Prove or disprove: a countable set of parabolas (arbitrarily oriented and placed) can completely cover (every point inside) the unit square in the plane (i.e., the interior and boundary of a square of side 1)
3. Prove or disprove: an uncountable set of pairwise-disjoint line segments can completely cover (every point in) the unit disk in the plane (i.e., the interior and boundary of a circle of diameter 1). What if the segments could intersect each other, but must all have unique slopes?
4. What is the cardinality of the set of all finite-sized matrices with rational entries?
5. What is the cardinality of the set of all infinite matrices (i.e., matrices with a countably-infinite number of rows and columns) with Boolean entries?
6. Does every regular language have a proper regular subset? Does every regular language have a proper regular superset?
7. Is every subset of a regular language necessarily regular?
Is every superset of a regular language necessarily non-regular?
8. Are the regular languages closed under infinite union? Infinite intersection?
9. Is a countable union of regular languages necessarily regular? Decidable?
Is a countable union of decidable languages necessarily decidable?
10. Prove or disprove: every regular language is countable.
11. Prove or disprove: the set of all regular languages is countable.
12. Prove or disprove: the set of all languages is countable.