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Quiz: Sets

Page history last edited by WJ 7 years, 3 months ago

This quiz is designed to test your knowledge of basic concepts in sets.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula and Formula be sets. What does it mean if we say that Formula is an element of Formula?
    1. Formula is an element of Formula and is also an element of Formula.
      • INCORRECT. This is what it means for Formula to lie in Formula.
    2. Formula is an element of Formula.
      • INCORRECT. It is true that if Formula lies in Formula, then Formula also lies in Formula; but it is possible to lie in Formula without lying in Formula.
    3. Formula is an element of Formula.
      • INCORRECT. It is true that if Formula lies in Formula, then Formula also lies in Formula; but it is possible to lie in Formula without lying in Formula.
    4. Formula is an element of Formula, or is an element of Formula, or both.
      • CORRECT.
    5. Formula is equal to Formula, and Formula is an element of both.
      • INCORRECT. One does not need the two sets Formula and Formula to be equal in order to form the union Formula.
    6. Formula is an element of Formula, or is an element of Formula, but not both.
      • INCORRECT. If Formula lies in both Formula and Formula then it still qualifies to lie in Formula.
    7. Formula is equal to some element a of Formula plus some other element Formula of Formula: Formula.
      • INCORRECT. This is what it means for Formula to lie in Formula.
  2. Let Formula and Formula be sets. What does it mean if we say that Formula is NOT an element of Formula?
    1. Formula is not an element of Formula, and Formula is not an element of Formula.
      • CORRECT.
    2. Either Formula is not an element of Formula, or Formula is not an element of Formula.
      • INCORRECT. This is what it means for Formula to not be an element of Formula.
    3. Formula does not belong to both Formula and Formula at the same time.
      • INCORRECT. This is what it means for Formula to not be an element of Formula.
    4. There is some element of Formula which is not equal to Formula, and there is some element of Formula which is not equal to Formula.
    5. There is some element of either Formula or Formula which is not equal to Formula.
    6. Formula and Formula are the same sets, and Formula is not an element of either.
  3. Let Formula and Formula be sets. What does it mean if we say that Formula is an element of Formula?
    1. Formula is an element of Formula and Formula is also an element of Formula.
      • CORRECT.
    2. Formula is an element of Formula.
    3. Formula is an element of Formula.
    4. Formula is an element of Formula, or is an element of Formula, or both.
      • INCORRECT. This is what it means for Formula to lie in Formula.
    5. Formula is equal to Formula, and Formula is an element of both.
      • INCORRECT. One does not need the two sets Formula and Formula to be equal in order to form the intersection Formula.
    6. Formula is an element of Formula, or is an element of Formula, but not both.
      • INCORRECT. This is what it means for Formula to lie in Formula.
    7. Formula is equal to some element a of Formula plus some other element Formula of Formula: Formula.
      • INCORRECT. This is what it means for Formula to lie in Formula.
  4. Let Formula and Formula be sets. What does it mean if we say that Formula is NOT an element of Formula?
    1. Formula cannot belong to both Formula and B; it may belong to Formula, or to Formula, or to neither, but not both.
      • CORRECT.
    2. Formula is not an element of Formula, and Formula is not an element of Formula.
      • INCORRECT. This does not cover the possibility that Formula is an element of exactly one of Formula or Formula
    3. Every element of Formula and every element of Formula is different from Formula.
      • INCORRECT. This is what it means for Formula to not be an element of Formula.
    4. There is some element of Formula which is not equal to Formula, and there is some element of Formula which is not equal to Formula.
    5. Formula belongs to exactly one of Formula and Formula.
      • INCORRECT. This does not cover the possibility that Formula belongs to neither Formula nor Formula.
    6. Formula belongs to neither Formula nor Formula.
      • INCORRECT. This is what it means for Formula to not be an element of Formula.
    7. Formula and Formula are the same sets, and Formula is not an element of either.
  5. Let Formula and Formula be sets. What does it mean if we say that Formula is a subset of B?
    1. Every element Formula in Formula is also an element of Formula.
      • CORRECT.
    2. Every element Formula in Formula is also an element of Formula.
      • INCORRECT. This is what it means for Formula to be a subset of Formula.
    3. Every element Formula in Formula is equal to every element Formula in Formula.
    4. Some element Formula in Formula is also an element of Formula.
      • INCORRECT. This is what it means for Formula and Formula to have a non-empty intersection.
    5. Every element Formula in Formula is contained in some element Formula of Formula.
      • INCORRECT. We want the elements in Formula to be equal to elements in Formula, not _contained_ in them.
    6. Every element Formula of Formula is equal to some element Formula of Formula.
      • INCORRECT. This is what it means for Formula to be a subset of Formula.
    7. Formula is an element of Formula.
