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Quiz: Countable sets
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last edited
by CAG 13 years, 11 months ago
This quiz is designed to test your knowledge of finite, countable, and uncountable sets.
Discuss this quiz
(Key: correct, incorrect, partially correct.)
- If is a countable set, and is an uncountable set, then the most we can say about is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is a countable set, and is a finite set, then the most we can say about is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is an uncountable set, and is a finite set, then the most we can say about is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is a finite set, and is a finite set, then the most we can say about is that it is
- Empty.
- Finite.
- CORRECT. Empty sets are considered finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or Uncountable.
- Finite, countable, or uncountable.
- If is a countable set, and is an uncountable set, then the most we can say about is that it is
- Empty.
- Finite.
- Countable.
- INCORRECT. The intersection of and could be smaller than countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is a finite set, and is an uncountable set, then the most we can say about is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or Uncountable.
- Finite, countable, or uncountable.
- If is a countable set, and is an uncountable set, then the most we can say about the Cartesian product is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is a nonempty finite set, and is an uncountable set, then the most we can say about the Cartesian product is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is a countable set, and is a countable set, then the most we can say about the Cartesian product is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is an uncountable set, and is a countable set, then the most we can say about the set (the elements of which are not in ) is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- If is a countable set, and is an uncountable set, then the most we can say about the set (the elements of which are not in ) is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or uncountable.
- Finite, countable, or uncountable.
- INCORRECT. Note that we did not say at any stage that had to be a subset of .
- If is a countable set, and is a countable set, then the most we can say about the set (the elements of which are not in ) is that it is
- Empty.
- Finite.
- Countable.
- At most countable.
- Uncountable.
- Countable or Uncountable.
- Finite, countable, or uncountable.
- Let be a set. What does it mean for to be finite?
- There exists a natural number and a bijection from to .
- There exists a natural number and a bijection from to .
- PARTIALLY. This does not quite work when is empty.
- Every element of is finite.
- is a proper subset of the natural numbers.
- There is a bijection from to a proper subset of the natural numbers.
- Every element of is bounded.
- is not countable.
- Let and be sets. What does it mean for and to have the same cardinality?
- is not a subset of , and is not a subset of .
- Every element of is an element of , and vice versa.
- There is a function which is both one-to-one and onto.
- and are both finite, or both countable, or both uncountable.
- There is a function which is one-to-one.
- There is a function which is onto.
- and are both finite, or both infinite.
- Let be a set. What does it mean for to be countable?
- is not finite or empty.
- INCORRECT. This does not exclude the possibility that is uncountable.
- is a subset of the natural numbers.
- is of the form for some sequence
- PARTIALLY. Such a set might be finite, if the sequence has enough repeats.
- There is a way to assign a natural number to every element of , such that each natural number is assigned to exactly one element of .
- One can assign a different element of to each natural number in N.
- Each element of is countable.
- One can assign a different natural number to each element of .
- INCORRECT. This does not exclude the possibility that is finite.
- Let be a set. What does it mean for to be uncountable?
- is not countable.
- PARTIALLY. This does not exclude the possibility that is finite.
- There is a bijection from to the real numbers .
- PARTIALLY. The real numbers are one type of uncountable set, but it turns out there are other uncountable sets of different cardinality than .
- There is no way to assign a distinct natural number to each element of .
- There is no way to assign a distinct element of to each natural number.
- INCORRECT. This is only true when is finite!
- There exist elements of which cannot be assigned to any natural number at all.
- contains irrational numbers.
- There is no bijection f from the natural numbers to .
- PARTIALLY. This does not exclude the possibility that is finite.
- Let be a set, and let be a proper subset of (so that is not equal to ). Is it possible for to have the same cardinality as ?
- Yes, but only when is infinite.
- Yes, but only when is countable.
- Yes, but only when is uncountable.
- No, unless is empty.
- No, unless is finite.
- No, it is not possible for any .
- Yes, it is possible for any .
Score:
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Quiz: Countable sets
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