If G ∩ H = ∅, then H is not connected.
- This statement suggests that if the intersection of G and H is empty, then H cannot be connected.
- This statement holds true in general. Consider the following proof by contradiction:
- Assume G ∩ H = ∅ and H is connected.
- Let A and B be two nonempty disjoint open sets in R such that H = A ∪ B.
- Since G is connected, it must be contained in either A or B (but not both).
- Without loss of generality, assume G ⊆ A.
- Since G U H is not connected, there must exist a point x in G and a point y in H such that there is no connected subset of R containing both x and y.
- However, since G ⊆ A
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Let G and H be nonempty subsets of R , where G is connected and G U H is not connected.Which one of the following statements is true for all such G and H ?a)If G H = , then H is connectedb)If G H = , then H is not connectedc)If GH , then H is connectedd)If GH ,then H is not connectedCorrect answer is option 'D'. Can you explain this answer?
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Let G and H be nonempty subsets of R , where G is connected and G U H is not connected.Which one of the following statements is true for all such G and H ?a)If G H = , then H is connectedb)If G H = , then H is not connectedc)If GH , then H is connectedd)If GH ,then H is not connectedCorrect answer is option 'D'. 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 Let G and H be nonempty subsets of R , where G is connected and G U H is not connected.Which one of the following statements is true for all such G and H ?a)If G H = , then H is connectedb)If G H = , then H is not connectedc)If GH , then H is connectedd)If GH ,then H is not connectedCorrect answer is option 'D'. 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 Let G and H be nonempty subsets of R , where G is connected and G U H is not connected.Which one of the following statements is true for all such G and H ?a)If G H = , then H is connectedb)If G H = , then H is not connectedc)If GH , then H is connectedd)If GH ,then H is not connectedCorrect answer is option 'D'. Can you explain this answer?.
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