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# Quiz: Infimum and supremum

last edited by 11 years, 9 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.)

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

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