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Quiz: Infimum and supremum

Page history last edited by RH 11 years, 7 months ago

This quiz is designed to test your knowledge of order, and specifically on infimum and supremum.

 

All sets in this quiz are subsets of the real line Formula.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula and Formula be bounded non-empty sets. Which of the following statements would be equivalent to saying that Formula?
    1. For every Formula and Formula there exists an Formula such that Formula.
      • CORRECT.
    2. For every Formula there exists an Formula such that Formula.
      • PARTIALLY. This will imply that Formula but is not necessary (can you think of a counterexample?).
    3. For every Formula there exists a Formula such that Formula.
    4. There exists Formula and Formula such that Formula.
    5. For every Formula and every Formula, we have Formula.
    6. There exists Formula such that Formula for all Formula.
      • PARTIALLY. This will imply that Formula but is not necessary (can you think of a counterexample?).
    7. There exists Formula such that Formula for all Formula.
  2. Let Formula and Formula be bounded non-empty sets such that Formula. Which of the following statements must be true?
    1. There exists Formula and Formula such that Formula.
      • PARTIALLY. Close, but not quite. Can you think of a counterexample?
    2. For every Formula there exists Formula and Formula such that Formula.
      • CORRECT.
    3. For every Formula and every Formula, we have Formula.
    4. For every Formula there exists a Formula such that Formula.
    5. For every Formula there exists an Formula such that Formula.
    6. There exists Formula such that Formula for all Formula.
    7. There exists Formula such that Formula for all Formula.
  3. Let Formula and Formula be non-empty sets. Which of the following statements would be equivalent to saying that Formula?
    1. There exists Formula and Formula such that Formula.
    2. For every Formula and every Formula, we have Formula.
      • CORRECT.
    3. For every Formula there exists a Formula such that Formula.
    4. For every Formula there exists an Formula such that Formula.
    5. There exists Formula such that Formula for all Formula.
    6. There exists Formula such that Formula for all Formula.
    7. For every epsilon there exists Formula and Formula such that Formula.
  4. Let Formula and Formula be non-empty sets. Which of the following statements would be equivalent to saying that Formula?
    1. For every Formula and epsilon > 0 there exists a Formula such that Formula.
      • CORRECT.
    2. For every Formula and every Formula, we have Formula.
    3. For every Formula there exists a Formula such that Formula.
      • PARTIALLY. This implies that Formula but is not necessary (can you think of a counterexample)?
    4. For every Formula there exists an Formula such that Formula.
    5. There exists Formula such that Formula for all Formula.
    6. There exists Formula such that Formula for all Formula.
    7. For every epsilon there exists Formula and Formula such that Formula.
  5. Let Formula and Formula be non-empty sets. The best we can say about Formula is that
    1. It is the maximum of Formula and Formula.
      • CORRECT.
    2. It is the minimum of Formula and Formula.
    3. It is strictly greater than Formula and strictly greater than Formula.
    4. It is greater than or equal to Formula and greater than or equal to Formula.
    5. It is less than or equal to Formula and less than or equal to Formula.
    6. It is strictly greater than at least one of Formula and Formula.
    7. It is equal to at least one of Formula and Formula.
  6. Let Formula and Formula be non-empty sets. The best we can say about Formula is that
    1. It is less than or equal to Formula, and less than or equal to Formula.
      • CORRECT.
    2. It is the maximum of Formula and Formula.
    3. It is the minimum of Formula and Formula.
    4. It is equal to at least one of Formula and Formula.
    5. It is strictly less than Formula and strictly less than Formula.
    6. It is greater than or equal to Formula and greater than or equal to Formula.
    7. It is greater than or equal to at least one of Formula and Formula.
  7. Let Formula be a non-empty set. If Formula, this means that
    1. For every real number Formula, there exists an Formula such that Formula.
      • CORRECT.
    2. There exists Formula and a real number Formula such that Formula.
    3. There exists an Formula such that Formula for every real number Formula.
