1) The Cantor set, a subset of the real numbers:
A. is not compact.
B. is not contained in an interval.
C. does not contain a non-trivial interval.
D. does not have uncountably many elements.
The facts about Cantor set assures me that options A, B are not correct. Also I know that Cantor set cannot contain any interval of non-zero length and it contains uncountably infinite no of points.
Option D and option C are confusing me. I think option C is correct but I am not sure.
Also I am not able to figure out the difference between uncountably many elements and uncountably infinite elements.
2) Let $F$ be a real valued function of real numbers such that $F(x)= \sin x, S = F \text{inverse of} {(-0.5,0.5)}$. Then $S$ is:
I) a connected set.
II) finite union of disjoint open intervals.
III) a closed set.
IV) an infinite union of disjoint open intervals.
I guess the set $S$ is $(-30,30)$ if I am not wrong. Also every interval of real numbers is connected. So correct option is I).
I request to correct me and explain what's wrong with my calculation.
$F^{-1}(S)\cap[-{\pi\over2},{\pi\over2}]=(-{\pi\over6},{\pi\over6})$. As $F(x+2\pi)=F(x),$ $$ F^{-1}(x)=\bigcup_{n\in\mathbb Z}\left(2\pi n+(-{\pi\over6},{\pi\over6})\right)=\bigcup_{n\in\mathbb Z}(2\pi n-{\pi\over6},2\pi n+{\pi\over6}) $$ So the correct choice is $(IV)$.
An uncountable set is never finite. So $D$ is wrong. Trivial interval is either $\varnothing$ or a singleton set. Any other interval is non-trivial. So $C$ is correct.