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Quiz: Countable sets

Page history last edited by CAG 10 years, 4 months ago

This quiz is designed to test your knowledge of finite, countable, and uncountable sets.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. If Formula is a countable set, and Formula is an uncountable set, then the most we can say about Formula is that it is
    1. Empty.
    2. Finite.
    3. Countable.
    4. At most countable.
    5. Uncountable.
      • CORRECT.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  2. If Formula is a countable set, and Formula is a finite set, then the most we can say about Formula is that it is
    1. Empty.
    2. Finite.
    3. Countable.
      • CORRECT.
    4. At most countable.
    5. Uncountable.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  3. If Formula is an uncountable set, and Formula is a finite set, then the most we can say about Formula is that it is
    1. Empty.
    2. Finite.
    3. Countable.
    4. At most countable.
    5. Uncountable.
      • CORRECT.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  4. If Formula is a finite set, and Formula is a finite set, then the most we can say about Formula is that it is
    1. Empty.
    2. Finite.
      • CORRECT. Empty sets are considered finite.
    3. Countable.
    4. At most countable.
    5. Uncountable.
    6. Countable or Uncountable.
    7. Finite, countable, or uncountable.
  5. If Formula is a countable set, and Formula is an uncountable set, then the most we can say about Formula is that it is
    1. Empty.
    2. Finite.
    3. Countable.
      • INCORRECT. The intersection of Formula and Formula could be smaller than countable.
    4. At most countable.
      • CORRECT.
    5. Uncountable.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  6. If Formula is a finite set, and Formula is an uncountable set, then the most we can say about Formula is that it is
    1. Empty.
    2. Finite.
      • CORRECT.
    3. Countable.
    4. At most countable.
    5. Uncountable.
    6. Countable or Uncountable.
    7. Finite, countable, or uncountable.
  7. If Formula is a countable set, and Formula is an uncountable set, then the most we can say about the Cartesian product Formula is that it is
    1. Empty.
    2. Finite.
    3. Countable.
    4. At most countable.
    5. Uncountable.
      • CORRECT.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  8. If Formula is a nonempty finite set, and Formula is an uncountable set, then the most we can say about the Cartesian product Formula is that it is
    1. Empty.
    2. Finite.
    3. Countable.
    4. At most countable.
    5. Uncountable.
      • CORRECT.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  9. If Formula is a countable set, and Formula is a countable set, then the most we can say about the Cartesian product Formula is that it is
    1. Empty.
    2. Finite.
    3. Countable.
      • CORRECT.
    4. At most countable.
    5. Uncountable.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  10. If Formula is an uncountable set, and Formula is a countable set, then the most we can say about the set Formula (the elements of Formula which are not in Formula) is that it is
    1. Empty.
    2. Finite.
    3. Countable.
    4. At most countable.
    5. Uncountable.
      • CORRECT.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
  11. If Formula is a countable set, and Formula is an uncountable set, then the most we can say about the set Formula (the elements of Formula which are not in Formula) is that it is
    1. Empty.
    2. Finite.
    3. Countable.
    4. At most countable.
      • CORRECT.
    5. Uncountable.
    6. Countable or uncountable.
    7. Finite, countable, or uncountable.
      • INCORRECT. Note that we did not say at any stage that Formula had to be a subset of Formula.
  12. If Formula is a countable set, and Formula is a countable set, then the most we can say about the set Formula (the elements of Formula which are not in Formula) is that it is
    1. Empty.
    2. Finite.
    3. Countable.
    4. At most countable.
      • CORRECT.
    5. Uncountable.
    6. Countable or Uncountable.
    7. Finite, countable, or uncountable.
  13. Let Formula be a set. What does it mean for Formula to be finite?
    1. There exists a natural number Formula and a bijection Formula from Formula to Formula.
      • CORRECT.
    2. There exists a natural number Formula and a bijection Formula from Formula to Formula.
      • PARTIALLY. This does not quite work when Formula is empty.
    3. Every element of Formula is finite.
    4. Formula is a proper subset of the natural numbers.
    5. There is a bijection from Formula to a proper subset of the natural numbers.
    6. Every element of Formula is bounded.
    7. Formula is not countable.
  14. Let Formula and Formula be sets. What does it mean for Formula and Formula to have the same cardinality?
    1. Formula is not a subset of Formula, and Formula is not a subset of Formula.
    2. Every element of Formula is an element of Formula, and vice versa.
    3. There is a function Formula which is both one-to-one and onto.
      • CORRECT.
    4. Formula and Formula are both finite, or both countable, or both uncountable.
    5. There is a function Formula which is one-to-one.
    6. There is a function Formula which is onto.
    7. Formula and Formula are both finite, or both infinite.
  15. Let Formula be a set. What does it mean for Formula to be countable?
    1. Formula is not finite or empty.
      • INCORRECT. This does not exclude the possibility that Formula is uncountable.
    2. Formula is a subset of the natural numbers.
    3. Formula is of the form Formula for some sequence Formula
      • PARTIALLY. Such a set might be finite, if the sequence has enough repeats.
    4. There is a way to assign a natural number to every element of Formula, such that each natural number is assigned to exactly one element of Formula.
      • CORRECT.
    5. One can assign a different element of Formula to each natural number in N.
    6. Each element of Formula is countable.
    7. One can assign a different natural number to each element of Formula.
      • INCORRECT. This does not exclude the possibility that Formula is finite.
  16. Let Formula be a set. What does it mean for Formula to be uncountable?
    1. Formula is not countable.
      • PARTIALLY. This does not exclude the possibility that Formula is finite.
    2. There is a bijection Formula from Formula to the real numbers Formula.
      • PARTIALLY. The real numbers are one type of uncountable set, but it turns out there are other uncountable sets of different cardinality than Formula.
    3. There is no way to assign a distinct natural number to each element of Formula.
      • CORRECT.
    4. There is no way to assign a distinct element of Formula to each natural number.
      • INCORRECT. This is only true when Formula is finite!
    5. There exist elements of Formula which cannot be assigned to any natural number at all.
    6. Formula contains irrational numbers.
    7. There is no bijection f from the natural numbers to Formula.
      • PARTIALLY. This does not exclude the possibility that Formula is finite.
  17. Let Formula be a set, and let Formula be a proper subset of Formula (so that Formula is not equal to Formula). Is it possible for Formula to have the same cardinality as Formula?
    1. Yes, but only when Formula is infinite.
      • CORRECT.
    2. Yes, but only when Formula is countable.
    3. Yes, but only when Formula is uncountable.
    4. No, unless Formula is empty.
    5. No, unless Formula is finite.
    6. No, it is not possible for any Formula.
    7. Yes, it is possible for any Formula.

 

 

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