    4. There exists a real number Formula such that Formula for every Formula.
    5. Formula is the empty set.
    6. For every real number Formula and every Formula, we have Formula.
    7. There exists an Formula such that Formula for every real number Formula.
  8. Let Formula be a set. If Formula, this means that
    1. For every real number Formula, there exists an Formula such that Formula.
    2. There exists Formula and a real number Formula such that Formula.
    3. There exists an Formula such that Formula for every real number Formula.
    4. There exists a real number Formula such that Formula for every Formula.
    5. Formula is the empty set.
      • CORRECT.
    6. For every real number Formula and every Formula, we have Formula.
      • CORRECT. This answer is technically correct, but there is a simpler way to state it.
    7. There exists an Formula such that Formula for every real number Formula.
  9. Let Formula be a non-empty set. If Formula is not bounded, this means that
    1. For every real number Formula, there exists an Formula such that Formula.
      • CORRECT.
    2. For every positive number Formula, there exists an Formula such that Formula, and for every negative number Formula, there exists Formula such that Formula.
    3. Formula and Formula.
    4. Formula and Formula.
    5. For every real number Formula, there exists an Formula such that Formula.
    6. There exists an Formula such that Formula for every real number Formula.
    7. There exists a real number Formula such that Formula for every Formula.
  10. Let Formula and Formula be bounded non-empty sets. Which of the following statements would be equivalent to saying that Formula?
    1. For every Formula and every Formula, we have Formula.
    2. For every Formula and every Formula, we have Formula. Also, for every Formula, there exists Formula and Formula such that Formula.
      • CORRECT.
    3. For every Formula, there exists Formula and Formula such that Formula.
    4. There exists a real number Formula such that Formula for all Formula and Formula.
    5. There exists Formula and Formula such that Formula. However, for any Formula, there does not exist Formula and Formula for which Formula.
    6. For every Formula there exists a Formula such that Formula.
    7. For every Formula there exists a Formula such that Formula. Also, for every Formula there exists an Formula such that Formula.
  11. Let Formula be a set, and let Formula be a real number. If Formula, this means that
    1. Formula for every Formula. Also, for every Formula, there exists an Formula such that Formula.
      • CORRECT.
    2. Formula for every Formula.
    3. There exists an Formula such that every real number a between Formula and Formula lies in Formula.
    4. Every number less than Formula lies in Formula, and every number greater than Formula does not lie in Formula.
    5. Formula lies in Formula, and Formula is larger than every other element of Formula.
      • INCORRECT. This implies that Formula, but is not necessary.
    6. There exists a sequence Formula of elements in Formula which converges to Formula.
    7. There exists a sequence Formula of elements in Formula which are less than Formula, but converges to Formula.
  12. Let Formula be a set, and let Formula be a real number. If Formula, this means that
    1. Formula for every Formula. Also, for every Formula, there exists an Formula such that Formula.
    2. Formula for every Formula.
      • CORRECT.
    3. There exists an Formula such that every element of Formula is less than Formula.
    4. For every Formula and every Formula, we have Formula.
    5. For every Formula there exists an Formula such that Formula.
    6. There exists an Formula such that Formula for every Formula.
    7. Every number less than or equal to Formula lies in Formula.
  13. Let Formula be a set, and let Formula be a real number. If Formula, this means that
    1. Formula for every Formula. Also, for every Formula, there exists an Formula such that Formula.
    2. Formula for every Formula.
      • INCORRECT. Close, but this is not quite sufficient to imply Formula (can you think of a counterexample?)
    3. There exists an Formula such that every element of Formula is less than Formula.
      • CORRECT.
    4. For every Formula and every Formula, we have Formula.
    5. For every Formula there exists an Formula such that Formula.
    6. There exists an Formula such that Formula for every Formula.
    7. Every number less than or equal to Formula lies in Formula.

 

 

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