Mathematics Exam > Mathematics Questions > Let S = {x ∈R : x6 – x5 ≤10...
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Let S = {x ∈ R : x6 – x5 ≤ 100} and T = {x2 – 2x : x ∈ (0, ∞)}. Then set S ∩ T is
- a)closed and bounded in R
- b)closed but NOT bounded in R
- c)bounded but NOT closed in R
- d)neither closed nor bounded in R
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Let S = {x ∈R : x6 – x5 ≤100} and T = {x2 – 2x...
To determine whether the set S ∩ T is closed and bounded in R, we need to analyze the properties of each set individually.
1. Set S:
The set S is defined as {x ∈ R : x^6 - x^5 ≤ 100}. Let's analyze this set.
- Boundedness: To check whether S is bounded, we need to determine if there exists a positive real number M such that |x| ≤ M for all x ∈ S.
- As x^6 - x^5 ≤ 100, we can rewrite this as x^6 - x^5 - 100 ≤ 0.
- The function f(x) = x^6 - x^5 - 100 is a polynomial function, and all polynomial functions are continuous.
- As f(x) is continuous, it means that the set of all x for which f(x) ≤ 0 is a closed set.
- Since the set of all x for which f(x) ≤ 0 is a closed set, it is also bounded.
- Therefore, set S is bounded.
2. Set T:
The set T is defined as {x^2 - 2x : x > 0}. Let's analyze this set.
- Boundedness: To check whether T is bounded, we need to determine if there exists a positive real number M such that |x^2 - 2x| ≤ M for all x ∈ T.
- The function f(x) = x^2 - 2x is a quadratic function.
- As f(x) is a continuous function, it means that the set of all x for which f(x) ≤ M is a closed set.
- Since the set of all x for which f(x) ≤ M is a closed set, it is also bounded.
- Therefore, set T is bounded.
3. Set S ∩ T:
Set S ∩ T is the intersection of sets S and T. Since both S and T are bounded sets, their intersection S ∩ T is also bounded.
- Closedness: To check whether S ∩ T is closed, we need to determine if its complement in R, denoted by (S ∩ T)', is open.
- The complement of S ∩ T is (S ∩ T)' = (S') ∪ (T'), where S' is the complement of S and T' is the complement of T.
- The complement of a closed set is an open set. Since both S and T are closed sets, their complements S' and T' are open sets.
- The union of two open sets is also an open set.
- Therefore, (S ∩ T)' = (S') ∪ (T') is an open set, implying that S ∩ T is closed.
Based on the above analysis, we can conclude that set S ∩ T is both closed and bounded in R. Therefore, the correct answer is option 'A'.
1. Set S:
The set S is defined as {x ∈ R : x^6 - x^5 ≤ 100}. Let's analyze this set.
- Boundedness: To check whether S is bounded, we need to determine if there exists a positive real number M such that |x| ≤ M for all x ∈ S.
- As x^6 - x^5 ≤ 100, we can rewrite this as x^6 - x^5 - 100 ≤ 0.
- The function f(x) = x^6 - x^5 - 100 is a polynomial function, and all polynomial functions are continuous.
- As f(x) is continuous, it means that the set of all x for which f(x) ≤ 0 is a closed set.
- Since the set of all x for which f(x) ≤ 0 is a closed set, it is also bounded.
- Therefore, set S is bounded.
2. Set T:
The set T is defined as {x^2 - 2x : x > 0}. Let's analyze this set.
- Boundedness: To check whether T is bounded, we need to determine if there exists a positive real number M such that |x^2 - 2x| ≤ M for all x ∈ T.
- The function f(x) = x^2 - 2x is a quadratic function.
- As f(x) is a continuous function, it means that the set of all x for which f(x) ≤ M is a closed set.
- Since the set of all x for which f(x) ≤ M is a closed set, it is also bounded.
- Therefore, set T is bounded.
3. Set S ∩ T:
Set S ∩ T is the intersection of sets S and T. Since both S and T are bounded sets, their intersection S ∩ T is also bounded.
- Closedness: To check whether S ∩ T is closed, we need to determine if its complement in R, denoted by (S ∩ T)', is open.
- The complement of S ∩ T is (S ∩ T)' = (S') ∪ (T'), where S' is the complement of S and T' is the complement of T.
- The complement of a closed set is an open set. Since both S and T are closed sets, their complements S' and T' are open sets.
- The union of two open sets is also an open set.
- Therefore, (S ∩ T)' = (S') ∪ (T') is an open set, implying that S ∩ T is closed.
Based on the above analysis, we can conclude that set S ∩ T is both closed and bounded in R. Therefore, the correct answer is option 'A'.
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Let S = {x ∈R : x6 – x5 ≤100} and T = {x2 – 2x...
S ∩ T ={0}
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Let S = {x ∈R : x6 – x5 ≤100} and T = {x2 – 2x : x ∈ (0, ∞)}. Then set S ∩ T isa)closed and bounded in Rb)closed but NOT bounded in Rc)bounded but NOT closed in Rd)neither closed nor bounded in RCorrect answer is option 'A'. Can you explain this answer?
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