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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