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

Score:  

.