      • INCORRECT. We want the elements of Formula to be elements of B; we don't what Formula itself to be an element of Formula.
  6. Let Formula and Formula be sets. What does it mean if we say that Formula is not a subset of B?
    1. Formula is a subset of Formula.
      • INCORRECT. It is possible for Formula and Formula to not be subsets of each other.
    2. Formula is equal to Formula.
    3. Formula and Formula are disjoint.
      • INCORRECT. It is possible for Formula and Formula to partially intersect without being subsets of each other.
    4. There is an element Formula of Formula which does not lie in Formula.
      • CORRECT.
    5. Every element Formula of Formula does not lie in Formula.
      • INCORRECT. It is possible for Formula and Formula to still have common elements without Formula being a subset of Formula.
    6. There is an element Formula of Formula which does not lie in Formula.
      • INCORRECT. This is what it means for Formula to not be a subset of Formula.
    7. Formula is not an element of Formula.
  7. Let Formula and Formula be sets. What does it mean if we say that Formula is equal to Formula?
    1. Every element Formula in Formula is also an element of Formula, and every element Formula in Formula is also an element of Formula.
      • CORRECT.
    2. Every element Formula in Formula is equal to some element of Formula.
      • INCORRECT. This only shows that Formula is a subset of Formula.
    3. Every element Formula in Formula is equal to every element Formula in Formula.
      • INCORRECT. This only shows that Formula is a subset of Formula.
    4. Some element Formula in Formula is equal to some element of Formula.
      • INCORRECT. This only shows that Formula and Formula have some non-empty intersection.
    5. Formula is not contained in Formula, and Formula is not contained in Formula.
      • INCORRECT. If Formula and Formula are equal, then they are automatically contained in each other.
    6. Formula is not strictly contained in Formula, and Formula is not strictly contained in Formula.
      • INCORRECT. It is possible for Formula and Formula to be unequal, and to not be strictly contained in each other.
    7. Every element in Formula is equal to every element in Formula.
      • INCORRECT. This can only be true if Formula and Formula have at most one element.
  8. Let Formula and Formula be sets. What does it mean if we say that Formula and Formula are disjoint?
    1. There does not exist any element Formula which belongs to both Formula and Formula.
      • CORRECT.
    2. Formula is not a subset of Formula, and Formula is not a subset of Formula.
    3. Formula is not equal to Formula.
    4. The union of Formula and Formula is empty.
    5. There exists an element Formula of Formula and an element Formula of Formula such that Formula is not equal to Formula.
    6. There is an element Formula of Formula which is not in Formula, and there is an element Formula of Formula which is not in Formula.
  9. Let Formula and Formula be sets. What does it mean if we say that Formula and Formula are not disjoint?
    1. There exists an element Formula which belongs to both Formula and Formula.
      • CORRECT.
    2. Either Formula is a subset of Formula, or Formula is a subset of Formula.
    3. Formula is a subset of Formula, and Formula is a subset of Formula.
    4. Formula is equal to Formula.
    5. Every element of Formula is equal to every element of Formula.
    6. Every element of Formula is equal to some element of Formula, and vice versa.
    7. The union of Formula and Formula is non-empty.
  10. Let Formula and Formula be sets. What does it mean if we say that Formula is not equal to Formula?
    1. Either there is some element Formula of Formula which is not in Formula, or there is some element Formula in Formula which is not in Formula, or both.
      • CORRECT.
    2. There is some element Formula of Formula which is not in Formula, and there is some element Formula in Formula which is not in Formula.
    3. Either Formula is a subset of Formula, or Formula is a subset of Formula.
    4. Either Formula is a proper subset of Formula, or Formula is a proper subset of Formula.
    5. For every Formula in Formula and every Formula, Formula is not equal to Formula.
    6. For every Formula in Formula there is some Formula such that Formula is not equal to Formula.
    7. There is some Formula in Formula and some Formula such that Formula is not equal to Formula.
  11. Let Formula be the set Formula. What does it mean if we say that Formula is an element of Formula? (It turns out that Formula is in fact the half-open interval Formula (why?). But you did not need to know that to work out this problem.)
    1. Formula is equal to Formula for some Formula.
      • CORRECT.
    2. Formula is equal to Formula for every Formula.
    3. Formula is between Formula and Formula.
    4. Formula is between Formula and Formula.
    5. Formula is between Formula and Formula.
    6. Formula is an element of Formula.
  12. Let Formula be the set Formula. What does it mean if we say that Formula is not an element of Formula?
    1. There exists Formula such that Formula is not equal to Formula.
    2. For every Formula, Formula is not equal to Formula.
      • CORRECT.
    3. Formula is not between Formula and Formula.
    4. Formula is not between Formula and Formula.
    5. Formula is not between Formula and Formula.
    6. Formula is not an element of Formula.

 

Score:  

 

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