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

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

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

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