Question 1: 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)
Question 2: 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?
I have the answer already but i don't get it; I need expert to explain it step by step. thanks!