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Quiz: Infimum and supremum
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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 .
Discuss this quiz
(Key: correct, incorrect, partially correct.)
 Let and be bounded nonempty sets. Which of the following statements would be equivalent to saying that ?
 For every and there exists an such that .
 For every there exists an such that .
 PARTIALLY. This will imply that but is not necessary (can you think of a counterexample?).
 For every there exists a such that .
 There exists and such that .
 For every and every , we have .
 There exists such that for all .
 PARTIALLY. This will imply that but is not necessary (can you think of a counterexample?).
 There exists such that for all .
 Let and be bounded nonempty sets such that . Which of the following statements must be true?
 There exists and such that .
 PARTIALLY. Close, but not quite. Can you think of a counterexample?
 For every there exists and such that .
 For every and every , we have .
 For every there exists a such that .
 For every there exists an such that .
 There exists such that for all .
 There exists such that for all .
 Let and be nonempty sets. Which of the following statements would be equivalent to saying that ?
 There exists and such that .
 For every and every , we have .
 For every there exists a such that .
 For every there exists an such that .
 There exists such that for all .
 There exists such that for all .
 For every epsilon there exists and such that .
 Let and be nonempty sets. Which of the following statements would be equivalent to saying that ?
 For every and epsilon > 0 there exists a such that .
 For every and every , we have .
 For every there exists a such that .
 PARTIALLY. This implies that but is not necessary (can you think of a counterexample)?
 For every there exists an such that .
 There exists such that for all .
 There exists such that for all .
 For every epsilon there exists and such that .
 Let and be nonempty sets. The best we can say about is that
 It is the maximum of and .
 It is the minimum of and .
 It is strictly greater than and strictly greater than .
 It is greater than or equal to and greater than or equal to .
 It is less than or equal to and less than or equal to .
 It is strictly greater than at least one of and .
 It is equal to at least one of and .
 Let and be nonempty sets. The best we can say about is that
 It is less than or equal to , and less than or equal to .
 It is the maximum of and .
 It is the minimum of and .
 It is equal to at least one of and .
 It is strictly less than and strictly less than .
 It is greater than or equal to and greater than or equal to .
 It is greater than or equal to at least one of and .
 Let be a nonempty set. If , this means that
 For every real number , there exists an such that .
 There exists and a real number such that .
 There exists an such that for every real number .
 There exists a real number such that for every .
 is the empty set.
 For every real number and every , we have .
 There exists an such that for every real number .
 Let be a set. If , this means that
 For every real number , there exists an such that .
 There exists and a real number such that .
 There exists an such that for every real number .
 There exists a real number such that for every .
 is the empty set.
 For every real number and every , we have .
 CORRECT. This answer is technically correct, but there is a simpler way to state it.
 There exists an such that for every real number .
 Let be a nonempty set. If is not bounded, this means that
 For every real number , there exists an such that .
 For every positive number , there exists an such that , and for every negative number , there exists such that .
 and .
 and .
 For every real number , there exists an such that .
 There exists an such that for every real number .
 There exists a real number such that for every .
 Let and be bounded nonempty sets. Which of the following statements would be equivalent to saying that ?
 For every and every , we have .
 For every and every , we have . Also, for every , there exists and such that .
 For every , there exists and such that .
 There exists a real number such that for all and .
 There exists and such that . However, for any , there does not exist and for which .
 For every there exists a such that .
 For every there exists a such that . Also, for every there exists an such that .
 Let be a set, and let be a real number. If , this means that
 for every . Also, for every , there exists an such that .
 for every .
 There exists an such that every real number a between and lies in .
 Every number less than lies in , and every number greater than does not lie in .
 lies in , and is larger than every other element of .
 INCORRECT. This implies that , but is not necessary.
 There exists a sequence of elements in which converges to .
 There exists a sequence of elements in which are less than , but converges to .
 Let be a set, and let be a real number. If , this means that
 for every . Also, for every , there exists an such that .
 for every .
 There exists an such that every element of is less than .
 For every and every , we have .
 For every there exists an such that .
 There exists an such that for every .
 Every number less than or equal to lies in .
 Let be a set, and let be a real number. If , this means that
 for every . Also, for every , there exists an such that .
 for every .
 INCORRECT. Close, but this is not quite sufficient to imply (can you think of a counterexample?)
 There exists an such that every element of is less than .
 For every and every , we have .
 For every there exists an such that .
 There exists an such that for every .
 Every number less than or equal to lies in .
Score:
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Quiz: Infimum and supremum

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