Comment. Countable sets quiz
Comment. This quiz is designed to test your knowledge of finite, countable, and uncountable sets.
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Don't Shuffle Answers.
Question. If A is a countable set, and B is an uncountable set, then the most we can say about (A union B) is that it is
Answer. Empty.
Answer. Finite.
Answer. Countable.
Answer. At most countable.
Correct Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a countable set, and B is a finite set, then the most we can say about (A union B) is that it is
Answer. Empty.
Answer. Finite.
Correct Answer. Countable.
Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is an uncountable set, and B is a finite set, then the most we can say about (A union B) is that it is
Answer. Empty.
Answer. Finite.
Answer. Countable.
Answer. At most countable.
Correct Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a finite set, and B is a finite set, then the most we can say about (A intersect B) is that it is
Answer. Empty.
Correct Answer. Finite.
Comment. Empty sets are considered finite.
Answer. Countable.
Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a countable set, and B is an uncountable set, then the most we can say about (A intersect B) is that it is
Answer. Empty.
Answer. Finite.
Answer. Countable.
Comment. The intersection of A and B could be smaller than countable.
Correct Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a finite set, and B is an uncountable set, then the most we can say about (A intersect B) is that it is
Answer. Empty.
Correct Answer. Finite.
Answer. Countable.
Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a countable set, and B is an uncountable set, then the most we can say about the Cartesian product (A x B) is that it is
Answer. Empty.
Answer. Finite.
Answer. Countable.
Answer. At most countable.
Correct Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a finite set, and B is an uncountable set, then the most we can say about the Cartesian product (A x B) is that it is
Answer. Empty.
Answer. Finite.
Answer. Countable.
Answer. At most countable.
Correct Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a countable set, and B is a countable set, then the most we can say about the Cartesian product A x B is that it is
Answer. Empty.
Answer. Finite.
Correct Answer. Countable.
Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is an uncountable set, and B is a countable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is
Answer. Empty.
Answer. Finite.
Answer. Countable.
Answer. At most countable.
Correct Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a countable set, and B is a finite set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is
Answer. Empty.
Answer. Finite.
Correct Answer. Countable.
Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a countable set, and B is an uncountable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is
Comment. Note that we did not say at any stage that B had to be a subset of A.
Answer. Empty.
Answer. Finite.
Answer. Countable.
Correct Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. If A is a countable set, and B is a countable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is
Answer. Empty.
Answer. Finite.
Answer. Countable.
Correct Answer. At most countable.
Answer. Uncountable.
Answer. Countable or Uncountable.
Answer. Finite, countable, or uncountable.
Question. Let A be a set. What does it mean for A to be finite?
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Correct Answer. There exists a natural number n and a bijection f from {i in N: i < n} to A.
Partially Correct Answer. There exists a natural number n and a bijection f from {i in N: i <= n} to A.
Comment. This does not quite work when A is empty.
Answer. Every element of A is finite.
Answer. A is a proper subset of the natural numbers.
Answer. There is a bijection from A to a proper subset of the natural numbers.
Answer. Every element of A is bounded.
Answer. A is not countable.
Question. Let A and B be sets. What does it mean for A and B to have the same cardinality?
Answer. A is not a subset of B, and B is not a subset of A.
Answer. Every element of A is an element of B, and vice versa.
Correct Answer. There is a function f: A -> B which is both one-to-one and onto.
Answer. A and B are both finite, or both countable, or both uncountable.
Answer. There is a function f: A -> B which is one-to-one.
Answer. There is a function f: A -> B which is onto.
Answer. A and B are both finite, or both infinite.
Question. Let A be a set. What does it mean for A to be countable?
Answer. A is not finite or empty.
Comment. This does not exclude the possibility that A is uncountable.
Answer. A is a subset of the natural numbers.
Partially Correct Answer. A is of the form {a_1, a_2, a_3, ...} for some sequence a_1, a_2, a_3, ...
Comment. Such a set might be finite, if the sequence has enough repeats.
Correct Answer. There is a way to assign a natural number to every element of A, such that each natural number is assigned to exactly one element of A.
Answer. One can assign a different element of A to each natural number in N.
Answer. Each element of A is countable.
Answer. One can assign a different natural number to each element of A.
Comment. This does not exclude the possibility that A is finite.
Question. Let A be a set. What does it mean for A to be uncountable?
Partially Correct Answer. A is not countable.
Comment. This does not exclude the possibility that A is finite.
Partially Correct Answer. There is a bijection f from A to the real numbers R.
Comment. The real numbers are one type of uncountable set, but it turns out there are other uncountable sets of different cardinality than R.
Correct Answer. There is no way to assign a distinct natural number to each element of A.
Answer. There is no way to assign a distinct element of A to each natural number.
Comment. This is only true when A is finite!
Answer. There exist elements of A which cannot be assigned to any natural number at all.
Answer. A contains irrational numbers.
Partially Correct Answer. There is no bijection f from the natural numbers to A.
Comment. This does not exclude the possibility that A is finite.
Question. Let A be a set, and let B be a proper subset of A (so that B is not equal to A). Is it possible for B to have the same cardinality as A?
Correct Answer. Yes, but only when A is infinite.
Answer. Yes, but only when A is countable.
Answer. Yes, but only when A is uncountable.
Answer. No, unless A is empty.
Answer. No, unless A is finite.
Answer. No, it is not possible for any A.
Answer. Yes, it is possible for any A.